Theoarithder 1

Shmei¸seic gia to mˆjhma
JEWRIAS ARIJMWN
(D. Derizi¸thc)

Perieqìmena
1 Diairetìthta 1
1.1 Diairetìthta . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Mègistoc Koinìc Diairèthc . . . . . . . . . . . . . . . . . . . . 9
EukleÐdioc Algìrijmoc . . . . . . . . . . . . . . . . . . . . . . . 19
Pr¸toi ArijmoÐ Jemelei¸dec Je¸rhma thc Arijmhtik c . . . . 23
To Pl joc kai h Katanom twn Pr¸twn . . . . . . . . . . . . . 30
Grammikèc Diofantikèc Exis¸seic . . . . . . . . . . . . . . . . . 39
H Katanom twn Pr¸twn . . . . . . . . . . . . . . . . . . . . . . 54
Pujagìreiec Triˆdec kai h Mèjodoc Kajìdou tou Fermat . . . 66
2 Arijmhtik UpoloÐpwn 83
2.1 IsotimÐec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.2 Prìsjesh kai Pollaplasiasmìc stic IsotimÐec . . . . . . . . . . 88
2.3 Nìmoc Apaloif c kai Antistrèyimec Klˆseic . . . . . . . . . . . 99
2.4 Grammikèc IsotimÐec . . . . . . . . . . . . . . . . . . . . . . . . . 119
iii

Kefˆlaio 1
Diairetìthta
1.1 Diairetìthta
H basikìterh ènnoia sthn opoÐa basÐzontai ìla ìsa akoloujoÔn eÐnai h ex c:
1.1.1 Orismìc. 'Estw α, β ∈ Z. Ja lème ìti o β diaireÐ ton α ( ìti o α
diaireÐtai dia tou β) an upˆrqei ènac akèraioc γ ètsi ¸ste
α = βγ
kai grˆfoume “β|α”. S aut thn perÐptwsh epÐshc lème ìti o α eÐnai èna pol-
laplˆsio tou β kai ìti o β eÐnai ènac diairèthc tou α.
'Otan o β den diaireÐ ton α grˆfoume “β α”.
Parat rhsh. Ston prohgoÔmeno orismì, an o β eÐnai = 0 tìte o γ orÐzetai
monadikˆ (giatÐ;)
1.1.2 Prìtash. (Basikèc idiìthtec)
i) 'Estw α ∈ Z me α = 0. Tìte α| ± α
ii) O akèraioc 0 diaireÐtai dia kˆje akèraio.
iii) O akèraioc 0 diaireÐ mìnon ton eautì tou.
1

2 Kefˆlaio 1. Diairetìthta
iv) O akèraioc 1 diaireÐ kˆje akèraio.
v) An α, β ∈ Z kai β|α, tìte β| − α, −β|α, −β| − α kai |β| |α|.
vi) An α, β, γ ∈ Z kai β|α, α|γ, tìte β|γ.
vii) An α, β, γ ∈ Z me γ = 0 kai βγ|αγ, tìte β|α.
viii) An α, β ∈ Z kai β|α, tìte β|αγ, gia kˆje γ ∈ Z.
ix) An α, β, γ, δ ∈ Z kai β|α, δ|γ, tìte βδ|αγ.
x) An α, β ∈ Z kai β|α, tìte βγ|αγ, gia kˆje γ ∈ Z.
xi) An α, β, γ ∈ Z kai β|α, β|γ, tìte β|ακ + γλ, gia kˆje κ, λ ∈ Z.
xii) An β|α kai α = 0, tìte |β| ≤ |α| (kai sunep¸c an β|α kai α|β tìte
|α| = |β|).
Apìdeixh.
i) IsqÔei α = α · 1 kai −α = α(−1).
ii) Prˆgmati, isqÔei 0 = α · 0, gia kˆje α ∈ Z.
iii) H sqèsh α = 0 · γ, dÐnei α = 0.
iv) An α = βγ, tìte −α = β(−γ), α = (−β)(−γ), −α = −β · γ kai sunep¸c
|α| = |βγ| = |β||γ|.
vi) 'Eqoume γ = αγ kai α = βγ , opìte γ = βγ γ .
vii) An αγ = βγγ , γ ∈ Z, autì shmaÐnei ìti β|α.
viii) An α = βγ , tìte kai αγ = βγ γ.
ix) An α = βγ kai γ = δγ , tìte αγ = βδγ γ .
x) An α = βγ , tìte αγ = βγγ .

1.1. Diairetìthta 3
ix) An α = βγ kai γ = βγ , tìte ακ = βγ κ kai γλ = βγ λ, opìte ακ+γλ =
β(γ κ + γ λ).
xii) 'Estw ìti α = βγ me α = 0, (opìte o γ prèpei na eÐnai = 0). Epeid o α
eÐnai akèraioc ja prèpei |γ| ≥ 1, opìte èqoume |α| = |β||γ| ≥ |β|.
Parat rhsh. H idiìthta v) anafèrei ìti “β|α an kai mìnon an |β| |α|”. Su-
nep¸c h diairetìthta stouc akèraiouc anˆgetai sth diairetìthta stouc mh ar-
nhtikoÔc akeraÐouc. Gi autì to lìgo suqnˆ, ìtan meletˆme probl mata diaire-
tìthtac, mporoÔme na periorizìmeja stouc mh arnhtikoÔc akeraÐouc.
ParadeÐgmata.
1. Na brejoÔn ìloi oi jetikoÐ akèraioi n gia touc opoÐouc isqÔei
n + 1|n2
+ 1.
Epeid n2 + 1 = (n + 1)(n − 1) + 2, gia na isqÔei n + 1|n2 + 1 prèpei
n + 1|2. Sunep¸c prèpei n + 1 = 1 n + 1 = 2 kai epeid o n prèpei na
eÐnai ≥ 1, h lÔsh eÐnai n = 1.
2. An o n eÐnai ˆrtioc tìte o 4 diaireÐ ton n2 + 2n + 4. Prˆgmati, to 2|n kai
2|n ˆra o 2·2 = 4|n2. EpÐshc isqÔei 2|n kai 2|2 ˆra 2·2 = 4|2n. Sunep¸c
4|n2 + 2n kai ˆra 4|n2 + 2n + 4.
3. To ginìmeno n diadoqik¸n akeraÐwn diaireÐtai dia tou n!. Upojètoume ìti
ìloi oi diadoqikoÐ akèraioi m + 1, m + 2, . . . , m + n eÐnai jetikoÐ. Opìte
èqoume
m + n
n
=
(m + n)(m + n − 1) · · · (m + 1)
n!
∈ Z.
An oi κ ap autoÔc den eÐnai jetikoÐ, tìte to prohgoÔmeno ginìmeno to
pollaplasiˆzoume me (−1)κ gia na gÐnei jetikì pou diaireÐtai dia n!, opìte
kai to arqikì diaireÐtai dia n!. Gia parˆdeigma, to 3! = 6 diaireÐ kˆje
arijmì thc morf c n(n − 1)(n + 1) = n3 − n, en¸ to 125 = 5! diaireÐ kˆje
arijmì thc morf c n5 − 5n3 + 4n (giatÐ;).

To epìmeno je¸rhma apoteleÐ th jemelei¸dh idiìthta thc diairetìthtac pˆnw
sthn opoÐa sthrÐzetai ìlh h anˆptuxh thc stoiqei¸douc jewrÐac arijm¸n.
1.1.3 Je¸rhma (EukleÐdhc). Gia α, β ∈ Z me β = 0, upˆrqoun monadikoÐ
akèraioi π kai v me 0 ≤ v |β| tètoioi ¸ste
α = πβ + v.
Apìdeixh. JewroÔme to sÔnolo Y = {α + κβ/κ ∈ Z}. To Y perièqei mh a-
rnhtikoÔc akeraÐouc, gia parˆdeigma o akèraioc α + (|α| + 1)|β| eÐnai ènac ap
autoÔc, (giatÐ;).
Apì thn Arq tou ElaqÐstou, to Y perièqei ènan elˆqisto mh arnhtikì
arijmì v. 'Estw v = α + κβ, opìte α = πβ + v, ìpou π = −κ ∈ Z. EpÐshc
isqÔei v |β| diìti diaforetikˆ o akèraioc v − |β| ja tan ènac mh arnhtikìc
akèraioc pou an kei sto Y kai eÐnai mikrìteroc apì ton v. Sunep¸c 0 ≤ v |β|.
T¸ra gia thn monadikìthta twn π kai v, èstw ìti èqoume kai α = βπ + v
me π , v ∈ Z kai 0 ≤ v |β|. Opìte β(π − π) = v − v . An tan v = v , èstw
v v , tìte, apì thn 1.1.2 xii), èqoume ìti |β| ≤ v − v . Allˆ epeid v, v |β|
eÐnai kai v − v |β|. Sunep¸c den mporeÐ na isqÔei v = v . 'Ara v = v , opìte
kai π = π , afoÔ β = 0.
1.1.4 Pìrisma. 'Estw α, β ∈ Z me β = 0. Tìte upˆrqoun monadikoÐ akèraioi
π, v me
−
1
2
|β| v ≤
1
2
|β|
tètoioi ¸ste α = πβ + v.
Apìdeixh. Apì to 1.1.3 upˆrqoun monadikoÐ akèraioi π1, v1 me 0 ≤ v1 |β|
tètoioi ¸ste
α = π1β + v1.
An v1 ≤
1
2
|β|, tìte jètoume v1 = v kai π1 = π. An
1
2
|β| v1 |β|, tìte jètou-
me v = v1 − |β| kai π =
π1 + 1 an |β| = β
π1 − 1 an |β| = −β
. H monadikìthta prokÔptei
ìpwc kai sto 1.1.3.

1.1.5 Orismìc. Oi akèraioi π kai v sto 1.1.3 onomˆzontai antÐstoiqa to
phlÐko kai to upìloipo thc EukleÐdeiac diaÐreshc ( aplˆ thc diaÐreshc) tou α
dia tou β.
Parathr seic.
1. An kai to Je¸rhma 1.1.3 eÐnai èna je¸rhma “Ôparxhc kai monadikìthtac”
pollèc forèc anafèretai wc o “Algìrijmoc DiaÐreshc” (Algìrijmoc eÐnai
mia mèjodoc sthn opoÐa epanalambˆnetai suneq¸c mia basik diadikasÐ-
a gia th lÔsh enìc probl matoc). Ed¸ aut h onomasÐa dikaiologeÐtai
apì thn apìdeixh tou 1.1.3 kaj¸c mporoÔme na kajorÐsoume to upìloipo
xekin¸ntac apì mia mh arnhtik diaforˆ α−κβ ≥ 0 kai diadoqikˆ na afai-
roÔme pollaplˆsia tou β tìsec forèc ìsec apaitoÔntai gia na fjˆsoume
se mia tètoia diaforˆ pou na eÐnai mikrìterh tou |β|, dhlad sto upìloipo
v = α − πβ.
2. UpenjumÐzoume ìti wc majhtèc sto sqoleÐo gia na diairèsoume ènan akè-
raio α dia enìc jetikoÔ akèraiou β efarmìzoume thn ex c algorijmhti-
k mèjodo: 'Estw ìti o α èqei n + 1 yhfÐa αn, αn−1, . . . , α0, dhlad
α = αnαn−1 · · · α0 eÐnai h dekadik parˆstash tou α. Sth diaÐresh tou
sqoleÐou ekteloÔme n+1 b mata kaj¸c kˆje yhfÐo tou α apaiteÐ akrib¸c
èna b ma pou dÐnei èna yhfÐo tou phlÐkou. Akribèstera, to b ma i eÐnai
to ex c: BrÐskoume to megalÔtero akèraio πi tètoion ¸ste o βπi na mhn
eÐnai megalÔteroc tou Ai ìpou o Ai orÐzetai wc ex c:
An = αn kai Ai = 10(Ai+1 − βπi+1) + αi, 0 ≤ i n.
Grˆfoume πi sta dexiˆ tou πi+1. Metˆ to b ma i = 0 autì pou mènei eÐnai
to upìloipo v. Sth diadikasÐa aut o diairejèntac α prèpei na isoÔtai me
to ˆjroisma tou upoloÐpou kai ìlwn twn arijm¸n pou èqoun afairejeÐ.
Allˆ oi arijmoÐ pou afairoÔntai eÐnai oi βπi epÐ 10i. Sunep¸c
α = βπn10n
+ βπn−110n−1
+ · · · + βπ110 + βπ0 + v
= β(πnπn−1 · · · π0) + v = βπ + v, ìpou 0 ≤ v β

'Etsi blèpoume ìti h diaÐresh tou sqoleÐou mac dÐnei to swstì phlÐko kai
upìloipo thc EukleÐdeiac diaÐreshc tou Jewr matoc 1.1.3.
Prèpei ed¸ na tonÐsoume ìti sto sqoleÐo de grˆfoume th diaÐresh tou α
dia tou β wc α = βπ + v allˆ sun jwc wc
α
β
= π +
v
β
enno¸ntac th
diaÐresh wc “prˆxh” paÐrnontac ènan rhtì arijmì. Ed¸ ìmwc lègontac
ìti diairoÔme ton α dia tou β den ennooÔme thn prˆxh thc diaÐreshc allˆ
jewroÔme akrib¸c autì pou anafèretai sto Je¸rhma 1.1.3, kaj¸c stouc
akèraiouc arijmoÔc den orÐzetai h prˆxh thc diaÐreshc.
3. Upojètoume ìti α 0 kai β 0. Tìte to pl joc twn jetik¸n polla-
plasÐwn tou β pou eÐnai mikrìtera apì ton α eÐnai akrib¸c π. Prˆgmati,
èna jetikì pollaplˆsio λβ eÐnai mikrìtero apì ton α an kai mìnon an
0 λ ≤
α
β
. Allˆ
α
β
= π +
v
β
. 'Ara to pl joc twn fusik¸n arijm¸n pou
eÐnai mikrìteroi apì ton
α
β
eÐnai π, afoÔ 0 ≤ v β, dhlad 0 ≤
v
β
1.
EpÐshc, autì mac lèei ìti o π eÐnai o mikrìteroc fusikìc arijmìc pou eÐnai
mikrìteroc tou
α
β
, dhlad ìpwc lème, eÐnai to akèraio mèroc tou klˆsmatoc
α
β
kai to sumbolÐzoume me
α
β
. Gia to akèraio mèroc enìc rhtoÔ arijmoÔ
ja anaferjoÔme sta epìmena.
ParadeÐgmata.
1. Na brejeÐ to phlÐko kai to upìloipo thc diaÐreshc tou 59 dia tou 7. 'Eqou-
me 59 = 8·7+3, To upìloipo eÐnai 3, 0 ≤ 3 7 kai to phlÐko eÐnai 8. Sth
diaÐresh tou −59 dia tou 7, epeid −59 = (−9) · 7 + 4, to upìloipo eÐnai
to 4 kai to phlÐko eÐnai to −9. An diairèsoume to 59 dia tou −7, tìte to
upìloipo eÐnai 3 kai to phlÐko eÐnai −8 kai an diairèsoume to −59 dia tou
−7, tìte to upìloipo eÐnai 4 kai to phlÐko eÐnai 9.
Kˆnontac tic prohgoÔmenec diarèseic sÔmfwna me to Pìrisma 1.1.4, èqoume
59 = 8 · 7 + 3, 3
1
2
7.
−59 = (−8) · 7 − 3, −3 = 4 − 7 −
1
2
7 kai − 8 = −9 + 1 afoÔ 7 0.

59 = (−8) · 7 + 3, 3
1
2
7.
−59 = 8(−7) − 3, −3 = 4 − 7 −
1
2
7 kai 8 = 9 − 1 afoÔ − 7 0.
2. SÔmfwna me th diaÐresh tou EukleÐdh 1.1.3, an diairèsoume ènan akèraio
α dia tou 4, paÐrnoume tèssera dunatˆ upìloipa: to 0, to 1, to 2 kai to
3. Autì shmaÐnei ìti o α mporeÐ na grafteÐ se mia apì tic ex c morfèc:
α = 4π + 0, α = 4π + 1, α = 4π + 2, α = 4π + 3.
Genikˆ, an diairèsoume to α dia enìc akeraÐou n tìte to upìloipo ja eÐnai
ènac apì touc n arijmoÔc: 0, 1, 2, . . . , n − 1. Autì shmaÐnei ìti h diaÐre-
sh tou EukleÐdh taxinomeÐ ìlouc touc akèraiouc s autoÔc pou af noun
upìloipo 0, s autoÔc pou af noun upìloipo 1, . . . , kai s autoÔc pou
af noun upìloipo n − 1. Gia parˆdeigma, an to n = 2, tìte oi akèraioi
taxinomoÔntai stouc ˆrtiouc, dhlad s autoÔc pou af noun upìloipo 0,
kai stouc perittoÔc dhlad s autoÔc pou af noun upìloipo 1. 'Ara ènac
akèraioc eÐnai ˆrtioc perittìc. PÐsw ap aut thn idèa thc taxinìmhshc
twn akeraÐwn, brÐsketai h arijmhtik twn isotimi¸n pou anaptÔqjhke apì
ton Gauss kai ja melet soume sto epìmeno kefˆlaio.
3. Ac jewr soume ènan perittì arijmì α = 2κ + 1. An κ = 2κ1, tìte
α = 4κ1 + 1 kai an κ = 2κ1 + 1, tìte α = 4κ1 + 3 = 4(κ1 + 1) − 1.
'Ara α2 = 8(2κ ± 1)κ + 1. Dhlad to tetrˆgwno enìc akèraiou perittoÔ
arijmoÔ eÐnai pˆnta thc morf c 8π + 1 gia kˆpoio π ∈ Z.
Sunep¸c an α kai β eÐnai dÔo perittoÐ akèraioi, tìte h diaforˆ α2 − β2
diaireÐtai pˆnta dia tou 8.
4. Ac deÐxoume ìti metaxÔ tri¸n akèraiwn arijm¸n pˆnta mporoÔme na dialè-
xoume dÔo apì autoÔc ètsi ¸ste h diaforˆ α3β −αβ3 na diaireÐtai dia tou
10. Prˆgmati èqoume
α3
β − αβ3
= αβ(α − β)(α + β)

pou eÐnai ènac ˆrtioc arijmìc. An ènac apì touc treic arijmoÔc diaireÐtai
dia 5, dhlad eÐnai thc morf c 5κ, tìte o (5κ)3β − 5κβ3 = 5(κβ(5κ −
β)(5κ + β)) diaireÐtai dia tou 10 afoÔ o κβ(5κ − β)(5κ + β) eÐnai pˆnta
ˆrtioc. An kanènac apì touc treic arijmoÔc den diaireÐtai dia tou 5 tìte
autìc eÐnai thc morf c 5κ ± 1 5κ ± 2 kai sunep¸c oi dÔo apì autoÔc
anagkastikˆ ja prèpei na eÐnai thc morf c 5κ ± 1 5κ ± 2. Sunep¸c h
diaforˆ touc to ˆjroismˆ touc ja prèpei na eÐnai pollaplˆsio tou 5.
5. To teleutaÐo yhfÐo tou tetrag¸nou enìc akèraiou arijmoÔ eÐnai ènac apì
touc arijmoÔc 0, 1, 4, 5, 6 9. Prˆgmati, kˆje akèraioc α grˆfetai wc
α = 10π + v, 0 ≤ v ≤ 9, ìpou to v eÐnai to teleutaÐo yhfÐo tou. 'Eqoume
α2 = 100π2 + 20πv + v2. Sunep¸c to teleutaÐo yhfÐo tou α2 eÐnai to
Ðdio me ekeÐno tou v2. Allˆ ta tetrˆgwna twn arijm¸n 0, 1, 2, . . . , 9 èqoun
teleutaÐo yhfÐo ènan apì touc arijmoÔc 0,1,4,5,6 9.
6. UpenjumÐzoume ìti ènac pragmatikìc arijmìc r eÐnai rhtìc an kai mìnon
an r =
α
β
, α, β ∈ Z, β = 0. 'Enac pragmatikìc arijmìc pou den eÐnai rhtìc
lègetai ˆrrhtoc. EÐnai gnwstì ìti o pr¸toc pou apèdeixe thn Ôparxh
twn ˆrrhtwn arijm¸n tan o Pujagìrac ( h Sqol tou) o opoÐoc mèsw
tou Pujagìriou Jewr matoc apèdeixe ìti o
√
2 den eÐnai rhtìc. Ed¸
dÐnoume wc pr¸th apìdeixh (ˆllec dÐdontai sta epìmena) tou gegonìtoc
autoÔ, qrhsimopoi¸ntac to diaqwrismì twn fusik¸n arijm¸n se ˆrtiouc
kai perittoÔc. Upojètoume ìti
√
2 =
α
β
, α, β ∈ Z. ApaleÐfontac touc
koinoÔc parˆgontec twn α kai β mporoÔme na jewr soume ìti o α kai o
β den èqoun koinoÔc parˆgontec. Sunep¸c an o α eÐnai ˆrtioc ( o β
eÐnai ˆrtioc) tìte o β eÐnai perittìc (antÐstoiqa o α eÐnai perittìc). Allˆ
èqoume 2β2 = α2 kai sunep¸c o α2 eÐnai ˆrtioc. 'Ara o α eÐnai ˆrtioc
(giatÐ;), èstw α = 2κ. Opìte β2 = 2κ2, dhlad kai o β eÐnai ˆrtioc pou
eÐnai ˆtopo sÔmfwna me thn upìjesh.
'Alloi gnwstoÐ ˆrrhtoi arijmoÐ eÐnai o π(= 3, 14159 · · · ) kai o e(= 2, 71828 · · · ).
Gia ton e eÐnai gnwst h ex c sÔntomh apìdeixh (oi gnwstèc apodeÐxeic

gia to π eÐnai perissìtero polÔplokec). O e orÐzetai wc
e =
∞
n=1
1
n!
.
Upojètoume ìti e =
α
β
, ìpou α kai β eÐnai akèraioi pou den èqoun koinoÔc
parˆgontec. 'Estw
p = 1 +
1
2!
+
1
3!
+ · · · +
1
β!
kai
q =
1
(β + 1)!
+
1
(β + 2)!
+ · · ·
opìte e = p + q kai β!e = β!p + β!q. Oi arijmoÐ β!e kai β!p eÐnai akèraioi
kai sunep¸c kai o β!q = β!e − β!p eÐnai akèraioc. Allˆ èqoume
0 β!q =
1
β + 1
+
1
(β + 1)(β + 2)
+
1
(β + 1)(β + 2)(β + 3)
+ · · ·

1
2
+
1
4
+
1
8
+ · · · = 1
(gewmetrik prìodoc). Autì eÐnai ˆtopo. 'Ara o e eÐnai ˆrrhtoc.
Mègistoc Koinìc Diairèthc
Apì thn idiìthta 1.1.2 xii) prokÔptei ìti to pl joc twn diairet¸n enìc = 0
akeraÐou arijmoÔ eÐnai peperasmèno. Sunep¸c metaxÔ ìlwn twn koin¸n diairet¸n
dÔo akeraÐwn arijm¸n α kai β apì touc opoÐouc toulˆqiston ènac eÐnai = 0,
upˆrqei ènac mègistoc (ˆra monadikìc) pou eÐnai jetikìc (afoÔ o 1 eÐnai koinìc
diairèthc).
Parˆdeigma. Na brejoÔn ìloi oi jetikoÐ koinoÐ diairètec δ tou n2 + 1 kai
(n+1)2+1. 'Estw δ|n2+1 kai δ|n2+2n+2, opìte δ|n2+2n+2−(n2+1) = 2n+1.
'Ara δ|(2n+1)2 = 4n2 +4n+1 kai sunep¸c δ|4(n2 +2n+2)−(4n2 +4n+1) =
4n + 7. Opìte δ|4n + 7 − 2(2n + 1) = 5. Dhlad o δ mporeÐ mìno na eÐnai o 1
o 5. Gia n = 1 o δ eÐnai o 1, en¸ gia n = 2 o δ eÐnai o 1 kai o 5.

1.1.6 Orismìc. 'Estw α, β ∈ Z, me toulˆqiston ènan = 0. O mègistoc koinìc
diairèthc twn α kai β pou ja ton sumbolÐzoume m.k.d.(α, β) h aplˆ me (α, β),
eÐnai o jetikìc akèraioc δ pou ikanopoieÐ tic dÔo idiìthtec
1. δ|α kai δ|β
2. an γ|α kai γ|β tìte γ ≤ δ.
Gia parˆdeigma, oi arijmoÐ ±1, ±2, ±7, ±14 eÐnai oi diairètec tou 14 kai oi
±1, ±5, ±7, ±35 eÐnai oi diairètec tou −35. Sunep¸c oi koinoÐ diairètec tou 14
kai −35 eÐnai oi ±1, ±7. 'Ara m.k.d.(14, −35) = 7.
T¸ra ac jewr soume ìla ta koinˆ pollaplˆsia dÔo akèraiwn arijm¸n α kai
β (pou eÐnai kai oi dÔo = 0). Tètoia upˆrqoun, gia parˆdeigma oi arijmoÐ αβκ,
κ ∈ Z. Apì thn arq tou elˆqistou, upˆrqei èna elˆqisto koinì pollaplˆsio
twn α kai β sto sÔnolo twn jetik¸n koin¸n pollaplasÐwn twn α kai β.
1.1.7 Orismìc. To elˆqisto koinì pollaplˆsio twn α kai β, α, β ∈ Z, α = 0,
β = 0, eÐnai o jetikìc akèraioc ε pou sumbolÐzetai me [α, β] kai ikanopoieÐ tic
ex c idiìthtec:
1. α|ε kai β|ε
2. an α|m kai β|m tìte ε ≤ m.
Gia parˆdeigma, [5, −15] = 15, [5, 21] = 105.
ShmeÐwsh. Ap ton orismì prokÔptei ˆmesa ìti (α, β)|[α, β].
To epìmeno je¸rhma qarakthrÐzei ton mègisto koinì diairèth kai to elˆqisto
koinì pollaplˆsio dÔo akèraiwn arijm¸n.
1.1.8 Je¸rhma.
i) Ta koinˆ pollaplˆsia dÔo akèraiwn arijm¸n α = 0, β = 0 eÐnai ta Ðdia me
autˆ tou [α, β].
ii) 'Enac koinìc diairèthc dÔo arijm¸n α, β ∈ Z me toulˆqiston ton ènan = 0
eÐnai o m.k.d.(α, β) an kai mìnon an autìc diaireÐtai dia kˆje koinì diairèth
twn α kai β. Dhlad oi koinoÐ diairètec twn α kai β eÐnai akrib¸c ekeÐnoi
tou m.k.d(α, β).

iii) IsqÔei
[α, β] =
|α||β|
(α, β)
.
Apìdeixh. 'Estw ε = [α, β] kai m èna koinì pollaplˆsio twn α kai β. Apì to
1.1.3 upˆrqoun monadikoÐ π, v ∈ Z me 0 ≤ v ε tètoia ¸ste
m = επ + v.
'Ara v = m − επ kai sunep¸c α|v kai β|v. An tan v = 0, tìte ja up rqe èna
mikrìtero apì to ε jetikì koinì pollaplˆsio twn α kai β. Autì eÐnai ˆtopo kai
ˆra prèpei ε|m.
ii) An α = 0 kai β = 0, tìte (0, β) = |β| kai an γ|0 kai γ|β tìte γ|(0, β).
An α = 0 kai β = 0, mporoÔme na upojèsoume ìti kai oi dÔo eÐnai jetikoÐ, afoÔ
o α èqei touc Ðdiouc diairètec me ton |α| kai to Ðdio isqÔei gia ton β. Apì thn
idiìthta i) prèpei ε|αβ, èstw αβ = δε, δ ∈ Z. Ja deÐxoume ìti an γ|α kai γ|β
tìte γ|δ kai ìti δ = (α, β). Profan¸c α|α
β
γ
kai β|β
α
γ
, dhlad o
αβ
γ
eÐnai èna
koinì pollaplˆsio twn α kai β. Sunep¸c apì thn idiìthta i) èqoume ε|
αβ
γ
. 'Ara
o
αβ
γ
αβ
δ
=
δ
γ
∈ Z, dhlad γ|δ. EpÐshc
α
δ
=
ε
β
∈ Z kai
β
δ
=
ε
α
∈ Z, pou
shmaÐnei ìti δ|α kai δ|β. 'Ara o δ eÐnai o megalÔteroc koinìc diairèthc twn α kai
β, dhlad δ = (α, β).
(Tautìqrona èqoume deÐxei kai thn iii)). To antÐstrofo eÐnai profanèc.
'Ena shmantikì apotèlesma pou qarakthrÐzei to m.k.d(α, β) kai qrhsimopoieÐ-
tai suqnˆ gia th lÔsh problhmˆtwn eÐnai to ex c.
1.1.9 Je¸rhma (Bachet–Bezout, Grammik Morf tou m.k.d.). O akèraioc
arijmìc δ eÐnai o m.k.d.(α, β) (ìpou α = 0 kai β = 0) an kai mìnon an o δ eÐnai o
mikrìteroc jetikìc akèraioc metaxÔ ìlwn twn jetik¸n arijm¸n pou mporoÔn na
ekfrasjoÔn sth grammik morf
αx + βy, x, y ∈ Z.
Dhlad oi arijmoÐ thc morf c αx + βy, x, y ∈ Z, eÐnai ta pollaplˆsia tou
m.k.d(α, β).

Apìdeixh. 'Estw γ o mikrìteroc jetikìc akèraioc metaxÔ ìlwn twn jetik¸n
akèraiwn thc morf c αx + βy, x, y ∈ Z. Apì thn arq tou elˆqistou, tètoioc γ
upˆrqei afoÔ to sÔnolo
= {αx + βy ∈ N/x, y ∈ Z}
eÐnai = ∅ (gia parˆdeigma o |α| ∈ , an α = 0). 'Estw γ = αx0 + βy0.
Diair¸ntac to α dia tou γ èqoume
α = γπ + v, 0 ≤ v γ
opìte v = α − γπ = α(1 − x0π) + β(−y0π). An v = 0, blèpoume ìti v ∈ ,
pou eÐnai ˆtopo giatÐ v γ. Sunep¸c prèpei v = 0. 'Ara γ|α kai to Ðdio isqÔei
gia to β, dhlad γ|β. Prèpei sunep¸c γ|(α, β) = δ. Fusikˆ isqÔei (α, β)|γ kai
telikˆ (α, β) = γ.
T¸ra kˆje pollaplˆsio κδ eÐnai thc morf c α(κx0) + β(κy0) = αx + βy,
x, y ∈ Z kai fusikˆ kˆje akèraioc thc morf c αx + βy eÐnai pollaplˆsio tou
δ.
Parat rhsh. An α, β, x, y ∈ Z, tìte upˆrqoun ˆpeira to pl joc zeugˆria
(x , y ), x , y ∈ Z, tètoia ¸ste αx + βy = αx + βy . Prˆgmati, an γ eÐnai ènac
koinìc diairèthc twn α kai β (p.q. o 1) èstw α = α γ kai β = β γ tìte
αx + βy = α(x − β t) + β(y + α t) = αx + βy
ìpou x = x − β t kai y = y + α t.
1.1.10 Orismìc. 'Estw α, β ∈ Z, (ìpou α = 0 β = 0). Tìte lème ìti autoÐ
eÐnai sqetikˆ pr¸toi metaxÔ touc an
m.k.d(α, β) = 1
(dhlad oi mìnoi koinoÐ diairètec touc eÐnai to 1 kai to 1).
Apì to prohgoÔmeno je¸rhma, an α kai β eÐnai sqetikˆ pr¸toi metaxÔ touc,
upˆrqoun x, y ∈ Z tètoioi ¸ste αx + βy = 1. Ap thn ˆllh meriˆ, an gia dÔo

akeraÐouc α kai β upˆrqoun akèraioi x, y ètsi ¸ste αx + βy = 1 tìte epeid
(α, β)|1 kai (α, β) 0 prèpei (α, β) = 1. Sunep¸c dÔo sqetikˆ pr¸toi arijmoÐ
qarakthrÐzontai wc ex c:
1.1.11 Pìrisma. DÔo akèraioi α kai β pou den eÐnai kai oi dÔo mhdèn, eÐnai
sqetikˆ pr¸toi metaxÔ touc an kai mìnon an upˆrqoun x, y ∈ Z tètoioi ¸ste
αx + βy = 1.
ShmeÐwsh. Autì to Pìrisma, ìpwc ja doÔme sta epìmena, eÐnai polÔ qr simo
sth lÔsh problhmˆtwn pou anafèrontai sth diairetìthta. Wc mia pr¸th efarmo-
g autoÔ, deÐqnoume to gnwstì apotèlesma tou Pujagìra: o arijmìc
√
2 den eÐ-
nai rhtìc. Diìti diaforetikˆ ja eÐqame
√
2 =
α
β
me (α, β) = 1, opìte αx+βy = 1
gia x, y ∈ Z. 'Etsi ja eÐqame
√
2 =
√
2(αx + βy) =
√
2αx +
√
2βy = 2βx + αy,
dhlad
√
2 ∈ Z pou eÐnai ˆtopo.
Wc èna epiplèon parˆdeigma deÐqnoume ìti
m.k.d.(n! + 1, (n + 1)! + 1) = 1
gia kˆje n ∈ N. Prˆgmati, epeid èqoume
n = n + 1 + (n + 1)! − (n + 1)! − 1 = (n + 1)(n! + 1) − ((n + 1)! + 1),
an δ eÐnai o m.k.d. twn n! + 1 kai (n + 1)! + 1 gia kˆpoio n ∈ N, tìte δ|n. Allˆ
δ|n! + 1 kai sunep¸c o δ ja prèpei na diaireÐ kai ton m.k.d.(n, n! + 1). Allˆ
m.k.d.(n, n! + 1) = 1, afoÔ 1 = (n! + 1) · 1 + n(−(n − 1)!). Sunep¸c δ = 1.
Sthn epìmenh prìtash diatup¸nontai merikèc aplèc allˆ basikèc idiìthtec
pou aforoÔn ton m.k.d. kai to e.k.p.
1.1.12 Prìtash. 'Estw α, β, γ ∈ Z me α2 + β2 = 0 (dhlad toulˆqiston
ènac apì touc α kai β eÐnai = 0). Tìte isqÔoun ta ex c:
i) (α, β) = (|α|, |β|) kai [α, β] = [|α|, |β|].
Idiaitèrwc isqÔei α|β an kai mìnon an (α, β) = |α| an kai mìnon an [α, β] =
|β|. EpÐshc (α, β) = [α, β] an kai mìnon an |α| = |β|.

ii) Ap to 1.1.9, upˆrqoun x, y ∈ Z ètsi ¸ste αx + βy = (α, β). EpÐshc
gnwrÐzoume ìti γ|(α, β) an kai mìnon an γ|α kai γ|β. Sunep¸c èqoume
α
γ
x +
β
γ
y =
(α, β)
γ
.
Pˆli apì to 1.1.9 prokÔptei ìti
α
γ
,
β
γ
(α, β)
γ
. Epeid ìmwc
(α, β)
γ
α
γ
kai
(α, β)
γ
β
γ
, apì to 1.1.8 èqoume
(α, β)
γ
α
γ
,
β
γ
kai sunep¸c
α
γ
,
β
γ
=
(α, β)
γ
.
iii) Epeid (α, β)|α kai (α, β)|κβ èqoume (α, β)|α+κβ. 'Ara (α, β)|(α+κβ)
( arkeÐ na sumperˆnoume ìti (α, β) ≤ (α + κβ, β) ìpwc apaiteÐ o Orismìc
1.1.6). EpÐshc èqoume (α + κβ, β)|κβ kai (α + κβ, β)|α + κβ − κβ = α. 'Ara
(α + κβ, β)|(α, β). Sunep¸c (α, β) = (α + κβ, β).
iv) Apì to 1.1.9, upˆrqoun x1, x2 ∈ Z ètsi ¸ste (γα, γβ) = γαx1 +γβx2 =
γ(αx1+βx2). Pˆli apì to 1.1.9, prèpei (α, β)|αx1+βx2, opìte γ(α, β)|γ(αx1+
βx2) = (γα, γβ) kai sunep¸c |γ|(α, β)|(γα, γβ). EpÐshc upˆrqoun x1, x2 ∈ Z
ètsi ¸ste (α, β) = αx1 + βx2 kai ˆra |γ|(α, β) = |γ|αx1 + |γ|βx2. Sunep¸c
apì to 1.1.9, (γα, γβ)||γ|(α, β). 'Ara (γα, γβ) = |γ|(α, β).
v) Apì to iv), èqoume (αγ, βγ) = |γ|. Profan¸c α|αγ kai apì thn upìjesh
α|βγ. 'Ara α||γ|, opìte α|γ.
Mia ˆllh ˆmesh apìdeixh eÐnai h ex c. Apì to 1.1.10 upˆrqoun x, y ∈ Z ètsi
¸ste αx + βy = 1 opìte γαx + γβy = γ kai epeid α|αγ kai α|βγ ja prèpei
α|γ.
vi) Epeid (α, (α, β)γ)|(α, β)|γ| = (αγ, βγ) ja prèpei (α, (α, β)γ)|βγ. Al-
lˆ (α, (α, β)γ)|α opìte (α, (α, β)γ)|(α, βγ). 'Eqoume ìmwc (α, βγ)|α opìte
(α, βγ)|αγ kai (α, βγ)|βγ. Sunep¸c (α, βγ)|(αγ, βγ) = (α, β)|γ| kai epeid
(α, βγ)|α ja prèpei (α, βγ)|(α, (α, β)γ).
vii) Epeid (β, γ)|γ kai γ|α prèpei (β, γ)|α. Allˆ (β, γ)|β. Sunep¸c (β, γ) =
1, afoÔ oi mìnoi koinoÐ diairètec tou α kai β eÐnai to ±1.
viii) Prˆgmati, upˆrqoun x, y ∈ Z ètsi ¸ste αx+βy = 1, opìte γαx+γβy =
γ. Allˆ α|α kai apì thn upìjesh β|γ, ˆra αβ|αγ. Gia ton Ðdio lìgo αβ|βγ.
Sunep¸c αβ|αγx + βγy = γ. Autì prokÔptei epÐshc ˆmesa apì to 1.1.8 iii)
afoÔ [α, β]|γ kai [α, β] = αβ.

ix) 'Estw (α, γ) = αx1 + βy1, (β, γ) = βx2 + γy2, x1, x2, y1, y2 ∈ Z. Opì-
te (α, γ)(β, γ) = αβx1x2 + γ(αx1y2 + βy1x2 + γy1y2) kai epeid γ|αβ prèpei
γ|(α, γ)(β, γ). x) Epeid γ|α, β, apì thn ix) prèpei γ|(α, γ)(β, γ). 'Omwc (α, γ)|γ
kai (β, γ)|γ. ParathroÔme epÐshc ìti an ρ eÐnai ènac koinìc diairèthc twn (α, γ)
kai (β, γ), dhlad ènac diairèthc tou m.k.d((α, γ), (β, γ)), autìc prèpei na eÐnai
ènac koinìc diairèthc twn α kai β. Allˆ ap thn upìjesh oi mìnoi koinoÐ diairètec
twn α kai β eÐnai ±1. Sunep¸c ((α, γ), (β, γ)) = 1. 'Ara ap thn viii) prokÔptei
ìti ((α, γ), (β, γ))|γ opìte γ = (α, γ)(β, γ) = γ1γ2, γ1 = (α, γ), γ2 = (β, γ).
DeÐqnoume t¸ra thn monadikìthta. 'Estw γ1, γ2 ∈ Z me γ1|α, γ2|β kai γ = γ1γ2.
Profan¸c γ1|(α, γ) kai γ2|(β, γ). An tan γ1 = (α, γ), tìte γ1 (α, γ), opìte
γ = γ1γ2 (α, γ)(β, γ) = γ, ˆtopo. To Ðdio prokÔptei fusikˆ kai an tan
γ2 = (β, γ). 'Ara prèpei γ1 = (α, γ) kai γ2 = (β, γ).
Shmei¸noume ìti ta prohgoÔmena apodeiknÔontai kai me ˆllouc trìpouc qw-
rÐc th qr sh tou Jewr matoc 1.1.9 (p.q. bl. Landau [?]).
ParadeÐgmata.
1. Na deiqjeÐ ìti gia kˆje n ∈ N kai α, β ∈ Z me α2 + β2 = 0 isqÔei
(α, β)n
= (αn
, βn
).
Kat arqˆc upojètoume ìti (α, β) = 1. Efarmìzoume epagwg sto n. Gia
n = 1 profan¸c isqÔei. Upojètoume ìti (αn, βn) = 1. Tìte lìgw thc
1.1.12.vi) èqoume
(αn+1
, βn+1
) = (αn
α, βn
β) = (αn
α, (αn
α, βn
)β)
= (αn
α, (α(αn
, βn
), βn
)β) = (αn
α, (α, βn
)β).
Allˆ (α, βn) = 1, kaj¸c an δ|(α, βn), tìte δ|αn kai δ|βn opìte δ|(αn, βn) =
1, dhlad δ = ±1. 'Ara
(αn+1
, βn+1
) = (αn
α, β) = (α(αn
, β), β) = (α, β) = 1
(afoÔ kai pˆli (αn, β) = 1). Sunep¸c an (α, β) = 1, tìte (αn, βn) =
(α, β)n = 1, gia kˆje n ∈ N.

GnwrÐzoume apì to 1.1.12.ii) ìti
α
(α, β)
,
β
(α, β)
= 1
opìte lìgw tou prohgoÔmenou apotelèsmatoc èqoume
αn
(α, β)n
,
βn
(α, β)n
=
α
(α, β)
n
,
β
(α, β)
n
= 1.
Lìgw tou 1.1.12.iv) paÐrnoume
(α, β)n
= (α, β)n αn
(α, β)n
,
βn
(α, β)n
= (αn
, βn
).
2. Na deiqjeÐ ìti gia kˆje n ∈ N kai α, β ∈ Z, α 0, β 0, isqÔei: αn|βn
an kai mìnon an α|β.
Profan¸c an α|β tìte αn|βn. 'Estw ìti αn|βn. Tìte (α, β)n αn
(α, β)n
γ =
(α, β)n βn
(α, β)n
gia kˆpoio γ ∈ Z. Opìte
αn
(α, β)n
γ =
βn
(α, β)n
, ˆra
αn
(α, β)n
= 1, afoÔ
αn
(α, β)n
,
βn
(α, β)n
= 1.
'Ara αn = (α, β)n, opìte α = (α, β) (giatÐ;) kai sunep¸c α|β.
3. Na deiqjeÐ ìti
αn − βn
α − β
, α − β = (n(α, β)n−1
, α − β).
IsqÔei
αn − βn
α − β
= αn−1 + αn−2β + · · · + αβn−2 + βn−1. (giatÐ;).
Prosjètoume kai afairoÔme apì to dexÐ mèloc to βn−1. Opìte
αn − βn
α − β
= (αn−1
− βn−1
) + (αn−2
β − βn−1
) + · · · + (αβn−2
− βn−1
)
+ nβn−1
= (α − β)
n−2
0
αn−2−i
βi
+ (α − β)β
n−3
0
αn−3−i
βi
+ (α − β)βn−2
+ nβn−1
= (α − β)κ + nβn−1
.

Apì to 1.1.12 iii) èqoume
αn − βn
α − β
, α−β = (α − β)κ + nβn−1, α − β
= (nβn−1, α−β). 'Omoia èqoume ìti
αn − βn
α − β
, α−β = (nαn−1, β−α) =
(nαn−1, α−β). 'Estw d =
αn − βn
α − β
, α−β kai d = n(α, β)n−1, α − β .
Apì to prohgoÔmeno parˆdeigma èqoume ìti d = n(αn−1, βn−1), α − β =
(nαn−1, nβn−1), α − β . Epeid d |(nαn−1, nβn−1), prèpei d |nαn−1 kai
d |nβn−1. 'Ara d |d = (nαn−1, α − β) = (nβn−1, α − β). EpÐshc epeid
d|nαn−1 kai d|nβn−1 ja prèpei d|(nαn−1, nβn−1), opìte d|d . 'Ara d = d .
Gia parˆdeigma, isqÔei
αn − 1
α − 1
, α − 1 = (n, α − 1)
gia kˆje n ∈ N. Poio genikˆ, an (α, β) = 1 tìte
αn − βn
α − β
, α − β =
(n, α − β).
4. Na deiqjeÐ ìti
(nα
− 1, nβ
− 1) = n(α,β)
− 1,
gia kˆje α, β, n ∈ N me n = 1.
'Estw α = (α, β)γ kai β = (α, β)δ. Tìte (nα−1) = (n(α,β)−1)
γ−1
i=0
n(α,β)i ,
dhlad n(α,β) − 1|nα − 1. 'Omoia isqÔei n(α,β) − 1|nβ − 1. Sunep¸c
n(α,β) − 1|(nα − 1, nβ − 1). Apì to 1.1.9, upˆrqoun x, y ∈ Z ètsi ¸ste
(α, β) = αx + βy.
Profan¸c toulˆqiston ènac apì touc x kai y prèpei na eÐnai = 0. EpÐshc
an ènac ap touc dÔo eÐnai jetikìc, èstw x 0, tìte y ≤ 0, diìti an
tan kai y 0, tìte ja eÐqame (α, β) = αx + βy ≥ α + β, en¸ èqoume
(α, β) ≤ α kai (α, β) ≤ β. Fusikˆ oi x kai y den mporoÔn na eÐnai kai oi
dÔo arnhtikoÐ, afoÔ o (α, β) 0. Upojètoume loipìn ìti x 0 kai y ≤ 0.

Tìte
kai
(nα
− 1, nβ
− 1)|nαx
− 1 = (nα
− 1)
x−1
i=0
nαi
(nα
− 1, nβ
− 1)|n−βy
− 1 = (nβ
− 1)
−y−1
i=0
nβi
.
Opìte (nα − 1, nβ − 1)|nαx − 1 − n(α,β)(n−βy − 1) = n(α,β) − 1.
ShmeÐwsh. MporoÔme epÐshc na apodeÐxoume to prohgoÔmeno efarmìzontac
thn EukleÐdeia diaÐresh: 'Estw α ≥ β, tìte α = βπ + v, 0 ≤ v β. Epeid
(nβπ − 1)nv = (nβ − 1)κnv, lìgw thc 1.1.12 iii) èqoume
(nα
− 1, nβ
− 1) = (nα
− 1 − (nβπ
− 1)nv
, nβ
− 1) = (nv
− 1, nβ
− 1).
An suneqÐsoume thn Ðdia diadikasÐa gia ton (nv −1, nβ −1) k.o.k., ìpwc ja doÔme
amèswc metˆ, apì ton algìrijmo tou EukleÐdh, h diadikasÐa aut ja termatÐsei
ston (n(α,β) − 1, 0) = n(α,β) − 1.
EukleÐdeioc Algìrijmoc
'Estw α, β ∈ Z me α2 +β2 = 0. Apì to Je¸rhma 1.1.9 gnwrÐzoume ìti upˆrqoun
x, y ∈ Z tètoioi ¸ste (α, β) = αx + βy. H apìdeixh autoÔ tou jewr matoc den
parèqei ènan trìpo upologismoÔ twn x kai y kai kat epèktash tou m.k.d(α, β).
Mia praktik mèjodoc ˆmesou upologismoÔ tou m.k.d(α, β) allˆ kai thc eÔreshc
enìc zeugarioÔ x, y ∈ Z ètsi ¸ste (α, β) = αx + βy èdwse prin 2400 qrìnia o
EukleÐdhc sto biblÐo tou “Ta StoiqeÐa tou EukleÐdh”. H mèjodoc aut sth-
rÐzetai stic dÔo basikèc idiìthtec 1.1.12 i) kai iii) kai onomˆzetai EukleÐdeioc
Algìrijmoc.
O algìrijmoc basÐzetai sta ex c dÔo b mata.
Upojètoume ìti α β 0 (kaj¸c mporoÔme). Apì thn EukleÐdeia diaÐresh
upˆrqoun monadikoÐ π, v ∈ Z me 0 ≤ v β ètsi ¸ste α = βπ + v.
1o b ma: An v = 0, dhlad an β|α tìte (α, β) = β (idiìthta 1.1.12 i)).

2o b ma: An v = 0, tìte (α, β) = (β, v) (idiìthta 1.1.12 iii)).
Sthn perÐptwsh pou eÐnai v = 0, epanalambˆnoume thn Ðdia diadikasÐa: Efar-
mìzoume thn EukleÐdeia diaÐresh gia touc β kai v, èstw β = vπ1 + v1, ìpou
0 ≤ v1 v. An v1 = 0, tìte (α, β) = (β, v) = v, diaforetikˆ (α, β) = (β, v) =
(v, v1).
SuneqÐzontac aut th diadikasÐa diadoqik¸n EukleÐdeiwn diairèsewn èqoume
α = βπ + v
β = vπ1 + v1
v = v1π2 + v2
v1 = v2π3 + v3
...
vi−1 = viπi+1 + vi+1
ìpou α β v v1 · · · vi+1 0
kai (α, β) = (β, v) = (v, v1) = · · · = (vi, vi+1).
Aut h diadikasÐa ìmwc prèpei na termatÐzei metˆ apì èna peperasmèno pl -
joc bhmˆtwn kaj¸c h austhrˆ fjÐnousa akoloujÐa α β v · · · vi+1 0
apoteleÐtai apì fusikoÔc arijmoÔc kai ˆra ja upˆrqei kˆpoio n tètoio ¸ste
vn+1 = 0, opìte (α, β) = (vn, vn+1) = vn. Dhlad o mègistoc koinìc diairè-
thc twn α kai β eÐnai to teleutaÐo mh mhdenikì upìloipo pou prokÔptei apì tic
diadoqikèc prohgoÔmenec EukleÐdeiec diairèseic.
Parˆdeigma. 'Estw α = 356 kai β = 156. Oi EukleÐdeiec diadoqikèc diairè-
seic dÐnoun
356 = 156 · 2 + 44
156 = 44 · 3 + 24
44 = 24 · 1 + 20
24 = 20 · 1 + 4
20 = 4 · 5.

Opìte (356, 156) = 4.
Parat rhsh. Qrhsimopoi¸ntac thn prohgoÔmenh diadikasÐa mporoÔme na dì-
soume mia ˆllh apìdeixh tou 1.1.9 wc ex c.
Ston prohgoÔmeno EukleÐdeio Algìrijmo gia touc α kai β, ton fusikì arij-
mì n + 1 (gia ton opoÐo jewr same ìti vn+1 = 0) ton onomˆzoume m koc
tou EukleÐdeiou Algìrijmou twn α kai β kai ton sumbolÐzoume me (α, β), gia
parˆdeigma (356, 156) = 5.
Efarmìzontac epagwg sto m koc autì, èqoume:
An (α, β) = 1, dhlad an β|α, tìte (α, β) = α ·0+β1. Upojètoume ìti gia
ìlouc touc arijmoÔc α kai β me (α, β) n + 1 upˆrqoun x, y ∈ Z ètsi ¸ste
(α, β) = αx + βy. 'Estw ìti (α, β) = n + 1, tìte apì thn EukleÐdeia diaÐresh
α = βπ + v ja èqoume (β, v) = n kai sunep¸c upˆrqoun x, y ∈ Z ètsi ¸ste
(α, β) = (β, v) = βx + vy. Allˆ v = α − βπ, opìte (α, β) = β(x − πy) + α − y.
'Ara upˆrqoun x = (−y) ∈ Z kai y = x − πy ∈ Z tètoioi ¸ste (α, β) =
αx + βy .
Gia thn eÔresh enìc x kai enìc y pou ikanopoieÐ thn αx + βy = (α, β) mpo-
roÔme pˆli na qrhsimopoi soume ton EukleÐdeio Algìrijmo: Apì tic diadoqikèc
EukleÐdeiec diairèseic grˆfoume ta upìloipa wc
v = α − βπ
v1 = β − vπ1
v2 = v − v1π2
...
vn−2 = vn−4 − vn−3πn−2
vn−1 = vn−3 − vn−2πn−1
vn = vn−2 − vn−1πn.
Opìte èqoume
vn = vn−2 − (vn−3 − vn−2πn−1)πn
= vn−2(1 + πn−1πn) + vn−3(−πn).

Sth sunèqeia antikajistoÔme to vn−2 ap thn prohgoÔmenh èkfras tou kai
paÐrnoume
vn = (vn−4 − vn−3πn−2)(1 + πn−1πn) + vn−3(−πn)
= vn−3(−πn−2(1 + πn−1πn) − πn) + vn−4(1 + πn−1πn).
SuneqÐzontac, me ton Ðdio trìpo, fjˆnoume telikˆ se mia parˆstash thc morf c
pou jèloume, dhlad (α, β) = vn = αx + βy. Aut h diadikasÐa dÐnei mia nèa
(kataskeuastik ) apìdeixh tou 1.1.9.
Gia parˆdeigma, gia α = 356, β = 156, èqoume
44 = 356 − 156 · 2
24 = 156 − 44 · 3
20 = 44 − 24 · 1
4 = 24 − 20 · 1.
Opìte
4 = 24 − 20 · 1 = 24 − (44 − 24 · 1) = 24 · 2 − 44 = (156 − 44 · 3)2 − 44
= 156 · 2 + 44(−7) = 156 + (356 − 156 · 2)(−7)
= 356(−7) + 156 · (16).
'Ara oi x = −7 kai y = 16 dÐnoun mia apì tic (ˆpeirec) parastˆseic tou (356, 156)
thc morf c αx + βy, x, y ∈ Z.
ShmeÐwsh. Gia thn eÔresh twn x kai y upˆrqoun kai ˆlloi diˆforoi trìpoi
pou elaqistopoioÔn touc qronobìrouc upologismoÔc pou prokÔptoun apì tic
diadoqikèc antikatastˆseic twn upoloÐpwn (blèpe Oystein Ore [?] kai S.P. Gla-
sby: Extended Euclid’s algorithm via backward recurrence relations, Math.
Magazine 1999, 72(3), 228–230). EpÐshc axÐzei na anaferjeÐ ed¸ ìti to pl joc
twn diairèsewn ston EukleÐdeio algìrijmo gia dÔo jetikoÔc akèraiouc arijmoÔc
eÐnai mikrìtero pènte forèc apì to pl joc twn dekadik¸n yhfÐwn tou mikrìterou
twn dÔo arijm¸n. Autì eÐnai èna je¸rhma tou Gabriel Lam´e (1890).

Parat rhsh. 'Olec oi prohgoÔmenec idiìthtec sthn Prìtash 1.1.12 mporoÔn
na apodeiqjoÔn efarmìzontac ton algìrijmo tou EukleÐdh. Gia parˆdeigma,
ac apodeÐxoume thn 1.1.12 v), dhlad an (α, β) = 1 kai β|αγ tìte β|γ. Kaj¸c
(α, β) = 1 ja èqoume vn = 1. Pollaplasiˆzontac ìlec tic EukleÐdeiec diairèseic
epÐ γ, upojètontac ìti β|αγ, o β ja diaireÐ ìlouc touc akèraiouc viγ kai ˆra kai
ton γ = vnγ.
Pr¸toi ArijmoÐ Jemelei¸dec Je¸rhma thc Arijmhtik c
1.1.13 Orismìc. 'Enac fusikìc arijmìc p 1 ja kaleÐtai pr¸toc arijmìc
aplˆ pr¸toc an oi mìnoi diairètec tou eÐnai oi ±1 kai ±p. 'Enac fusikìc arijmìc
n 1 pou den eÐnai pr¸toc ja kaleÐtai sÔnjetoc.
Sunep¸c ènac jetikìc akèraioc p 1 eÐnai pr¸toc an kai mìnon an gia kˆje
n ∈ Z isqÔei h (p, n) = 1 (p, n) = p, dhlad o p eÐnai sqetikˆ pr¸toc proc
ton n o p diaireÐ ton n.
Epeid kˆje ˆrtioc arijmìc diaireÐtai me to 2, apì ton orismì prokÔptei ìti
ìloi oi pr¸toi ektìc apì ton 2 eÐnai perittoÐ arijmoÐ.
Oi pr¸toi pou eÐnai mikrìteroi tou 100 eÐnai oi ex c 25 arijmoÐ:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89 kai 97.
Shmei¸noume ìti o arijmìc 1 katˆ sunj kh den jewreÐtai oÔte pr¸toc oÔte
sÔnjetoc arijmìc. Apodeqìmaste aut th sunj kh diìti diaforetikˆ den ja
isqÔoun basikˆ apotelèsmata pou ja doÔme amèswc pio kˆtw.
1.1.14 L mma. Kˆje akèraioc 1 eÐnai ènac pr¸toc arijmìc to ginìmeno
pr¸twn arijm¸n.
Apìdeixh. 'Estw n ∈ N, n 1. Upojètoume ìti kˆje fusikìc m, 1 m n,
eÐte eÐnai pr¸toc eÐnai ginìmeno pr¸twn. An o n den eÐnai pr¸toc, dhlad eÐnai
sÔnjetoc, tìte èqei ènan diairèth α, 1 α n kai n = αβ gia kˆpoion β ∈ N
me 1 β n (giatÐ;). Opìte oi α kai β eÐnai eÐte pr¸toi ginìmeno pr¸twn.
'Ara o n an den eÐnai pr¸toc autìc eÐnai ginìmeno pr¸twn. Sunep¸c epagwgikˆ
to zhtoÔmeno isqÔei gia kˆje n ∈ N, n 1.

To 1.1.14 mac lèei ìti oi pr¸toi arijmoÐ eÐnai pollaplasiastikˆ oi “jemèlioi
lÐjoi” gia thn kataskeu twn fusik¸n arijm¸n. 'Etsi eÐnai fusikì èna megˆlo
mèroc thc JewrÐac Arijm¸n na epikentr¸netai sth melèth twn pr¸twn arijm¸n.
Apì ton orismì twn pr¸twn kai apì to 1.1.12 v) prokÔptei ˆmesa to ex c:
1.1.15 L mma (EukleÐdhc). An α, β ∈ Z kai p eÐnai ènac pr¸toc pou diaireÐ
to ginìmeno αβ tìte o p diaireÐ ton α o p diaireÐ ton β.
1.1.16 Pìrisma. An p eÐnai pr¸toc kai p|α1α2 · · · αn, α1, . . . , αn ∈ Z tìte
p|αi, gia kˆpoio i = 1, 2, . . . , n. Eidikìtera, an oi α1, α2, . . . , αn eÐnai pr¸toi tìte
p = pi, gia kˆpoio i.
Apìdeixh. Efarmìzoume epagwg sto n qrhsimopoi¸ntac to prohgoÔmeno l m-
ma.
T¸ra ja lème ìti ènac fusikìc arijmìc 1 analÔetai monadikˆ se ginìmeno
pr¸twn paragìntwn an gia dosmènouc pr¸touc p1, p2, . . . , pr kai q1, q2, . . . , qs
tètoiouc ¸ste
n = p1p2 · · · pr = q1q2 · · · qs
ìsec forèc emfanÐzetai ènac pr¸toc metaxÔ twn p1, p2, . . . , pr tìsec forèc em-
fanÐzetai autìc metaxÔ twn q1, q2, . . . , qs (Shmei¸noume ìti ap autì prokÔptei
r = s).
Me aut thn orologÐa apodeiknÔoume t¸ra to Jemelei¸dec Je¸rhma thc
Arijmhtik c (merikèc forèc autì anafèretai kai wc Je¸rhma Monadik c Para-
gontopoÐhshc).
1.1.17 Je¸rhma. Kˆje fusikìc arijmìc n 1 analÔetai monadikˆ se ginì-
meno pr¸twn paragìntwn.
1h Apìdeixh. S aut thn apìdeixh qrhsimopoioÔme to 1.1.16. MporoÔme na upo-
jèsoume ìti o n eÐnai sÔnjetoc arijmìc. Upojètoume ìti kˆje sÔnjetoc arijmìc
mikrìteroc tou n analÔetai monadikˆ se pr¸touc parˆgontec. DeÐqnoume ìti
tìte kai o n analÔetai monadikˆ se pr¸touc parˆgontec, opìte to apotèlesma
èpetai apì th majhmatik epagwg sto n.

'Estw loipìn ìti
n = p1p2 · · · pr = q1q2 · · · qs
ìpou p1, p2, . . . , pr kai q1, q2, . . . , qs eÐnai pr¸toi kai p1 ≤ p2 ≤ · · · ≤ pr, q1 ≤
q2 ≤ · · · ≤ qs. Prèpei na deÐxoume ìti r = s kai pi = qi gia kˆje i = 1, 2, . . . , r.
'Estw p o mikrìteroc pr¸toc pou diaireÐ ton n. Tìte lìgw tou 1.1.16,
p = pi, gia kˆpoio i = 1, 2, . . . , r, kai epeid p ≤ p1 ja prèpei p = p1. 'Omoia
p = q1, opìte p1 = q1. 'Estw m = n/p. Tìte
m = p2p3 · · · pr = q2 · · · qs.
Allˆ tìte r = s kai pi = qi, i = 2, . . . , r, afoÔ m n. 'Ara o n analÔetai
monadikˆ se pr¸touc parˆgontec. 'Etsi èqoume deÐxei ìti an ìloi oi sÔnjetoi
arijmoÐ m, 1 m n analÔontai monadikˆ se pr¸touc parˆgontec, to Ðdio
isqÔei kai gia ìlouc touc sÔnjetouc arijmoÔc m, 1 m n + 1.
'Ara ìloi oi sÔnjetoi arijmoÐ analÔontai monadikˆ se pr¸touc parˆgontec.
2h Apìdeixh. S aut thn apìdeixh den ja qrhsimopoi soume to L mma tou
EukleÐdh. Upojètoume ìti upˆrqoun sÔnjetoi arijmoÐ 1 pou èqoun dÔo dia-
foretikèc analÔseic se pr¸touc parˆgontec. 'Estw n o mikrìteroc tètoioc
arijmìc gia ton opoÐo èqoume
n = p1p2 · · · pr = q1q2 · · · qs
ìpou p1 ≤ p2 ≤ · · · ≤ pr, q1 ≤ q2 ≤ · · · qs.
Kˆje pi eÐnai diˆforoc apì kˆje qi, diìti diaforetikˆ an up rqe koinìc pr¸-
toc parˆgontac ja paÐrname kˆpoion n n me thn Ðdia idiìthta pou èqei o n
(pou den mporeÐ na isqÔei lìgw thc upìjeshc ìti o n eÐnai o mikrìteroc m aut
thn idiìthta). 'Estw ìti p1 q1. JewroÔme to sÔnjeto arijmì
m = p1q2q3 · · · qs.
O fusikìc arijmìc = n−m = (q1 −p1)q2 · · · qs pou eÐnai mikrìteroc apì ton n
diaireÐtai dia p1, afoÔ p1|n kai p1|m. Opìte o analÔetai monadikˆ se pr¸touc
parˆgontec ènac ek twn opoÐwn eÐnai o p1. 'Estw
= p1t2t3 · · · tk

h anˆlush tou se pr¸touc parˆgontec. An o arijmìc q1 − p1 = 1, tìte o ja
eÐqe dÔo analÔseic se pr¸touc parˆgontec, h mia ja perieÐqe ton p1 kai h ˆllh
den ja ton perieÐqe, opìte, epeid n, prèpei q1 − p1 = 1. Sunep¸c mporoÔme
na grˆyoume ton q1 − p1 wc ginìmeno pr¸twn arijm¸n, èstw
q1 − p1 = h1h2 · · · ht.
'Etsi èqoume = h1h2 · · · htq2 · · · qs. Aut eÐnai mia anˆlush tou se pr¸touc
pou den perièqei ton p1, afoÔ p1 q1 −p1 kai p1 = qi, i = 1, 2, . . . , s. 'Opwc ìmwc
eÐdame o èqei kai ˆllh anˆlush se pr¸touc pou perièqei ton p1. Epeid n,
autì eÐnai ˆtopo, afoÔ o n eÐnai o mikrìteroc arijmìc me perissìterec thc miac
analÔseic se pr¸touc parˆgontec. 'Ara den upˆrqei kanènac fusikìc sÔnjetoc
arijmìc me perissìterec thc miac analÔseic se pr¸touc parˆgontec.
ShmeÐwsh. H pr¸th apìdeixh dìjhke apì ton EukleÐdh en¸ h deÔterh apì ton
Zermelo. Upˆrqoun kai ˆllec apodeÐxeic pou metaxÔ aut¸n xeqwrÐzoun autèc
pou qrhsimopoioÔn th jewrÐa omˆdwn (bl. [?]) thn TopologÐa. (Ja d¸soume
argìtera mia ˆllh apìdeixh me th qr sh twn akolouji¸n tou Farey).
Parat rhsh. 1. An eÐqame sumperilˆbei ton arijmì 1 stouc pr¸touc arij-
moÔc, tìte ja èprepe na anadiatup¸noume to prohgoÔmeno je¸rhma epitrèpontac
diaforetikèc paragontopoi seic, gia parˆdeigma 6 = 2 · 3 = 1 · 2 · 3.
2. Upˆrqoun pollˆ paradeÐgmata diaktulÐwn sthn ˆlgebra pou den ika-
nopoioÔn to je¸rhma thc monadik c paragontopoÐshc. Epeid ed¸ den èqoume
anaferjeÐ se daktulÐouc, ja d¸soume to pio aplì parˆdeigma. jewroÔme touc
ˆrtiouc akèraiouc 2Z = {. . . , −4, −2, 0, 2, 4, . . .}. An α, β ∈ 2Z ja lème ìti
β|α an upˆrqei γ ∈ 2Z ètsi ¸ste α = βγ. 'Etsi 2|4 afoÔ 4 = 2 · 2, en¸ to
2 den diaireÐ ton eautì tou epeid to 1 /∈ 2Z. 'Ena stoiqeÐo p ∈ 2Z lègetai
“pr¸toc” an den upˆrqoun α, β ∈ 2Z tètoia ¸ste p = αβ. Gia parˆdeigma,
to 2, to 6, to 14 eÐnai “pr¸toi” sto 2Z. ParathroÔme epÐshc ìti en¸ to 2
diaireÐ to 4, to 2 de diaireÐ ènan apì touc parˆgontec tou, 4 = 2 · 2. Dhla-
d den isqÔei to L mma tou EukleÐdh. ParathroÔme epÐshc ìti den isqÔei to
je¸rhma monadik c paragontopoÐhshc, kaj¸c 2 · 18 = 6 · 6 kai oi 2, 6 kai 18

eÐnai “pr¸toi”. Ap thn ˆllh meriˆ kˆje mh mhdenikì jetikì stoiqeÐo tou 2Z
analÔetai (paragontopoieÐtai) se “pr¸touc”, kaj¸c to stoiqeÐo 2z analÔetai
se pr¸touc akèraiouc: 2z = 2kp1 · · · ps, ìpou p1, . . . , ps eÐnai perittoÐ pr¸toi,
opìte èqoume 2z = 2 · · · 2(2p1 · · · ps) ìpou to 2 kai to 2p1 · · · ps eÐnai “pr¸toi”
parˆgontec tou 2z.
'Ena ˆllo parˆdeigma eÐnai to sÔnolo S = {3k + 1/k ∈ N}.
An s1, s2 ∈ S, tìte kai s1s2 ∈ S. An α, β ∈ S, tìte lème ìti o β diaireÐ
ton α an α = βγ, gia kˆpoio γ ∈ S. 'Ena de stoiqeÐo p ∈ S lègetai S-pr¸toc
an p 1 kai gia r 1, r ∈ S me r|p tìte r = p. Gia parˆdeigma, oi arijmoÐ
4, 7, 10 kai 13 eÐnai S-pr¸toi en¸ o 1 kai o 16 den eÐnai S-pr¸toi. Kˆje de
pr¸toc thc morf c 3κ + 1 eÐnai S-pr¸toc ìpwc epÐshc to ginìmeno dÔo pr¸twn
thc morf c 3κ + 2 eÐnai S-pr¸toc (afoÔ (3κ + 2)(3κ + 2) = 3κ + 1). 'Estw
3λ + 1 = p1p2 · · · ps h anˆlush se pr¸touc enìc stoiqeÐou tou S. Epeid kˆje
pr¸toc = 3 eÐnai thc morf c 3κ + 1 3κ + 2, kˆje pi eÐnai thc morf c 3κ + 1
3κ + 2. Epeid de (3κ + 1)(3κ + 2) = 3κ + 2, sthn anˆlush tou 3λ + 1 prèpei
na upˆrqoun ˆrtio pl joc pr¸twn thc morf c 3κ+2 pou to ginìmenì touc eÐnai
ènac S-pr¸toc. 'Ara kˆje stoiqeÐo tou S analÔetai se ginìmeno S-pr¸twn.
ParathroÔme ìmwc ìti
100 = 3 · 33 + 1 = 4 · 25 = 10 · 10
ìpou to 4, to 10 kai to 25 eÐnai S-pr¸toi, pou shmaÐnei ìti den èqoume monadik
paragontopoÐhsh.
T¸ra kˆnoume thn paradoq ìti o arijmìc n = 1 eÐnai to “kenì” ginìmeno
pr¸twn. An de stic analÔseic twn fusik¸n arijm¸n se pr¸touc parˆgontec
sullèxoume touc Ðsouc pr¸touc se dunˆmeic pr¸twn tìte to 1.1.17 mporeÐ na
xanadiatupwjeÐ wc ex c.
1.1.18 Je¸rhma. Kˆje fusikìc arijmìc n 0 grˆfetai monadikˆ sth morf
n = pα1
1 pα2
2 · · · pακ
κ
ìpou p1, p2, . . . pκ eÐnai pr¸toi me p1 p2 · · · pκ kai αi ≥ 0, i = 1, 2, . . . , κ.

Merikèc forèc mac boleÔei na grˆfoume thn anˆlush tou n se ginìmeno
pr¸twn kai wc
n =
p∈P
pαp
ìpou P eÐnai to sÔnolo ìlwn twn pr¸twn, kˆje αp ≥ 0 kai mìno èna peperasmèno
pl joc ekjet¸n αp eÐnai = 0.
Oi ekjètec αp pou emfanÐzontai sthn prohgoÔmenh paragontopoÐhsh tou n
sun jwc sumbolÐzontai me vp(n) kai qarakthrÐzontai ap thn ex c idiìthta
αp = vp(n) ⇐⇒ pαp
|n kai pαp+1
n.
Gia parˆdeigma, èqoume 600 = 24 ·3−5, opìte v2(600) = 4, v3(600) = v5(600) =
1 kai vp(600) = 0 gia kˆje pr¸to p = 2, 3, 5.
EÐnai fanerì ìti kˆje n ∈ N orÐzetai monadikˆ apì touc ekjètec vp(n).
H apeikìnish vp : N → N èqei tic ex c aplèc idiìthtec.
1.1.19 Prìtash. Gia kˆje m, n ∈ N, m, n ≥ 1, isqÔoun ta ex c:
i) vp(n) = 0, gia kˆje pr¸to p an kai mìnon an n = 1
ii) vp(mn) = vp(m) + vp(n)
iii) m|n an kai mìnon an vp(m) ≤ vp(n), gia kˆje pr¸to p. Sunep¸c h isìthta
isqÔei an kai mìnon an m = n.
iv) δ = (m, n) an kai mìnon an vp(δ) = min{vp(n), vp(m)}, gia kˆje pr¸to p.
v) ε = (m, n) an kai mìnon an vp(ε) = max{vp(n), vp(m)} gia kˆje pr¸to p.
Apìdeixh. ApodeiknÔoume thn iv), oi upìloipec af nontai wc ask seic. 'Estw
δp = min{vp(m), vp(n)}. Gia kˆje p pr¸to, èqoume pδp |m kai pδp |n. 'Ara
pδp |(m, n). Opìte pδp |(m, n). An tan 1 α =
(m, n)
pδp
, tìte α|(m, n) kai
an p eÐnai ènac pr¸toc diairèthc tou α tìte p|m kai p|n. Autì shmaÐnei ìti
pδp+1|(m, n). Allˆ tìte pvp(n)+1|n kai pvp(m)+1|m pou eÐnai adÔnaton. 'Ara
α = 1.

Shmei¸noume ìti h idiìthta 1.1.19 ii) eÐnai h Ðdia m aut pou isqÔei stouc
logˆrijmouc.
Parat rhsh. H paragontopoÐhsh twn jetik¸n akeraÐwn se pr¸touc mporeÐ
na epektajeÐ kai stouc arnhtikoÔc akeraÐouc jètontac ±1 emprìc apì to ginì-
meno. EpÐshc mporeÐ na epektajeÐ kai stouc mh mhdenikoÔc rhtoÔc, epitrèpontac
oi ekjètec na eÐnai arnhtikoÐ. 'Etsi mporoÔme na epekteÐnoume thn apeikìnish vp
sto Q−{0} jètontac vp(−n) = vp(n) kai vp(n/m) = vp(n)−vp(m) gia
n
m
= 0,
(ed¸ prèpei na deiqjeÐ ìti h apeikìnish eÐnai kalˆ orismènh anexˆrthta apì thn
parˆstash tou klˆsmatoc
n
m
).
ParadeÐgmata.
1. Qrhsimopoi¸ntac to Pìrisma 1.1.11 eÐqame deÐxei ìti o arijmìc
√
2 eÐnai
ˆrrhtoc. To apotèlesma autì mporeÐ t¸ra na deiqjeÐ qrhsimopoi¸ntac to
jemelei¸dec je¸rhma thc arijmhtik c: 'Estw
√
2 =
α
β
, ìpou (α, β) = 1,
α, β ∈ N. Tìte 2β2 = α2. An α = pα1
1 pα2
2 · · · pαs
s kai β = qβ1
1 qβ2
2 · · · qβt
t ,
tìte o α2 kai o β2 analÔontai se ˆrtio pl joc pr¸twn paragìntwn en¸ o
2β2 se perittì. 'Ara den mporeÐ na isqÔei 2β2 = α2. Me ton Ðdio isqurismì
apodeiknÔetai ìti gia kˆje pr¸to p kai ˆrtio n o arijmìc n
√
p eÐnai ˆrrhtoc.
Allˆ epÐshc kai autì perilambˆnetai wc eidik perÐptwsh tou ex c genikoÔ
apotelèsmatoc: An o α ∈ N, den eÐnai h n-iost dÔnamh enìc fusikoÔ arij-
moÔ, gia kˆpoio n ∈ N, tìte o n
√
α eÐnai ˆrrhtoc. Prˆgmati upojètontac
ìti eÐnai rhtìc èstw n
√
α =
m
n
, (m, n) = 1, tìte α =
mκ
nκ
αnκ = mκ.
Opìte efarmìzontac thn apeikìnish vp èqoume κvp(m) = vp(α) + κvp(n).
Sunep¸c vp(n) ≤ vp(m). 'Ara, lìgw thc 1.1.19 iii), prèpei n|m pou eÐnai
ˆtopo.
2. 'Estw α, β ∈ N me αβ = γ2, gia kˆpoio γ ∈ N. Tìte upˆrqoun γ1, γ2 ∈ N
ètsi ¸ste
α
(α, β)
= γ2
1 kai
β
(α, β)
= γ2
2.
Prˆgmati, èstw
α
(α, β)
= pα1
1 · · · pas
s kai
β
(α, β)
= qβ1
1 · · · qβκ
κ oi analÔseic

se pr¸touc. Epeid o
α
(α, β)
eÐnai sqetikˆ pr¸toc proc ton
β
(α, β)
ta
sÔnola {p1, . . . , ps} kai {q1, . . . , qκ} eÐnai xèna metaxÔ touc. 'Eqoume de
pα1
1 · · · pαs
s qβ1
1 · · · qβκ
κ =
γ
(α, β)
2
, opìte ta αi kai βi prèpei na eÐnai ˆrtioi
arijmoÐ kai sunep¸c oi
α
(α, β)
kai
β
(α, β)
eÐnai tetrˆgwna akeraÐwn.
3. To ginìmeno tri¸n diadoqik¸n fusik¸n arijm¸n den mporeÐ na eÐnai mia
dÔnamh enìc fusikoÔ arijmoÔ. Prˆgmati, èstw ìti o akèraioc (n−1)n(n+
1) = (n2 − 1)n eÐnai h m-ost dÔnamh enìc akeraÐou. Allˆ to n2 − 1
kai to n eÐnai sqetikˆ pr¸toi arijmoÐ. Apì to jemelei¸dec je¸rhma thc
Arijmhtik c ja prèpei tìte to n2 − 1 kai to n2 na eÐnai m-iostèc dunˆmeic
akeraÐwn. (giatÐ;). Epeid ìmwc to n2 − 1 kai to n2 eÐnai diadoqikoÐ autì
den mporeÐ na isqÔei (giatÐ;).
4. An α, β ∈ N tètoioi ¸ste α|β2, β2|α3, α3|β4, β4|α5, . . . . Tìte α = β.
'Estw ìti
α = pα1
1 · · · pαs
s kai β = qβ1
1 · · · qβt
t
oi analÔseic twn α kai β se pr¸touc. Mac dÐnetai ìti gia n = 1, 2, . . .
isqÔei
kai
p
(2n−1)α1
1 · · · p(2n−1)αs
s q2nβ1
1 · · · q2nβt
t
q2nβ1
1 · · · q2nβt
t p
(2n+1)α1
1 · · · p(2n+1)αs
s .
'Ara to pi|β, gia i = 1, . . . , s kai qj|α gia j = 1, . . . , t. Opìte s = t kai me
mia katˆllhlh arÐjmhsh twn deikt¸n èqoume pi = qi, i = 1, . . . , s.
'Ara αi ≤
2n
2n − 1
βi kai βi ≤
2n + 1
2n
αi. Sunep¸c αi − βi ≤
βi
2n − 1
kai
βi − αi ≤
αi
2n
kaj¸c n → ∞ prokÔptei ìti αi = βi.
To Pl joc twn Pr¸twn
EÐnai fusikì na anarwthjoÔme pìsoi pr¸toi arijmoÐ upˆrqoun. Thn apˆnthsh
s autì to er¸thma èdwse o EukleÐdhc prin perÐpou 2300 qrìnia (Prìtash 20,
9o biblÐo, StoiqeÐa tou EukleÐdh).

1.1.20 Je¸rhma (EukleÐdhc). To pl joc twn pr¸twn arijm¸n eÐnai ˆpeiro.
Apìdeixh (EukleÐdhc). Upojètoume ìti autì to pl joc den eÐnai ˆpeiro kai
èstw ìti ìloi oi pr¸toi eÐnai oi 2 = p1, p2, . . . , pn. JewroÔme ton arijmì
N = p1p2 · · · pn + 1 ≥ 3. Epeid N pj, gia 1 ≤ j ≤ n, o N den eÐnai
pr¸toc kai kaj¸c N 1, lìgw tou 1.1.17, o N diaireÐtai di enìc pr¸tou, èstw
ton pκ. Allˆ tìte o pκ ja prèpei na diaireÐ kai to 1 = N − (N − 1) pou eÐnai
ˆtopo.
Parathr seic.
1. H prohgoÔmenh apìdeixh paramènei h Ðdia an antÐ tou N jewr soume ton
arijmì p1p2 · · · pn − 1.
2. H apìdeixh tou 1.1.20 eÐnai mia eidik perÐptwsh thc ex c apìdeixhc pou
ofeÐletai ston T.J. Stieljes. 'Estw A to ginìmeno opoiond pote r apì
touc n pr¸touc pi, 1 ≤ r ≤ n, pou jewr same sthn apìdeixh kai B =
p1p2 · · · pn/A. Tìte o A + B den diaireÐtai apì kanènan pi, i = 1, . . . , n.
Epeid A + B 1, o A + B prèpei na èqei ènan pr¸to diairèth diˆforo
twn pi, 1 ≤ i ≤ n.
3. An pn eÐnai o n-iostìc pr¸toc, tìte ìpwc sthn apìdeixh tou EukleÐdh,
gia kˆpoio m n, o m-iostìc pr¸toc pm ja diaireÐ ton 2 · 3 · · · pn + 1
kai eÐnai pm ≥ pn+1. Sunep¸c m autì ton trìpo, apì ènan pr¸to p
kajorÐzetai toulˆqiston ènac ˆlloc pr¸toc megalÔteroc apì ton p pou
eÐnai diairèthc tou 2 · 3 · · · p + 1. Gia parˆdeigma, o 7, o 31 kai o 211
kajorÐzontai wc diairètec antÐstoiqa tou 2·3+1 = 7, tou 2·3·5+1 = 31
kai tou 2 · 3 · 5 · 7 + 1 = 211. En¸ o 59 kai o 509 kajorÐzontai wc
diairètec tou 2 · 3 · 5 · 7 · 11 · 13 · 17 + 1 = 59 · 509. Paramènei anapˆnthto
to er¸thma an upˆrqoun ˆpeiroi to pl joc pr¸toi p gia touc opoÐouc oi
arijmoÐ 2·3 · · · p+1 eÐnai pr¸toi an upˆrqoun ˆpeiroi to pl joc sÔnjetoi
arijmoÐ aut c thc morf c. Sthn ptuqiak tou ergasÐa o Alan Borning (J.
Mathematics of Computations 1972) kˆnontac upologismoÔc br ke ìti

metaxÔ ìlwn twn pr¸twn p ≤ 307 mìno gia touc p = 2, 3, 5, 7, 11 kai 31
oi arijmoÐ 2 · 3 · · · p + 1 eÐnai pr¸toi, en¸ gia p = 3, 5, 11, 13, 41 kai 89 oi
arijmoÐ 2 · 3 · · · p − 1 eÐnai pr¸toi.
4. 'Opwc eÐdame, an o pn eÐnai o n-iostìc pr¸toc tìte, gia kˆpoio m n, o
pm diaireÐ ton
n
i=1
pi + 1, opìte
pn+1 ≤ pm ≤
n
i=1
pi + 1 ≤ pn
n + 1.
Ap aut th sqèsh kai th majhmatik epagwg prokÔptei ìti gia kˆje
n ≥ 1 isqÔei pn ≤ 22n−1
me pn 22n−1
, gia n 1. Sunep¸c upˆrqoun
toulˆqiston n pr¸toi pou eÐnai mikrìteroi tou 22n−1
. Prˆgmati, èqoume
p1 ≤ 2 kai upojètoume ìti p2 ≤ 22, p3 ≤ 24, . . . , pn ≤ 22n−1
. Tìte
pn+1 ≤ p1 · · · pn+1 ≤ 21+2+···+2n−1
+1 = 22n−1+1 22n−1+22n−1 = 22n
.
Ap autì sumperaÐnoume ìti gia n ≥ 1 upˆrqoun toulˆqiston n+1 pr¸toi
pou eÐnai mikrìteroi tou 22n
.
5. Jètoume n1 = 2 kai epagwgikˆ orÐzoume touc arijmoÔc nκ wc nκ+1 =
n2
κ −nκ +1. Opìte nκ+1 = n1 · · · nκ +1. An m eÐnai ènac koinìc diairèthc
tou ns kai n (èstw s ), tìte o m diaireÐ ton n1 · · · ns · · · n −1 kai ton
n = n1 · · · n −1 + 1, sunep¸c m = 1. 'Ara anˆ dÔo oi nκ eÐnai sqetikˆ
pr¸toi metaxÔ touc. Epeid oi arijmoÐ nκ (pou eÐnai ˆpeiroi se pl joc)
anˆ dÔo eÐnai sqetikˆ pr¸toi metaxÔ touc kai o kajènac èqei ènan pr¸to
diairèth, upˆrqoun ˆpeiroi se pl joc pr¸toi. Aut h apìdeixh eÐnai mia
diaforopoÐhsh tou sulogismoÔ thc apìdeixhc tou EukleÐdh allˆ ìmwc
upodeiknÔei ìti s autì to sulogismì ekeÐno pou paÐzei shmantikì rìlo
eÐnai ìti oi pi eÐnai pr¸toi anˆ dÔo kai ìqi tìso ìti autoÐ eÐnai pr¸toi
arijmoÐ.
S aut thn idèa sthrÐzetai kai h epìmenh apìdeixh tou 1.1.20 pou ofeÐletai
ston G. P´olya. JewroÔme tou arijmoÔc
Fn = 22n
+ 1, n ∈ N

kai apodeiknÔoume ìti anˆ dÔo autoÐ eÐnai sqetikˆ pr¸toi. Pr¸ta deÐqnoume
ìti isqÔei h sqèsh
Fn = F0F1 · · · Fn−1 + 2.
Prˆgmati gia n = 1 profan¸c isqÔei. Epagwgikˆ, upojètontac ìti Fn =
F0F1 · · · Fn−1 + 2, èqoume
F0F1 · · · Fn = (Fn − 2)Fn = (22n
− 1)(22n
+ 1) = 22n+1
− 1 = Fn+1 − 2,
ìpwc apaiteÐtai.
T¸ra an δ eÐnai ènac diairèthc twn Fk kai F (èstw k ), tìte o δ eÐnai
diairèthc kai tou F − F0F1 · · · Fk · · · F −1 = 2. 'Ara o δ eÐnai o 1 o 2.
Allˆ epeid oi arijmoÐ tou Fermat eÐnai perittoÐ, prèpei δ = 1. Sunep¸c,
gia kˆje n ∈ N, upˆrqoun toulˆqiston n+1 diakekrimènoi pr¸toi arijmoÐ,
afoÔ kˆje Fi, i ≤ n, èqei èna pr¸to diairèth (o opoÐoc den diaireÐ kanènan
ˆllo Fj, j = i, j ≤ n). Epeid upˆrqoun ˆpeiroi arijmoÐ tou Fermat,
upˆrqoun ˆpeiroi pr¸toi arijmoÐ. (Mia pio sÔntomh apìdeixh eÐnai h ex c.
Upojètoume ìti m n, kai èstw m = n + κ, κ 0. Jètoume x = 22n
,
opìte
Fn+κ − 2
Fn
=
22n2κ
− 1
22n
+ 1
=
x2κ
− 1
x + 1
= x2κ−1
− x2κ−2
+ · · · − 1 ∈ Z.
Autì shmaÐnei ìti Fn|Fn+κ − 2. Opìte an δ|Fn kai δ|Fn+κ ja prèpei δ|2.
Allˆ oi Fn kai Fn+κ eÐnai perittoÐ, ˆra δ = 1.)
AutoÐ oi arijmoÐ onomˆzontai arijmoÐ tou Fermat, kai fèroun aut thn
onomasÐa epeid o Fermat se mia epistol tou proc ton Pascal kai proc
ˆllouc ègrafe ìti autoÐ oi arijmoÐ eÐnai pr¸toi. O Fermat diatÔpwse aut
thn eikasÐa apì to gegonìc ìti oi pr¸toi 5 tètoioi arijmoÐ eÐnai pr¸toi:
F0 = 3, F1 = 5, F2 = 17, F3 = 257, F4 = 65537. O epìmenoc arijmìc
tou Fermat F5 = 4294967297 eÐnai arketˆ megˆloc kai thn epoq ekeÐnh
tan dÔskolo na paragontopoihjeÐ se pr¸touc. 100 qrìnia metˆ o Euler
to 1739 apèdeixe ìti kˆje diairèthc enìc arijmoÔ tou Fermat prèpei na
eÐnai thc morf c 2n+1k + 1. Sunep¸c, gia n = 5 ènac pr¸toc parˆgontac

tou F5 prèpei na eÐnai thc morf c 64k + 1. 'Etsi eÔkola brèjhke ìti o
641 = 26 · 10 + 1 diaireÐ ton F5 kai ˆra h eikasÐa tou Fermat den isqÔei.
To 1877 o J. Pepin apèdeixe ìti o Fn eÐnai pr¸toc an kai mìnon an autìc
diaireÐ ton 3
Fn−1
2 + 1. EpÐshc o Fn, n ≥ 1, eÐnai pr¸toc an kai mìnon an o
monadikìc trìpoc pou mporeÐ na grafteÐ o Fn wc ˆjroisma dÔo tetrag¸nwn
eÐnai o profan c:
Fn = (22n−1
)2
+ 22
.
Paramènei anapˆnthto to er¸thma an kˆje arijmìc tou Fermat den diai-
reÐtai apì to tetrˆgwno enìc arijmoÔ.
Upˆrqoun kai ˆllec anagkaÐec kai ikanèc sunj kec gia na elègqoume an o
Fn eÐnai pr¸toc, ìmwc den upˆrqei èwc s mera ènac genikìc kanìnac pou ja
odhgoÔse se mia kajoristik apˆnthsh sto anapˆnthto er¸thma: EÐnai o
F4 o megalÔteroc pr¸toc arijmìc tou Fermat; dhlad gia n 4, o Fn eÐnai
sÔnjetoc; an autì den isqÔei, upˆrqoun ˆpeiroi to pl joc pr¸toi arijmoÐ
tou Fermat; upˆrqoun ˆpeiroi to pl joc sÔnjetoi arijmoÐ tou Fermat;
H apˆnthsh s autˆ ta erwt mata eÐnai shmantik giatÐ, ìpwc apèdeixe
to 1801 o Gauss sto gnwstì Disquisitiones Arithmeticae, upˆrqei mia
axishmeÐwth sqèsh metaxÔ twn EukleÐdeiwn kataskeu¸n (dhlad , me th
qr sh kanìna kai diab th) twn kanonik¸n polug¸nwn kai twn arijm¸n
tou Fermat. Sugkekrimèna, o Gauss èdeixe ìti an to pl joc twn pleur¸n
enìc kanonikoÔ polug¸nou eÐnai thc morf c 2κFm1 · · · Fmr , ìpou κ ≥ 0,
r ≥ 0 kai Fmi eÐnai diakekrimènoi pr¸toi arijmoÐ tou Fermat, tìte autì to
polÔgwno mporeÐ na kataskeuasjeÐ me kanìna kai diab th. To antÐstrofo
autoÔ tou jewr matoc apedeÐqjei to 1837 apì ton P.L. Wantzel, an kai o
Gauss sto Disquisions Arithmeticae isqurÐzetai ìti isqÔei (qwrÐc ìmwc na
to apodeiknÔei). Oi ArqaÐoi 'Ellhnec gn¸rizan thn EukleÐdeia kataskeu
mìno gia to isìpleuro trÐgwno kai to kanonikì pentˆgwno ˆra kai gia
kˆje kanonikì polÔgwno pou èqei n pleurèc, ìpou n = 2κ, 2κ · 3, 2κ ·
5 kai 2κ · 15.∗ 'Etsi metˆ apì 2000 qrìnia, me to je¸rhma tou Gauss,
∗
Kˆje gwnÐa mporeÐ na diqotomhjeÐ. Sunep¸c an èna n-gwno eÐnai kataskeuˆsimo tìte

apant jhke pl rwc èna apì ta shmantikìtera erwt mata twn ArqaÐwn:
poia kanonikˆ polÔgwna eÐnai EukleÐdeia kataskeuˆsima. SÔmfwna me to
je¸rhma tou Gauss kai èqontac upìyin ìti oi mìnoi gnwstoÐ èwc s mera
pr¸toi arijmoÐ tou Fermat eÐnai oi 3, 5, 17, 257 kai 65537, to pl joc twn
gnwst¸n tètoiwn polug¸nwn pou èqoun perittì pl joc pleur¸n (ed¸
perilambˆnoume kai autì pou èqei mia pleurˆ) isoÔtai me to pl joc twn
diairet¸n tou arijmoÔ 1 · 3 · 5 · 17 · 257 · 65537 kai autì to pl joc eÐnai 32.
(Shmei¸noume ìti 232 − 1 = 1 · 3 · 5 · 17 · 257 · 65537).
EpÐshc apì to je¸rhma tou Gauss prokÔptei ìti genikˆ den eÐnai dunat
h triqotìmhsh miac gwnÐac, afoÔ diaforetikˆ, gia parˆdeigma, triqoto-
m¸ntac tic gwnÐec (60o) enìc isìpleurou trig¸nou ja tan EukleÐdeia
kataskeuˆsimo kai to kanonikì polÔgwno 9(= 21 + 1)2 pleur¸n.
Lègetai, ìti apì to je¸rhma autì o Gauss apofˆsise na afier¸sei th zw
tou sth melèth twn majhmatik¸n. 'Htan de polÔ uper fanoc gi autì to
apotèlesma kai z thse na topojethjeÐ èna 17-gwno ston tˆfo tou. H
epijumÐa tou aut den ekplhr¸jhke sto G¨ottingen ìpou etˆfh. Upˆr-
qei ìmwc sto mnhmeÐo tou pou èqei anegerjeÐ sth gennèthrˆ tou pìlh to
Brunswick.
H idèa sthn apìdeixh tou Jewr matoc tou EukleÐdh mporeÐ epÐshc na efarmo-
sjeÐ se orismènec peript¸seic gia na deÐqnoume ìti to pl joc miac sugkekrimènhc
morf c pr¸twn arijm¸n eÐnai ˆpeiro. Gia parˆdeigma, ènac perittìc pr¸toc arij-
mìc eÐnai thc morf c 4k +1 thc morf c 4k +3. Gia to pl joc twn pr¸twn thc
kai to kanonikì 2κ
n-gwno eÐnai kataskeuˆsimo. EpÐshc an dÔo kanonikˆ polÔgwna me n1 kai
n2 pleurèc, ìpou (n1, n2) = 1, eÐnai kataskeuˆsima tìte kai to kanonikì polÔgwno me n1n2
pleurèc eÐnai kataskeuˆsimo. Prˆgmati, upˆrqoun akèraioi x1, x2 tètoioi ¸ste

n1x1 − n2x2 = 1
360
n1n2
= x1
360
n2
− x2
360
n1
.
Opìte h gwnÐa
360
n1n2
eÐnai kataskeuˆsimh. P.q. to kanonikì 15-gwno eÐnai kataskeuˆsimo
afoÔ to isìpleuro trÐgwno kai to pentˆgwno eÐnai kataskeuˆsima.

morf c 4k+3 h mèjodo tou EukleÐdh mporeÐ na efarmosjeÐ wc ex c. Upojètoume
ìti upˆrqoun mìno peperasmènou pl jouc tètoioi pr¸toi p1 = 3, p2 = 7, . . . , pn
kai jewroÔme ton arijmì N = 4p1 · · · pn +3. Autìc o arijmìc den mporeÐ na eÐnai
pr¸toc (afoÔ N = pj kai ap thn upìjesh oi pj exantloÔn ìlouc touc pr¸touc
thc morf c 4k + 3). 'Ara autìc prèpei na èqei pr¸touc diairètec. An ìloi autoÐ
oi diairètec èqoun th morf 4k+1 tìte kai o N prèpei na eÐnai aut c thc morf c
kaj¸c (4k1 + 1)(4k2 + 1) = 4(4k1k2 + k1 + k2) + 1. Allˆ o N den èqei aut
th morf kai ˆra toulˆqiston ènac apì touc pr¸touc diairètec tou, èstw o p,
prèpei na eÐnai thc morf c 4n + 3. Epeid p|N, o p eÐnai diˆforoc twn pj, diìti
diaforetikˆ o p ja èprepe na eÐnai 1. 'Ara upˆrqoun ˆpeiroi pr¸toi thc morf c
4k + 3.
Kaj¸c gnwrÐzoume ìti to pl joc ìlwn twn pr¸twn eÐnai ˆpeiro kai ìti to
Ðdio isqÔei kai gia ekeÐnouc thc morf c 4k+3, to pl joc twn pr¸twn thc morf c
4k+1 mporeÐ na eÐnai ˆpeiro peperasmèno. 'Omwc den mporoÔme na apofanjoÔme
poio eÐnai autì to pl joc qrhsimopoi¸ntac thn Ðdia idèa me aut pou efarmìsame
gia touc pr¸touc thc morf c 4k + 3, kaj¸c tètoioi arijmoÐ mporoÔn na eÐnai
ginìmeno pr¸twn thc morf c 4k + 3, p.q. 21 = 3 · 7. Argìtera ja deÐxoume, sta
tetragwnikˆ upìloipa, ìti kai autì to pl joc twn pr¸twn eÐnai ˆpeiro. EpÐshc
shmei¸noume ìti h idèa tou EukleÐdh mporeÐ na efarmosjeÐ gia touc pr¸touc thc
morf c 6k+5 ìpwc kai gia touc pr¸touc thc morf c 3k+2. Den mporoÔme ìmwc
na mimhjoÔme aut thn idèa gia touc pr¸touc thc morf c 8k + 7. ParathroÔme
t¸ra ìti ìloi autoÐ oi pr¸toi eÐnai ìroi arijmhtik¸n proìdwn αn + β, n ∈ N.
An o m.k.d(α, β) eÐnai = 1 tìte ìloi oi ìroi miac tètoiac akoloujÐac diairoÔntai
dia tou (α, β) kai ˆra kanènac ap autoÔc den eÐnai pr¸toc. An (α, β) = 1,
tìte upˆrqoun ˆpeiroi to pl joc sÔnjetoi arijmoÐ thc morf c αn + β, afoÔ
upˆrqoun ˆpeiroi n me (n, β) = 1. Allˆ s aut thn perÐptwsh de mporoÔme na
apofanjoÔme, mimoÔmenoi th mèjodo tou EukleÐdh (ìpwc eÐdame p.q. gia α = 8,
β = 7), ìti upˆrqoun ˆpeiroi to pl joc ìroi miac tètoiac akoloujÐac pou eÐnai
pr¸toi arijmoÐ. 'Omwc o Lejeune-Dirichlet (1805 1859) to 1837, sthrizìmenoc
se proqwrhmènec mejìdouc thc majhmatik c anˆlushc, apèdeixe ìti prˆgmati

isqÔei:
1.1.21 Je¸rhma (Dirichlet). Kˆje arijmhtik prìodoc αn+β me (α, β) = 1,
perilambˆnei ˆpeiro pl joc ìrwn pou eÐnai pr¸toi arijmoÐ.
'Eqontac upìyin autì to apotèlesma, dhmiourgeÐtai eÔloga to er¸thma: EÐnai
to je¸rhma tou Dirichlet mia eidik perÐptwsh enìc eurÔterou fainomènou; dh-
lad parousiˆzetai to Ðdio fainìmeno se mia genikìterh oikogèneia ekfrˆsewn;
Diaisjhtikˆ anamenìtan ìti autì èprepe na sumbaÐnei, dhlad episteÔeto ìti,
ìpwc gia tic arijmhtikèc proìdouc, ja tan dunatìn na mporoÔsame na apodeÐ-
xoume ìti mia sunˆrthsh pou orÐzetai me aplì trìpo dÐnei ˆpeiro pl joc pr¸twn.
Dustuq¸c ìmwc mia tètoia diaÐsjhsh tan esfalmènh. Gia parˆdeigma, gia pˆ-
ra pollˆ qrìnia paramènei anoiktì prìblhma an oi arijmoÐ thc morf c n2 + 1,
n ∈ N, perilambˆnoun ˆpeiro pl joc pr¸twn, (ìpwc èqoun eikˆsei oi G. Hardy
kai J. Littlewood). To 1978 o H. Iwaniec apèdeixe ìti upˆrqoun ˆpeiroi to
pl joc arijmoÐ thc morf c n2 + 1 pou eÐnai eÐte pr¸toi ginìmeno dÔo pr¸-
twn. Mia ˆllh oikogèneia pr¸twn eÐnai autoÐ thc morf c 2p −1, ìpou p pr¸toc.
(Shmei¸noume ìti apì to Parˆdeigma 4 metˆ thn Prìtash 1.1.12, prokÔptei ìti
an o 2α − 1 eÐnai pr¸toc tìte anagkastikˆ o α eÐnai pr¸toc. To antÐstrofo
den isqÔei, p.q. 211 − 1 = 2047 = 23 · 89). AutoÐ onomˆzontai pr¸toi arijmoÐ
tou Mersenne. EÐnai ˆgnwsto mèqri s mera an upˆrqoun ˆpeiroi tètoioi pr¸toi.
Shmei¸noume ìti o megalÔteroc pr¸toc arijmìc pou eÐnai gnwstìc mèqri s mera
eÐnai o 44os pr¸toc arijmìc tou Mersenne. Brèjhke to Septèmbrio tou 2006
eÐnai o 232582657 − 1. Autìc èqei 9808358 dekadikˆ yhfÐa. O 43os pr¸toc tou
Mersenne brèjhke to Dekèmbrio tou 2005 kai eÐnai o 230402457 − 1. Autìc èqei
9152052 dekadikˆ yhfÐa, en¸ o prohgoÔmenoc pr¸toc èqei 7816230 yhfÐa kai
eÐnai o 225964951 − 1.
Thc Ðdiac fÔshc er¸thma pou paramènei ìmwc anapˆnthto aforˆ touc dÐdu-
mouc pr¸touc. 'Enac pr¸toc arijmìc onomˆzetai dÐdumoc pr¸toc an h diaforˆ
tou epìmenou tou prohgoÔmenou ap autìn pr¸tou arijmoÔ eÐnai 2, dhlad o p
eÐnai dÐdumoc pr¸toc an o p + 2 o p − 2 eÐnai pr¸toc. Sunep¸c autoÐ mporoÔn
na grafoÔn se zeÔgh, gia parˆdeigma, (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), . . . .

'Oloi oi dÐdumoi ektìc apì touc 3 kai 5 eÐnai thc morf c 6k±1, o de megalÔteroc
ap autoÔc pou eÐnai gnwstìc mèqri s mera (upologÐsjhke ton AÔgousto tou
2005) eÐnai o
100314512544015 · 217196
− 1
kai èqei 51779 yhfÐa.
Sqetikì me to prìblhma thc Ôparxhc ˆpeirou pl jouc didÔmwn pr¸twn eÐnai
to prìblhma thc Ôparxhc ˆpeirou pl jouc pr¸twn thc morf c 2p + 1, ìpou p
pr¸toc. Sqetikèc anaforèc gia ta erwt mata autˆ, allˆ kai gia pollˆ ˆlla
tou Ðdiou tÔpou, upˆrqoun se diˆfora biblÐa thc JewrÐac Arijm¸n allˆ kai sto
“Internet”.
AxÐzei na shmei¸soume ed¸ ìti arketoÐ majhmatikoÐ èqoun prospaj sei na
broun ènan aplì tÔpo pou na dÐnei wc timèc tou ìlouc touc pr¸touc ( akìmh
perissìtero mìno touc pr¸touc). 'Ena apì ta ikanopoihtikˆ apotelèsmata sthn
kateÔjunsh aut eÐnai tou W.H. Mills. Autìc to 1947 apèdeixe ìti upˆrqei ènac
ˆrrhtoc pragmatikìc arijmìc a 1 tètoioc ¸ste o n-iostìc pr¸toc pn isoÔtai
me [a3n
], ìpou me [x] sumbolÐzoume to megalÔtero akèraio pou eÐnai mikrìteroc
apì to x. Mèqri s mera den èqei kajorisjeÐ ènac sugkekrimènoc tètoioc arijmìc
α. EpÐshc to 1970 apì to axioshmeÐwto apotèlesma tou Yuri Matijasevich pou
èluse to 10o prìblhma tou Hilbert (sumplhr¸nontac tic ergasÐec twn Martin
Davis, Hilary Putman kai thc Julia Robinson) proèkuye ìti upˆrqoun polu¸-
numa me akèraiouc suntelestèc twn opoÐwn ìlec oi jetikèc timèc pou paÐrnoun
stouc fusikoÔc arijmoÔc eÐnai akrib¸c oi pr¸toi arijmoÐ. 'Ena tètoio polu¸nu-
mo kajorÐsjhke to 1976 apì touc J. Jones, D. Sato, H. Wada kai D. Wiens.
Autì to polu¸numo eÐnai 25oυ bajmoÔ kai 26 metablht¸n.
EÐnai eukairÐa ed¸ na apodeÐxoume to ex c je¸rhma.
Je¸rhma (Goldbach). Den upˆrqei èna mh-stajerì polu¸numo f(x) me akè-
raiouc suntelestèc pou ìlec oi timèc tou stouc fusikoÔc arijmoÔc na eÐnai pr¸toi
arijmoÐ.
Apìdeixh. 'Estw ìti to je¸rhma den isqÔei kai
f(x) = αrxr
+ · · · + α1x + α0, αi ∈ Z

eÐnai èna polu¸numo bajmoÔ ≥ 1 pou oi timèc touc stouc fusikoÔc eÐnai pr¸toi
arijmoÐ.
'Estw ènac pr¸toc arijmìc kai gia kˆpoio n ∈ N, f(n) = p. Tìte gia kˆje
k ∈ Z, oi akèraioi arijmoÐ f(n + kp) diairoÔntai dia p, afoÔ èqoume
f(n + kp) =αr(n + kp)r
+ · · · + α1(n + kp) + α0 = αr
r
i=0
r
i
ni
(kp)r−i
+ αr−1
r−1
i=0
r − 1
i
ni
(kp)r−1−i
+ · · · + α1(n + kp) + α0
=αrnr
+ · · · + α1n + α0 + αr
r−1
i=0
r
i
ni
(kp)r−i
+ · · · + α1kp
=f(n) + pg(k) = p(1 + g(k)),
ìpou g(k) eÐnai èna polu¸numo tou k me akèraiouc suntelestèc. Epeid p|f(n+
kp), apì thn upìjesh ja prèpei f(n + kp) = p gia ìla ta k. Allˆ tìte autì ja
s maine ìti to polu¸numo F(k) = f(n + kp) − p bajmoÔ r èqei ˆpeiro pl joc
riz¸n. Autì ìmwc de mporeÐ na isqÔei kaj¸c apì to Jemelei¸dec Je¸rhma
thc 'Algebrac èna polu¸numo bajmoÔ r de mporeÐ na èqei perissìterec apì r
rÐzec.
Anafèroume epÐshc ìti eÐnai akìma anapˆnthto to er¸thma an upˆrqei èna
polu¸numo deutèrou bajmoÔ miac metablht c pou na paÐrnei stouc akèraiouc
ˆpeiro pl joc tim¸n oi opoÐec na eÐnai pr¸toi arijmoÐ. Autì den eÐnai gnwstì
an isqÔei oÔte gia mia eidik perÐptwsh tètoiou poluwnÔmou.
Grammikèc Diofantikèc Exis¸seic
H lèxh “diofantikèc” proèrqetai apì to ìnoma tou 'Ellhna Diìfantou thc Ale-
xˆndreiac o opoÐoc èzhse ton trÐto ai¸na m.Q. kai tan o pr¸toc pou melèthse
me susthmatikì trìpo tic akèraiec lÔseic exis¸sewn. To diaswjèn èrgo tou
Diìfantou “Ta Arijmhtikˆ” (prìkeitai gia ta èxi biblÐa apì ìlo to èrgo tou)
dhmosieÔjhkan se biblÐo apì ton Bachet de M´ezitiac to 1621 pou melèthse o

Pierre de Fermat (1608 1665). To biblÐo autì eÐnai s mera gnwstì kaj¸c sta
perij¸ria twn selÐdwn enìc antitÔpou tou o Fermat eÐqe diatup¸sei pollèc apì
tic idèec tou. Autèc oi idèec apotèlesan thn arq thc diamìrfwshc thc JewrÐac
Arijm¸n wc enìc xeqwristoÔ klˆdou twn majhmatik¸n.
Mia exÐswsh thc morf c
f(x, y, z, . . . ) = 0,
ìpou f(x, y, z, . . . ) eÐnai èna polu¸numo metablht¸n x, y, z, . . . me akèraiouc sun-
telestèc, onomˆzetai Diofantik exÐswsh kai to prìblhma thc eÔreshc twn akè-
raiwn lÔsewn miac tètoiac exÐswshc lègetai Diofantikì prìblhma. 'Otan lème
“èstw h Diofantik exÐswsh f(x, y, z, . . . ) = 0” ennooÔme to Diofantikì prì-
blhma.
Mia Diofantik exÐswsh mporeÐ na mhn èqei lÔseic, na èqei peperasmèno
pl joc lÔsewn na èqei ˆpeirec lÔseic. Sthn teleutaÐa perÐptwsh oi lÔseic
sun jwc dÐdontai sunart sei miac perissìterwn akèraiwn paramètrwn.
Gewmetrikˆ, oi akèraiec lÔseic thc Diofantik c exÐswshc f(x, y) = 0 pari-
stoÔn ta shmeÐa epÐ thc kampÔlhc f(x, y) = 0 pou èqoun akèraiec suntetagmènec.
Gia parˆdeigma, sthn perÐptwsh thc kampÔlhc x2 −2y2 = 0, h mình akèraia lÔsh
eÐnai profan¸c h (x, y) = (0, 0), dhlad to shmeÐo (0, 0) eÐnai to mìno shmeÐo epÐ
twn dÔo eujei¸n x2 − 2y2 = 0 me akèraiec suntetagmènec. Sthn perÐptwsh thc
kampÔlhc x + 2y − 1 = 0 èqoume tic ˆpeirec lÔseic (x, y) = (3 + 2k, −1 − k)
k ∈ Z, en¸ sthn perÐptwsh thc kampÔlhc 4x+6y = 11 den èqoume kamÐa akeraÐa
lÔsh (afoÔ gia kˆje akèraiouc x kai y to aristerì mèloc eÐnai ˆrtioc arijmìc
en¸ to dexiì eÐnai perittìc arijmìc).
'Estw α1, α2, . . . , αn, βn ∈ Z. H pio apl Diofantik exÐswsh n metablht¸n
x1, x2, . . . , xn eÐnai thc morf c
α1x1 + α2x2 + · · · + αnxn = βn(*)
Aut onomˆzetai grammik Diofantik exÐswsh n metablht¸n. H onomasÐa
proèrqetai apì to gegonìc ìti h grammik Diofantik exÐswsh dÔo metablh-
t¸n α1x1 + α2x2 = β paristˆ eujeÐa gramm sto epÐpedo.

Skopìc mac ed¸ eÐnai na kajorÐsoume ìlec tic akèraiec lÔseic thc (∗). Gia
to lìgo autì qreiazìmaste thn ènnoia twn m.k.d kai e.k.p perissìterwn twn dÔo
akèraiwn arijm¸n.
H ènnoia tou m.k.d. dÔo akèraiwn epekteÐnetai kai se perissìterouc twn dÔo.
An α1, α2, . . . , αn eÐnai akèraioi, ìqi ìloi Ðsoi me to mhdèn, tìte autoÐ èqoun koi-
noÔc diairètec, gia parˆdeigma touc ±1. To pl joc twn koin¸n diairet¸n aut¸n
eÐnai peperasmèno. O megalÔteroc twn koin¸n diairet¸n twn α1, α2, . . . , αn, ono-
mˆzetai mègistoc koinìc diairèthc kai sumbolÐzetai me (α1, α2, . . . , αn). Autìc
eÐnai ≥ 1. An (α1, . . . , αn) = 1, tìte lème ìti oi α1, α2, . . . , αn eÐnai sqetikˆ pr¸-
toi metaxÔ touc. An (αi, αj) = 1, i, j = 1, 2, . . . , n tìte lème ìti oi α1, α2, . . . , αn
eÐnai anˆ dÔo sqetikˆ pr¸toi metaxÔ touc. Profan¸c an oi α1, α2, . . . , αn eÐnai
anˆ dÔo sqetikˆ pr¸toi tìte eÐnai kai sqetikˆ pr¸toi metaxÔ touc. Prosoq , to
antÐstrofo den isqÔei.
Me anˆlogo trìpo epekteÐnetai kai h ènnoia tou e.k.p. dÔo akèraiwn se
perissìterouc twn dÔo mh mhdenik¸n akèraiwn α1, α2, . . . , an, orÐzontac wc elˆ-
qisto koinì pollaplˆsiì touc to mikrìtero jetikì akèraio (pou upˆrqei lìgw
thc arq c tou elaqÐstou) metaxÔ ìlwn twn koin¸n pollaplasÐwn touc (pou eÐnai
ˆpeirou pl jouc). Autì sumbolÐzoume me [α1, α2, . . . , αn].
EÔkola prokÔptei ìti isqÔei
(α1, α2, . . . , αn) = ((α1, α2, . . . , αn−1), αn) = ((α1, . . . , αk), . . . , (αm, . . . , αn))
kai
[α1, α2, . . . , αn] = [[α1, α2, . . . , αn−1], αn] = [[α1, . . . , αk], . . . , [αm, . . . , αn]].
Oi idiìthtec de pou isqÔoun gia ton m.k.d. kai to e.k.p. epekteÐnontai kai gia
perissìterouc twn dÔo. Gia parˆdeigma, èqoume ìti
• To e.k.p. [a1, a2, . . . , an] diaireÐ ìla ta koinˆ pollaplˆsia twn a1, a2, . . . , an.
• An δ = (α1, α2, . . . , αn) tìte upˆrqoun akèraioi x1, x2, . . . , xn ètsi ¸ste
δ = α1x1 + α2x2 + · · · + αnxn
kai kˆje diairèthc twn α1, α2, . . . , αn eÐnai diairèthc tou δ.

EpÐshc o δ eÐnai o mikrìteroc ìlwn twn ekfrˆsewn α1y1+α2y2+· · ·+αnyn
0, me y1, . . . , yn ∈ Z, kai mˆlista ìlec autèc oi ekfrˆseic eÐnai pollaplˆsia tou
δ.
H sqèsh 1.1.8 pou sundèei ton m.k.d. kai to e.k.p. dÔo akèraiwn epekteÐnetai
se perissìterouc twn dÔo mh mhdenik¸n jetik¸n akèraiwn α1, α2, . . . , αn wc
ex c.
[a1, a2, . . . , an] =
(αi1 , αi2 , . . . , αim )
(αj1 , αj2 , . . . , αjm )
ìpou to ginìmeno ston arijmht (ant. sto paronomast ) lambˆnetai wc proc
ìlec tic m-idec (i1, i2, . . . , im) (ant. (j1, . . . , jm)) ìpou 1 ≤ i1 i2 · · ·
im ≤ n gia m perittì (ant. 1 j1 j2 · · · jm n, gia m ˆrtio). Gia
parˆdeigma èqoume
[α1, α2, α3] =
α1α2α3(α1, α2, α3)
(α1, α2)(α1, α3)(α2, α3)
kai
[α1, α2, α3, α4] =
α1α2α3α4(α1, α2, α3)(α1, α2, α4)(α1, α3, α4)(α2, α3, α4)
(α1, α2)(α1, α3)(α1, α4)(α2, α3)(α2, α4)(α3, α4)(α1, α2, α3, α4)
.
Prˆgmati, grˆfoume touc αi se ginìmeno pr¸twn:
αi = p
vk(αi)
k .
Opìte arkeÐ na deÐxoume ìti
max{α1, . . . , αn} =
n
m=1
(−1)m+1
min{αi1 , αi2 , . . . , αim }.
ParathroÔme ìti aut h sqèsh eÐnai summetrik wc proc ta αi, dhlad an efar-
mosjeÐ opoiad pote metˆjesh sta α1, α2, . . . , αn h sqèsh aut paramènei h Ðdia.
Sunep¸c mporoÔme na upojèsoume ìti α1 ≥ α2 ≥ α3 ≥ · · · ≥ αn. 'Ara aut h
sqèsh eÐnai isodÔnamh me thn
α1 =
n
m=1
(−1)m+1
αim

pou profan¸c isqÔei.
'Estw t¸ra δn = m.k.d.(α1, α2, . . . , αn) ìpou αi eÐnai oi suntelestèc pou
emfanÐzontai sth Diofantik ExÐswsh (∗). Sta prohgoÔmena eÐdame ìti o δn
eÐnai o mikrìteroc akèraioc tou sunìlou {α1x1+· · ·+αnxn 0/x1, . . . , xn ∈ Z}.
Epiplèon kˆje stoiqeÐo tou sunìlou
M = {α1x1 + · · · + αnxn/x1, . . . , xn ∈ Z}
eÐnai èna pollaplˆsio tou δn, dhlad èqoume
M = δnZ = {δnm/m ∈ Z}.
Sunep¸c èqoume
1.1.22 L mma. H (∗) èqei lÔsh an kai mìnon an o βn diaireÐtai dia tou m.k.d.
(α1, α2, . . . , αn).
T¸ra prosdiorÐzoume ìlec tic lÔseic thc (∗).
1.1.23 Je¸rhma. 'Estw α1, α2 dÔo mh mhdenikoÐ akèraioi kai èstw δ2 =
(α1, α2). Tìte h exÐswsh
α1x1 + α2x2 = β2
èqei lÔsh an kai mìnon an δ2|β2. Epiplèon, an (u1, u2) eÐnai mia lÔsh, tìte kˆje
ˆllh lÔsh eÐnai thc morf c
x1 = u1 + k
α2
δ2
, x2 = u2 − k
α1
δ2
.
Apìdeixh. O pr¸toc isqurismìc dÐdetai sto prohgoÔmeno L mma. 'Estw loipìn
ìti δ2|β2 kai ìti (u1, u2) eÐnai mia lÔsh. Tìte h antikatˆstash twn x1 kai x2 sthn
exÐswsh me touc akèraiouc u1 + k
α2
δ2
kai u2 − k
α1
δ2
antÐstoiqa, deÐqnei ìti autoÐ
thn epalhjeÔoun. Autì apodeiknÔetai kai gewmetrikˆ kaj¸c o −
α1
α2
= −
α1/δ2
α2/δ2
eÐnai h klÐsh thc eujeÐac α1x1 + α2x2 = β2 sto epÐpedo.
AntÐstrofa, t¸ra apodeiknÔoume ìti an u1, u2 eÐnai mia lÔsh tìte kˆje ˆllh
lÔsh èqei th morf pou anafèretai sto je¸rhma. 'Estw (x1, x2) mia opoiad pote

lÔsh, opìte èqoume
α1x1 + α2x2 = β2 kai α1u1 + α2u2 = β2.
'Ara
α1(x1 − u1) = α2(u2 − x2)
α1
δ2
(x1 − u1) =
α2
δ2
(u2 − x2).
Allˆ
α1
δ2
,
α2
δ2
= 1, opìte apì to L mma tou EukleÐdh prèpei
α2
δ2
x1 − u1 kai
α1
δ2
u2 − x2.
'Ara
x1 − u1 = k
α2
δ2
x1 = u1 + k
α2
δ2
,
gia kˆpoio k ∈ Z. 'Etsi èqoume
α1
δ2
u1 + k
α2
δ2
− u1 =
α2
δ2
(u2 − x2) k
α1
δ2
= u2 − x2,
dhlad x2 = u2 − k
α1
δ2
.
To Ðdio apotèlesma mporeÐ na exaqjeÐ kai qwrÐc th qr sh tou L mmatoc tou
EukleÐdh wc ex c. Upojètoume ìti (v1, v2) eÐnai mia lÔsh thc
α1x1 + α2x2 = δ2(1)
dhlad
α1v1 + α2v2 = δ2(2)
kai èstw (x1, x2) mia opoiad pte lÔsh thc (1).
Pollaplasiˆzontac thn (1) kai thn (2) epÐ v1 kai epÐ x1 antÐstoiqa, paÐrnoume
α1x1v1 + α2x2v1 = v1δ2
α1x1v1 + α2x1v2 = x1δ2,
opìte
α2(x2v1 − x1v2) = δ2(v1 − x1).

EpÐshc pollaplasiˆzontac thn (1) kai thn (2) epÐ tou v2 kai epÐ x2 antÐstoiqa,
èqoume
α1v2x1 + α2v2x2 = v2δ2
α1v1x2 + α2v2x2 = x2δ2
opìte
α1(v2x1 − v1x2) = δ2(v2 − x2).
Jètontac x2v1 − x1v2 = k, paÐrnoume
kai
δ2(v1 − x1) = α2k, dhlad x1 = v1 +
α2
δ2
k
δ2(v2 − x2) = −α1k, dhlad x2 = v2 −
α1
δ2
k.
An β2 = β2δ2, tìte h (u1, u2), u1 = β2v1, u2 = β2v2, eÐnai mia lÔsh thc arqik c
exÐswshc kai h opoiad pote ˆllh lÔsh ja eÐnai (β2x1, β2x2), ìpou
β2x1 = u1 +
α2
δ2
β2k = u1 +
α2
δ2
k
k ∈ Z.
β2x2 = u2 −
α1
δ2
β2k = u2 −
α1
δ2
k
Parat rhsh. Grˆfontac thn exÐswsh α1x1 + α2x2 = β2 sth morf
x2 = −
α1
α2
x1 +
β2
α2
,
blèpoume ìti h klÐsh thc eujeÐac pou paristˆ aut h exÐswsh eÐnai jetik ìtan ta
α1 kai α2 eÐnai eterìshma kai sunep¸c èna tm ma thc eujeÐac brÐsketai sto pr¸to
tetarthmìrio kai perilambˆnei ˆpeiro pl joc shmeÐwn autoÔ tou tetarthmorÐou.
An (x1, x2) eÐnai èna shmeÐo autoÔ tou tetarthmorÐou, dhlad x1 0, x2 0,
pou brÐsketai epÐ thc eujeÐac kai eÐnai x1, x2 ∈ Z, dhlad eÐnai mia jetik akèraia
lÔsh thc exÐswshc, tìte upˆrqoun ˆpeiro pl joc jetikèc akèraiec lÔseic thc
exÐswshc. An h klÐsh eÐnai arnhtik , dhlad ta α1 kai α2 eÐnai kai ta dÔo jetikˆ

kai ta dÔo arnhtikˆ, kai èna tm ma thc eujeÐac brÐsketai sto pr¸to tetarthmìria
tìte h exÐswsh èqei to polÔ èna peperasmèno pl joc jetik¸n akèraiwn lÔsewn.
Argìtera ja broÔme autì to pl joc.
Parˆdeigma. 1. Na lujeÐ h Diofantik exÐswsh
392x1 − 21x2 = 14.
BrÐskoume ton m.k.d. (392, 21), me ton EukleÐdeio algìrijmo. 'Eqoume
392 = 21 · 18 + 14
21 = 14 · 1 + 7
14 = 7 · 2 + 0.
'Ara
(392, 21) = 7 = 21 − 14 · 1 = 21 − (392 − 21 · 18) · 1
= −392 + 21 · 19 = 392(−1) + 21(19).
Sunep¸c 392(−2)−21(−38) = 14, dhlad mia lÔsh thc dosmènhc exÐswshc eÐnai
h (−2, −38). Opìte ìlec oi lÔseic eÐnai oi (−2 − 3t, −38 − 56t), t ∈ Z. EpÐshc
parathroÔme ìti gia t = −1 paÐrnoume th mikrìterh jetik akèraia lÔsh aut c
thc exÐswshc, dhlad thn (1, 18). Opìte ìlec oi lÔseic thc exÐswshc mporoÔn
na dÐdontai kai wc oi (1 − 3k, 18 − 56k), k ∈ Z. EpÐshc shmei¸noume ìti kaj¸c
o k diatrèqei ìlouc touc akèraiouc, o −k pˆli diatrèqei ìlouc touc akèraiouc,
opìte ìlec oi lÔseic mporoÔn na ekfrasjoÔn kai wc (1 + 3k, 18 + 56k), k ∈ Z.
Ap autì blèpoume ˆmesa ìti upˆrqoun ˆpeirec jetikèc akèraiec lÔseic, dhlad
blèpoume autì pou dhl¸netai kai apì th jetik klÐsh thc eujeÐac.
To prohgoÔmeno parˆdeigma ja mporoÔse na tejeÐ kai wc ex c. Na brejeÐ
o mikrìteroc jetikìc akèraioc arijmìc o opoioc ìtan diairejeÐ dia tou 392 kai
dia tou 21 af nei upìloipo 3 kai 17 antÐstoiqa. Dhlad ja prèpei na isoÔte me
392x1 + 3 kai me 21x2 + 17 kai ˆra
392x1 − 21x2 = 14.

'Opwc eÐdame o mikrìteroc tètoioc akèraioc eÐnai o
392 · (1) + 3 = 21(18) + 17 = 395.
Parˆdeigma. 2. Na brejeÐ to pl joc twn jetik¸n akèraiwn lÔsewn thc
Diofantik c exÐswshc
392x1 + 21x2 = 14.
'Opwc eÐdame eÐnai (392, 21) = 7 = 392(−1) + 21(19). Sunep¸c 392(−2) +
21(38) = 14 kai h genik lÔsh thc exÐswshc t¸ra eÐnai h (−2 + 3t, 38 − 56t),
t ∈ Z. Oi jetikèc lÔseic eÐnai autèc gia tic opoÐec t
3
2
kai t
38
56
, dhlad
3
2
t
38
56
pou den mporeÐ na isqÔei, dhlad den upˆrqei tètoio t kai ˆra den
upˆrqei kamiˆ jetik akèraia lÔsh, ìpwc ˆllwste ˆmesa faÐnetai gewmetrikˆ
sto sq ma:
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
CC
x2
•
2
3
•
2
56
x1
1.1.24 Je¸rhma. H exÐswsh (∗) èqei lÔsh an kai mìnon an δn|βn. Epiplèon
an δi eÐnai o m.k.d. (α1, α2, . . . , αi) kai x
(i)
1 , x
(i)
2 , . . . , x
(i)
i ) eÐnai mia lÔsh thc
δi = α1x1 + α2x2 + · · · + αixi, i = 2, . . . , n tìte ìlec oi lÔseic thc (∗) eÐnai thc

morf c















x1
x2
x3
x4
...
xn−1
xn















=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
α2
δ2
α3
δ3
x
(2)
1
α4
δ4
x
(3)
1 ···
αn−1
δn−1
x
(n−2)
1
αn
δn
x
(n−1)
1 x
(n)
1
−α1
δ2
α3
δ3
x
(2)
2
α4
δ4
x
(3)
2 ···
αn−1
δn−1
x
(n−2)
2
αn
δn
x
(n−1)
2 x
(n)
2
0 −δ2
δ3
α4
δ4
x
(3)
3 ···
αn−1
δn−1
x
(n−2)
3
αn
δn
x
(n−1)
3 x
(n)
3
0 0 −δ3
δ4
···
αn−1
δn−1
x
(n−2)
4
αn
δn
x
(n−1)
4 x
(n)
4
...
...
... ···
...
...
...
0 0 0 ··· −δn−2
δn−1
αn
δn
x
(n−1)
n−1 x
(n)
n−1
0 0 0 ··· 0 −δn−1
δn−1
x
(n)
n
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
t1
t2
t3
t4
...
tn−1
tn
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
ìpou βn = tnδn kai ti ∈ Z, i = 1, 2, . . . , n − 1.
Apìdeixh. To pr¸to mèroc tou jewr matoc eÐnai to 1.1.22. Gia to deÔtero mèroc
tou jewr matoc efarmìzoume epagwg sto n. H perÐptwsh n = 2 dÐdetai sto
Je¸rhma 1.1.23. Upojètoume ìti to je¸rhma isqÔei gia n = k ≥ 2 kai èstw
x
(k+1)
1 , x
(k+1)
2 , . . . , x
(k+1)
k+1 eÐnai mia lÔsh thc δk+1 = α1x1 + · · · + αk+1xk+1 kai
ìti x1, . . . , xk+1 eÐnai mia lÔsh thc tk+1δk+1 = α1x1 + · · · αk+1xk+1, ìpou tk+1,
α1, α2, . . . , αk+1 eÐnai mh mhdenikoÐ akèraioi. Opìte èqoume
α1(x1 − tk+1x
(k+1)
1 ) + · · · + αk+1(xk+1 − tk+1x
(k+1)
k+1 ) = 0

α1
δk+1
(x1−tk+1x
(k+1)
1 )+· · ·+
αk
δk+1
(xk−tk+1x
(k+1)
k ) = −
αk+1
δk+1
(xk+1−tk+1x
(k+1)
k+1 ).
T¸ra to ˆjroisma α1(x1 − tk+1x
(k+1)
1 ) + · · · αk(xk − tk+1x
(k+1)
k ) eÐnai èna pol-
laplˆsio tou m.k.d. δk twn α1, α2, . . . , αk, èstw λδk. Opìte to aristerì mèloc
thc prohgoÔmenhc isìthtac eÐnai λδk/δk+1.
Allˆ oi akèraioi
α1
δk+1
,
α2
δk+1
, . . . ,
αk
δk+1
,
αk+1
δk+1
eÐnai sqetikˆ pr¸toi metaxÔ
touc kai sunep¸c
δk
δk+1
,
αk+1
δk+1
= 1. 'Ara
αk+1
δk+1
tk = λ. Opìte h teleutaÐa
isìthta isqÔei an kai mìnon an λδk/δκ+1 =
αk+1
δk+1
tk+1
δk
δk+1
= −
αk+1
δk+1
(xk −
tk+1x
(k+1)
k+1 ) xk+1 − tk+1x
(k+1)
k+1 = −tk
δk
δk+1
kai
α1
δk+1
(x1 − tk+1x
(k+1)
1 ) + · · · +
αk
δk+1
(xk − tk+1x
(k+1)
k+1 ) = tk
αk+1
δk+1
δk.

Apì thn upìjesh thc epagwg c, gia kˆje eklog tou tk, èqoume
x1 − tk+1x
(k+1)
1 = tk
αk+1
δk+1
x
(k)
1 + tk−1
αk δk+1
δk δk+1
x
(k−1)
1 + · · · + t1
α2 δk+1
δ2 δk+1
x2 − tk+1x
(k+1)
2 = tk
αk+1
δk+1
x
(k)
2 + tk−1
αk δk+1
δk δk+1
x
(k−1)
2 + · · · − t1
α1 δk+1
δ2 δk+1
............................................................................................................
xk − tk+1x
(k+1)
k = tk
αk+1
δk+1
x
(k)
k − tk−1
δk−1 δk+1
δk δk+1
,
ìpou t1, t2, . . . , tk−1 eÐnai kˆpoioi akèraioi kai
α1
δk+1
x
(i)
1 + · · · +
αk
δk+1
x
(i)
k =
δi
δk+1
gia i = 2, . . . , k.
Metafèrontac touc ìrouc pou perilambˆnoun ta tk+1 sto dexiì mèloc twn
prohgoÔmenwn isot twn kai aplopoi¸ntac touc upìloipouc ìrouc paÐrnoume to
epijumhtì apotèlesma.
Sunep¸c an broÔme mia lÔsh thc (∗) tìte gnwrÐzoume ìlec tic lÔseic. Gia
thn eÔresh miac lÔshc thc (∗) ergazìmaste wc ex c: JewroÔme thn exÐswsh
α1x1 + α2x2 + · · · + αnxn = δn(**)
ìpou ìloi oi suntelestèc αi eÐnai = 0 (diaforetikˆ ja eÐqame mia exÐswsh m(
n) metablht¸n). MporoÔme na upojèsoume ìti ìloi oi suntelestèc αi eÐnai 0,
diìti diaforetikˆ mporoÔme na antikatast soume kˆje arnhtikì αi me ton −αi
(pou den allˆzei thn tim tou m.k.d. δn) kai to antÐstoiqo xi me to −xi.
GnwrÐzoume ìti δn = (α1, α2, . . . , αn) = ((α1, . . . , αn−1), , αn) = (δn−1, δn).
Sunep¸c an gnwrÐzoume mia lÔsh (x
(n−1)
1 , . . . , x
(n−1)
n−1 ) thc
α1x1 + · · · + αn−1xn−1 = δn−1
tìte brÐskoume (me ton algìrijmo tou EukleÐdh me ˆllo trìpo) mia lÔsh thc
δn−1x1 + αnxn = δn, èstw thn (y
(n)
1 , x
(n)
n ), opìte h (x
(n)
1 , x
(n)
2 , . . . , x
(n)
n ), ìpou
x
(n)
1 = x
(n−1)
1 y
(n)
1 , . . . , x
(n)
n−1 = x
(n−1)
n−1 y
(n)
1 , eÐnai lÔsh thc (∗∗). 'Ara epagwgikˆ,
xekin¸ntac apì thn eÔresh miac lÔshc thc α1x1 +α2x2 = δ2, brÐskoume mia lÔsh
thc α1x1 + α2x2 + α3x3 = δ3, katìpin thc α1x1 + α2x2 + α3x3 + α4x4 = δ4

k.o.k. èwc ìtou na broÔme mia lÔsh thc (∗∗). An βn = tnδn, tìte mia lÔsh thc
(∗) eÐnai h (tnx
(n)
1 , . . . , tnx
(n)
n ).
Parˆdeigma. Na lujeÐ h Diofantik exÐswsh
5x1 + 35x2 + 8x3 + 7x4 = 2.
'Eqoume δ4 = (5, 35, 8, 7) = 1. 'Ara aut èqei lÔsh.
JewroÔme thn
5x1 + 35x2 + 8x3 + 7x4 = 1.
BrÐskoume mia lÔsh thc 5x1 + 35x2 = 5 thc x1 + 7x2 = 1 pou profan¸c mia
lÔsh thc eÐnai h (8, −1).
Katìpin brÐskoume mia lÔsh thc 5q1 + 8q2 = 1 pou profan¸c mia lÔsh thc
eÐnai h (−3, 2), opìte h (−24, 3, 2) eÐnai mia lÔsh thc 5x1 + 35x2 + 8x3 = 1.
Tèloc jewroÔme thn exÐswsh x1 + 7x4 = 1. Mia lÔsh aut c eÐnai h (−6, 1)
kai ˆra h (2 · 144, −2 · 18, −2 · 12, 2 · 1) eÐnai lÔsh thc arqik c exÐswshc.
Sunep¸c, sÔmfwna me to 1.1.24, h genik thc lÔshc (x1, x2, x3, x4) eÐnai h







x1
x2
x3
x4







=







7 64 −168 144
−1 −8 21 −18
0 −5 14 −12
0 0 −1 1














t1
t2
t3
2







Mèjodoc Weinstock. 'Eqoume dei ìti o m.k.d. δn mh mhdenik¸n akeraÐwn
α1, α2, . . . , αn eÐnai o mikrìteroc jetikìc akèraioc thc morf c
α1x1 + α2x2 + · · · + αnxn, x1, x2, . . . , xn ∈ Z.
Qrhsimopoi¸ntac autì to gegonìc o R. Weinstock (1960) prìteine ton ex c
algìrijmo gia thn eÔresh tou δ allˆ kai thn eÔresh akeraÐwn x1, . . . , xn ètsi
¸ste δ = α1x1 + · · · + αnxn.
'Estw β
(0)
1 , β
(0)
2 , . . . , β
(0)
n tuqaÐoi akèraioi tètoioi ¸ste
α1β
(0)
1 + α2β
(0)
2 + · · · + αnβ(0)
n = γ 0.

An γ|α1, γ|α2, . . . , γ|αn, tìte γ = δ, (afoÔ δ ≤ γ kai γ|δ, dhlad γ ≤ δ), opìte
èqoume brei ton δ. An ìqi, dhlad an γ αi gia kˆpoio i, 1 ≤ i ≤ n, tìte
αi = qiγ + v1, 0 v1 γ. 'Etsi èqoume
ìpou
v1 = αi − q1γ = α1β
(1)
1 + α2β
(1)
2 + · · · + αnβ(1)
n
β
(1)
j = −q1β
(0)
j , j = i, kai β
(1)
i = 1 − q1β
(0)
i .
Epanalambˆnontac aut th diadikasÐa paÐrnoume mia austhrˆ fjÐnousa akolou-
jÐa {vi} akèraiwn arijm¸n thc morf c
α1x1 + α2x2 + · · · + αnxn ≥ δ
kai ˆra metˆ apì èna peperasmèno pl joc bhmˆtwn aut c thc diadikasÐac ja fjˆ-
soume se ènan deÐkth k gia ton opoÐon gia pr¸th forˆ ja èqoume vk|α1, vk|α2, . . . , vk|αn,
opìte δ = vk = α1β
(k)
1 + α2β
(k)
2 + · · · + αnβ
(k)
n . Dhlad h (β
(k)
1 , β
(k)
2 , . . . , β
(k)
n )
eÐnai mia lÔsh thc δ = α1x1 + · · · + αnxn.
Parˆdeigma. Ac jewr soume pˆli ton m.k.d. δ = (5, 35, 8, 7). Dialègoume
tuqaÐouc β
(0)
1 , β
(0)
2 , β
(0)
3 , β
(0)
4 ètsi ¸ste
γ = 5β
(0)
1 + 35β
(0)
2 + 8β
(0)
3 + 7β
(0)
4 0.
FerepeÐn, β
(0)
1 = 0, β
(0)
2 = 0, β
(0)
3 = 1, β
(0)
4 = 0. EÐnai γ = 7 kai epeid γ 5,
prèpei γ δ. DiairoÔme to 5 dia 7 : 5 = 7 · 0 + 5, opìte
0 5 = 5 − 7 · 0 = 5 · 1 + 35 · 0 + 8 · 0 + 7 · 0 7.
Epeid 5 7, prèpei 5 δ kai èqoume 7 = 5 · 1 + 2. 'Ara 2 = 7 − 5 · 1 =
5(−1) + 35 · 0 + 8 · 0 + 7(1 − 0). Pˆli epeid 2 5, prèpei 2 δ kai èqoume
5 = 2 · 2 + 1. Opìte 1 = 5(1 − 2 · (−1)) + 35 · 0 + 8 · 0 + 7(−2).
T¸ra profan¸c δ = 1 kai h (β
(3)
1 = 3, β
(3)
2 = 0, β
(3)
3 = 0, β
(3)
4 = −2) eÐnai
mia lÔsh thc 1 = 5x1 + 35x2 + 8x3 + 7x4.
Mèjodoc Euler. 'Estw h exÐswsh
51x1 + 31x2 + 14x3 = 1.

O Leonard Euler sto biblÐo tou “Vollst¨andige Anleitung zur Algebra. St.
Petersburg 1770” gia na brei tic akèraiec lÔseic grammik¸n Diofantik¸n exis¸-
sewn akoloujeÐ thn ex c diadikasÐa:
Dialègoume ton mikrìtero kat apìluth tim suntelest thc exÐswshc dhla-
d to 14 (An upˆrqoun ˆlloi suntelestèc sthn exÐswsh pou eÐnai pollaplˆsia
autoÔ tìte ant autoÔ jewroÔme to megalÔtero aut¸n). Grˆfoume
14x3 = −51x1 − 31x2 + 1 x3 = −3x1 − 2x2 +
−9x1 − 3x2 + 1
14
.
Epeid endiaferìmeja gia tic akèraiec lÔseic, an (x1, x2, x3) eÐnai mia tètoia,
tìte ja prèpei o arijmìc
−9x1 − 3x2 + 1
14
na eÐnai akèraioc. Jètoume
t =
−9x1 − 3x2 + 1
14
.
Opìte èqoume 9x1 + 3x2 + 14t = 1. Epeid o 3 eÐnai o mikrìteroc suntelest c
metaxÔ twn x1 kai x2 allˆ o suntelest c 9 eÐnai pollaplˆsiìc tou, jewroÔme
ton suntelest 9 kai epanalambˆnoume thn Ðdia diadikasÐa ìpwc prin. 'Eqoume
x1 = −t +
−5t − 3x2 + 1
9
ìpou t¸ra prèpei o u =
−5t − 3x2 + 1
9
na eÐnai akèraioc. 'Etsi èqoume
3x2 + 5t + 9u = 1
opìte
x2 = −t − 3u +
−2t + 1
3
.
Jètoume v =
−2t + 1
3
(ton opoÐo jewroÔme akèraio) kai èqoume 3v = −2t + 1
t = −v +
−v + 1
2
me ω =
−v + 1
2
∈ Z, 2ω = −v + 1 v = 1 − 2ω.
Antikajist¸ntac to v sto t kai autì sto x2, èqoume t = −1 + 3ω kai
x2 = −(−1 + 3ω) − 3u + 1 − 2ω, opìte x2 = 2 − 5ω − 3u. EpÐshc eÐnai
x1 = 1 − 3ω + u

kai
x3 = −3(1 − 3ω + u) − 2(2 − 5ω − 3u) − 1 + 3ω = −8 + 22ω + 3u.
Opìte h genik lÔsh dÐnetai wc h
x1 = u − 3ω + 1
x2 = −3u − 5ω + 2
x3 = 3u + 22ω − 8.
An jewr soume th lÔsh (1, 2, −8), tìte apì to Je¸rhma 1.1.24 h genik lÔsh
dÐdetai wc h




x1
x2
x3



 =




31 196 1
−51 −322 2
0 −1 −8








t1
t2
1




afoÔ 51(14)+31(−23) = 1, dhlad x
(2)
1 = 14 kai x
(2)
2 = −23, x
(3)
1 = 1, x
(3)
2 = 2
kai x
(3)
3 = −8. 'Ara ja prèpei na isqÔei
kai
31t1 + 196t2 = u − 3ω
−51t1 − 322t2 = −3u − 5ω
kai −t2 = 3u + 22ω
gia t1, t2, u, ω ∈ Z. Prˆgmati, blèpoume ìti gia t2 = −3u + 22ω ∈ Z èqoume
−51t1 + 322(3u + 22ω) = −3u − 5ω
−51t1 = −969u − 7089ω
t1 = 19u + 139ω ∈ Z.
EpÐshc to Ðdio prokÔptei apì thn pr¸th isìthta, afoÔ
31t1 = 589u + 4309ω
t1 = 19u + 139ω.

Parat rhsh. H orÐzousa |A| tou n × n pÐnaka A sto Je¸rhma 1.1.24 isoÔ-
tai me 1. Prˆgmati an pollaplasiˆsoume thn i-gramm epÐ αi gia kˆje i =
1, 2, . . . , n, tìte h orÐzousa tou pÐnaka pou ja prokÔyei isoÔte me α1α2 · · · αn|A|.
Allˆ h orÐzousa tou pÐnaka autoÔ eÐnai Ðsh me thn orÐzousa tou pÐnaka pou pro-
kÔptei an sthn i-gramm prosjèsoume ìlec tic j-grammèc gia j i. O pÐnakac
autìc eÐnai ˆnw trigwnikìc me diag¸nio Ðsh me
α1α2
δ2
,
δ2α3
δ3
,
δ3α4
δ4
, . . . ,
δn−1αn
δn
, δn .
Allˆ h orÐzousa autoÔ tou pÐnaka eÐnai Ðsh me
α1α2
δ2
·
δ2α3
δ3
· · ·
δn−1αn
δn
, δn = α1α2α3 · · · αn.
Sunep¸c èqoume α1α2 · · · αn|A| = α1α2 · · · αn. Opìte |A| = 1. ShmeÐwse ìti







t1
t2
...
1







= A−1







x1
x2
...
xn







.
Gia parˆdeigma, prohgoumènwc eÐqame A =




31 196 1
−51 −322 2
0 −1 −8



 opìte




t1
t2
1



 =




2578 1567 714
−408 −248 −113
−51 −31 14








x1
x2
x3



 .
H Katanom twn Pr¸twn
S mera, me th bo jeia isqur¸n hlektronik¸n upologist¸n, èqoun katagrafeÐ
makroskeleÐc katˆlogoi pr¸twn arijm¸n (blèpe google: The list of prime num-
bers).

Melet¸ntac ènan tètoio katˆlogo, to pr¸to qarakthristikì pou diakrÐnoume
sthn akoloujÐa
2 = p1 3 = p2 p3 · · ·
twn pr¸twn arijm¸n eÐnai h apousÐa enìc kanìna o opoÐoc kajorÐzei ton epìme-
no pr¸to arijmì apì ènac dosmèno pr¸to arijmì. 'Oqi mìno den mporoÔme na
diakrÐnoume ènan tètoio kanìna allˆ oÔte kan na mantèyoume kˆpoia eikasÐa gia
ton trìpo pou oi pr¸toi arijmoÐ ekteÐnontai katˆ m koc twn fusik¸n arijm¸n.
Exetˆzontac arketˆ megˆlouc arijmoÔc blèpoume ìti oi pr¸toi arijmoÐ spanÐ-
zoun allˆ tautìqrona mporeÐ na emfanÐzontai kontˆ o ènac me ton ˆllo. Gia
parˆdeigma, stouc 100 arijmoÔc amèswc prin to 10.000.000, dhlad metaxÔ tou
9.999.900 kai 10.000.000, upˆrqoun ennèa pr¸toi pou eÐnai oi
9.999.901, 9.999.907, 9.999.929, 9.999.931, 9.999.937
9.999.943, 9.999.971, 9.999.973, kai 9.999.991
en¸ upˆrqoun mìno dÔo pr¸toi stouc epìmenouc ekatì arijmoÔc apì ton 10.000.000
èwc ton 10.000.100 pou eÐnai oi 10.000.019 kai 10.000.079. EpÐshc den upˆrqei
kanènac pr¸toc metaxÔ twn arijm¸n 20.831.323 kai 20.831.533 allˆ oi arijmoÐ
1.000.000.000.061 kai 1.000.000.000.063 eÐnai pr¸toi. Autˆ ta arijmhtikˆ apo-
telèsmata dikaiologoÔntai kai apì to gegonìc ìti gia kˆje fusikì arijmì n oi
diadoqikoÐ arijmoÐ
(n + 1)! + 2, (n + 1)! + 3, . . . , (n + 1)! + (n + 1)
eÐnai ìloi sÔnjetoi. Autì shmaÐnei ìti h akoloujÐa twn pr¸twn arijm¸n perièqei
diadoqikoÔc pr¸touc pou h diaforˆ touc eÐnai polÔ megˆlh. EpÐshc isqÔei h
“Arq tou Bertrand” pou anafèrei ìti gia kˆje fusikì arijmì n ≥ 2 upˆrqei
toulˆqiston ènac pr¸toc p me n ≤ p ≤ 2n (opìte pn+1 ≤ 2pn). (H “arq ”
aut eÐnai je¸rhma kaj¸c apedeÐqjh apì ton Tshebycheﬀ to 1952. O Bertrand
upojètei aut thn arq gia na deÐxei ìti to pl joc twn pr¸twn eÐnai ˆpeiro wc
ex c: De mporeÐ na upˆrqei arijmìc p pou eÐnai o megalÔteroc pr¸toc, afoÔ
metaxÔ tou p + 1 kai tou 2(p + 1) ja upˆqei ˆlloc pr¸toc megalÔteroc tou p).

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