Proceedings - NCUR VIII. (1994), Vol. II, pp. 794-798. Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah In this paper we present some preliminary results on a conjecture by Paul Erdos concerning covering sets of congruences. A covering set consists of a finite system of congruences with distinct moduli, such that every integer satisfies as a minimum one of the congruences. An interesting consequence of this conjecture is the dependence of the solution on abundant numbers; an abundant number is an integer whose sum of its proper divisors exceeds the integer.
Proceedings - NCUR VI. (1992), Vol. II, pp. 1036-1041.
Jeffrey F. Gold
Department of Mathematics, Department of Physics
University of Utah
Don H. Tucker
Department of Mathematics
University of Utah
Introduction
Remodulization introduces a new method applied to congruences and systems of congruences. We prove the Chinese Remainder Theorem using the remodulization method and establish an efficient method to solve linear congruences. The following is an excerpt of Remodulization of Congruences and its Applications [2].
Proceedings - NCUR VI. (1992), Vol. II, pp. 1036-1041.
Jeffrey F. Gold
Department of Mathematics, Department of Physics
University of Utah
Don H. Tucker
Department of Mathematics
University of Utah
Introduction
Remodulization introduces a new method applied to congruences and systems of congruences. We prove the Chinese Remainder Theorem using the remodulization method and establish an efficient method to solve linear congruences. The following is an excerpt of Remodulization of Congruences and its Applications [2].
Journal Molecular Crystals and Liquid Crystals (1994), Vol. 256, pp. 555-561.
S. A. Jeglinski, M. E. Hollier, J. F. Gold, and Z. V. Vardeny
University of Utah, Physics Department, Salt Lake City, UT 84112
Y. Ding and T. Barton
Iowa State University, Chemistry Department, Ames, IA 50011
Abstract
A diode has been fabricated with poly(phenylene acetylene) [PPA] as the electroluminescent polymer. The device exhibited an unusual symmetric (positive and negative bias) I-V characteristic and electroluminescent output. These experimental results are discussed in terms of tunneling of electrons and holes via localized states.
Is Gravitation A Result Of Asymmetric Coulomb Charge Interactions?Jeffrey Gold
Journal of Undergraduate Research (JUR), University of Utah (1992), Vol. 3, No. 1, pp. 56-61.
Jeffrey F. Gold
Department of Physics, Department of Mathematics
University of Utah
Abstract
Many attempts have been made to equate gravitational forces with manifestations of other phenomena. In these remarks we explore the consequences of formulating gravitational forces as asymmetric Coulomb charge interactions. This is contrary to some established theories, for the model predicts differential accelerations dependent on the elemental composition of the test mass. The
predicted di erentials of acceleration of various elemental masses are compared to those differentials that have been obtained experimentally. Although the model turns out to fail, the construction of this model is a useful intellectual and pedagogical exercise.
Complementary Sets of Systems of CongruencesJeffrey Gold
Proceedings - NCUR VII. (1993), Vol. II, pp. 793-796. Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah Introduction We introduce remodulization and use it to characterize the complementary sets of systems of congruences. The following is an excerpt of a continuing effort to characterize systems of congruences.
Vector Products Revisited: A New and Efficient Method of Proving Vector Ident...Jeffrey Gold
Proceedings - NCUR X. (1996), Vol. II, pp. 994-998 Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah Introduction The purpose of these remarks is to introduce a variation on a theme of the scalar (inner, dot) product and establish multiplication in R^{n}. If a = (a_{1},...,a_{n}) and b = (b_{1},...,b_{n}), we define the product ab = (a_{1}b_{1},...,a_{n}b_{n }). The product is the multiplication of corresponding vector components as in the scalar product; however, instead of summing the vector components, the product preserves them in vector form. We define the inner sum (or trace) of a vector a = (a_{1},...,a_{n}) by (a) = a_{1}+...+a_{n}. If taken together with an additional definition of cyclic permutations of vectors, we are able to prove complicated vector products (combinations of dot and cross products) extremely efficiently, without appealing to the traditional (and cumbersome) epsilon-ijk proofs. When applied to determinants, this method hints at the rudiments of Galois theory.
A Novel Solution Of Linear CongruencesJeffrey Gold
Proceedings - NCUR IX. (1995), Vol. II, pp. 708-712
Jeffrey F. Gold
Department of Mathematics, Department of Physics
University of Utah
Salt Lake City, Utah 84112
Don H. Tucker
Department of Mathematics
University of Utah
Salt Lake City, Utah 84112
Introduction
Although the solutions of linear congruences have been of interest for a very long time, they still remain somewhat pedagogically di cult. Because of the importance of linear congruences in fields such as public-key cryptosystems, new and innovative approaches are needed both to attract interest and to make them more accessible. While the potential for new ideas used in future research
is difficult to assess, some use may be found here. In this paper, the authors make use of the remodulization method developed in [1] as a vehicle to characterize the conditions under which solutions exist and then determine the solution space. The method is more efficient than those cited in the standard references. This novel approach relates the solution space of cx = a mod b to the Euler totient function for c rather than that of b, which allows one to develop an alternative and somewhat more efficient
approach to the problem of creating enciphering and deciphering keys in public-key cryptosystems.
A Characterization of Twin Prime PairsJeffrey Gold
Proceedings - NCUR V. (1991), Vol. I, pp. 362-366. Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah The basic idea of these remarks is to give a tight characterization of twin primes greater than three. It is hoped that this might lead to a decision on the conjecture that infinitely many twin prime pairs exist; that is, number pairs (p; p+ 2) in which both p and p + 2 are prime integers.
This book of Biology has the definite purpose of refuting the theory of evolution and exalt God as the one responsible for creating. We dedicate this volume to the Creator that is very close to the men, as the Scripture says in Acts 17.24-28, through this book we tax the God the title of THE LORD OF LIFE. Men announced this literature that their existence here originated from a CREATION ACT and not a PROCESS DEVELOPMENT. The true believers in God, this book will reinforce their beliefs regarding the origin of LIFE and MAN. Scientists to request reconsideration of theories that hinder the development of science, noting that no belief should divert research at its sole purpose: TRUTH.
A carta de Paulo a igreja de Corinto visava coibir várias práticas pecaminosas, eram problemas de ordem doutrinaria, como imoralidade sexual, facções, abusos na ceia, recusa das mulheres de usar véu, desordem no culto quanto a manifestação dos dons espirituais e até dúvidas sobre a ressurreição. Como fundador daquela igreja e como apóstolo respeitado entre todos os cristãos, Paulo manda uma extensa carta para orientar a igreja e até mesmo exigir que fosse expulso alguns membros que estão corrompendo moralmente aquela comunidade.
This talk concerning The Importance of Arts in Education was delivered at Westminster College on March 23, 2009. My personal favorite part of the talk was revealing the percentage of GNP the arts represent---a real sabot for that certain kind of philistine social (and economic) conservative who wants to crush arts funding in schools. For that individual, here's an economic argument that almost sounds like an artifact of some capitalist Utopia. Considering we don't produce much in the U.S. anymore---the legacy of Bretton-Woods---at least we still export the arts.
Short Lifetimes of Light Emitting PolymersJeffrey Gold
Jeffrey Frederick Gold
University of Cambridge
Microelectronics Research Centre
Cavendish Laboratory
Cambridge CB3 0HE
Introduction
The following manuscript was submitted as part of the MPhil program in Microelectronic Engineering and Semiconductor Physics at the Microelectronics Research Centre (MRC) of the Cavendish Laboratory at the University of Cambridge. The manuscript is a literature survey undertaken by the author as part of the MESP program during Lent term 1997 and is the basis of a talk of the same
title given at the MRC on January 20, 1997.
Presentation about Story Structure given April 20, 2009 at Westminster College. This presentation discusses the intrinsic three act structure and uses The Shawshank Redemption to show how two overlapping three-act structures gives rise to a four-act structure.
Journal Molecular Crystals and Liquid Crystals (1994), Vol. 256, pp. 555-561.
S. A. Jeglinski, M. E. Hollier, J. F. Gold, and Z. V. Vardeny
University of Utah, Physics Department, Salt Lake City, UT 84112
Y. Ding and T. Barton
Iowa State University, Chemistry Department, Ames, IA 50011
Abstract
A diode has been fabricated with poly(phenylene acetylene) [PPA] as the electroluminescent polymer. The device exhibited an unusual symmetric (positive and negative bias) I-V characteristic and electroluminescent output. These experimental results are discussed in terms of tunneling of electrons and holes via localized states.
Is Gravitation A Result Of Asymmetric Coulomb Charge Interactions?Jeffrey Gold
Journal of Undergraduate Research (JUR), University of Utah (1992), Vol. 3, No. 1, pp. 56-61.
Jeffrey F. Gold
Department of Physics, Department of Mathematics
University of Utah
Abstract
Many attempts have been made to equate gravitational forces with manifestations of other phenomena. In these remarks we explore the consequences of formulating gravitational forces as asymmetric Coulomb charge interactions. This is contrary to some established theories, for the model predicts differential accelerations dependent on the elemental composition of the test mass. The
predicted di erentials of acceleration of various elemental masses are compared to those differentials that have been obtained experimentally. Although the model turns out to fail, the construction of this model is a useful intellectual and pedagogical exercise.
Complementary Sets of Systems of CongruencesJeffrey Gold
Proceedings - NCUR VII. (1993), Vol. II, pp. 793-796. Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah Introduction We introduce remodulization and use it to characterize the complementary sets of systems of congruences. The following is an excerpt of a continuing effort to characterize systems of congruences.
Vector Products Revisited: A New and Efficient Method of Proving Vector Ident...Jeffrey Gold
Proceedings - NCUR X. (1996), Vol. II, pp. 994-998 Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah Introduction The purpose of these remarks is to introduce a variation on a theme of the scalar (inner, dot) product and establish multiplication in R^{n}. If a = (a_{1},...,a_{n}) and b = (b_{1},...,b_{n}), we define the product ab = (a_{1}b_{1},...,a_{n}b_{n }). The product is the multiplication of corresponding vector components as in the scalar product; however, instead of summing the vector components, the product preserves them in vector form. We define the inner sum (or trace) of a vector a = (a_{1},...,a_{n}) by (a) = a_{1}+...+a_{n}. If taken together with an additional definition of cyclic permutations of vectors, we are able to prove complicated vector products (combinations of dot and cross products) extremely efficiently, without appealing to the traditional (and cumbersome) epsilon-ijk proofs. When applied to determinants, this method hints at the rudiments of Galois theory.
A Novel Solution Of Linear CongruencesJeffrey Gold
Proceedings - NCUR IX. (1995), Vol. II, pp. 708-712
Jeffrey F. Gold
Department of Mathematics, Department of Physics
University of Utah
Salt Lake City, Utah 84112
Don H. Tucker
Department of Mathematics
University of Utah
Salt Lake City, Utah 84112
Introduction
Although the solutions of linear congruences have been of interest for a very long time, they still remain somewhat pedagogically di cult. Because of the importance of linear congruences in fields such as public-key cryptosystems, new and innovative approaches are needed both to attract interest and to make them more accessible. While the potential for new ideas used in future research
is difficult to assess, some use may be found here. In this paper, the authors make use of the remodulization method developed in [1] as a vehicle to characterize the conditions under which solutions exist and then determine the solution space. The method is more efficient than those cited in the standard references. This novel approach relates the solution space of cx = a mod b to the Euler totient function for c rather than that of b, which allows one to develop an alternative and somewhat more efficient
approach to the problem of creating enciphering and deciphering keys in public-key cryptosystems.
A Characterization of Twin Prime PairsJeffrey Gold
Proceedings - NCUR V. (1991), Vol. I, pp. 362-366. Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah The basic idea of these remarks is to give a tight characterization of twin primes greater than three. It is hoped that this might lead to a decision on the conjecture that infinitely many twin prime pairs exist; that is, number pairs (p; p+ 2) in which both p and p + 2 are prime integers.
This book of Biology has the definite purpose of refuting the theory of evolution and exalt God as the one responsible for creating. We dedicate this volume to the Creator that is very close to the men, as the Scripture says in Acts 17.24-28, through this book we tax the God the title of THE LORD OF LIFE. Men announced this literature that their existence here originated from a CREATION ACT and not a PROCESS DEVELOPMENT. The true believers in God, this book will reinforce their beliefs regarding the origin of LIFE and MAN. Scientists to request reconsideration of theories that hinder the development of science, noting that no belief should divert research at its sole purpose: TRUTH.
A carta de Paulo a igreja de Corinto visava coibir várias práticas pecaminosas, eram problemas de ordem doutrinaria, como imoralidade sexual, facções, abusos na ceia, recusa das mulheres de usar véu, desordem no culto quanto a manifestação dos dons espirituais e até dúvidas sobre a ressurreição. Como fundador daquela igreja e como apóstolo respeitado entre todos os cristãos, Paulo manda uma extensa carta para orientar a igreja e até mesmo exigir que fosse expulso alguns membros que estão corrompendo moralmente aquela comunidade.
This talk concerning The Importance of Arts in Education was delivered at Westminster College on March 23, 2009. My personal favorite part of the talk was revealing the percentage of GNP the arts represent---a real sabot for that certain kind of philistine social (and economic) conservative who wants to crush arts funding in schools. For that individual, here's an economic argument that almost sounds like an artifact of some capitalist Utopia. Considering we don't produce much in the U.S. anymore---the legacy of Bretton-Woods---at least we still export the arts.
Short Lifetimes of Light Emitting PolymersJeffrey Gold
Jeffrey Frederick Gold
University of Cambridge
Microelectronics Research Centre
Cavendish Laboratory
Cambridge CB3 0HE
Introduction
The following manuscript was submitted as part of the MPhil program in Microelectronic Engineering and Semiconductor Physics at the Microelectronics Research Centre (MRC) of the Cavendish Laboratory at the University of Cambridge. The manuscript is a literature survey undertaken by the author as part of the MESP program during Lent term 1997 and is the basis of a talk of the same
title given at the MRC on January 20, 1997.
Presentation about Story Structure given April 20, 2009 at Westminster College. This presentation discusses the intrinsic three act structure and uses The Shawshank Redemption to show how two overlapping three-act structures gives rise to a four-act structure.
1. Chapter 5
On A Conjecture Of Erdos
Proceedings|NCUR VIII. 1994, Vol. II, pp. 794 798.
Je rey F. Gold
Department of Mathematics, Department of Physics
University of Utah
Don H. Tucker
Department of Mathematics
University of Utah
In this paper we present some preliminary results on a conjecture by Paul
Erdos 1,2,5 concerning covering sets of congruences. A covering set consists
of a nite system of congruences with distinct moduli, such that every integer
satis es as a minimum one of the congruences. An interesting consequence
of this conjecture is the dependence of the solution on abundant numbers; an
abundant number is an integer whose sum of its proper divisors exceeds the
integer.
Complementary Sets
De nition 1 If a and b are integers, then
a mod b = fa; a b; a 2b; : : :g :
De nition 2 If a ,a ,: : :,an ,b 2 Z, then
1 2
n
a1 ; a2 ; : : : ; an mod b = fa1 mod bg fa2 mod bg fan mod bg = fai mod bg :
i=1
1
2. CHAPTER 5. ON A CONJECTURE OF ERDOS 2
The Remodulization Theorem 3 states that if a; b; c 2 Z and c 0, then
a mod b = a; a + b; : : : ; a + bc , 1 mod cb :
If we use De nition 2, the complementary set of fa mod bg is given by
fa mod bgc = Z n fa mod bg = 0; 1; 2; : : :; a , 1; a + 1; : : : ; b , 1 mod b :
In this case, the complementary set consists of b , 1 congruences modulo b. We
will always refer to the size of a set and its complement with respect to a speci c
modulus. The following theorem and its proof is found in 4 .
n o
Theorem 1 The complementary set of Sn ai; ; : : : ; ai; mod bi , where
i =1 1 bi
6 6=
ai;j = ai;k for j Qn k, and b bi , and the bi are Qn
i pairwise relatively prime,
contains exactly i bi , b congruences modulo i bi .
=1 i =1
Covering Sets of Congruences
In Davenport 1 , a problem has been proposed to construct a set of congruences
with distinct moduli, such that every integer is contained in at least one of the
congruences of the system. All moduli are 2, since modulo 1 constitutes its
own complete residue system. An extension has been proposed by Erdos 5 : If
given any integer N 1, does there exist a nite covering set of congruences
using only distinct moduli greater than N ?
The following system represents a set of covering congruences for N = 1:
8
x 0 mod 2
x 0 mod 3
x 1 mod 4
: x 1 mod 612
x 11 mod
Note that the moduli are all divisors of 12.
Using the remodulization method to remodulize each congruence to the mod-
ulus 12, we have 8
x 0; 2; 4; 6; 8; 10 mod 12
x 0; 3; 6; 9 mod 12
x 1; 5; 9 mod 12 5.1
x 1; 7 mod 12
: x 11 mod 12
By inspection, this system constitutes a covering system, because it is equivalent
to the complete residue system modulo 12, that is, 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11
3. CHAPTER 5. ON A CONJECTURE OF ERDOS 3
mod12 Z.
The question naturally arises as to the possibility of constructing a set of
covering congruences whose moduli are pairwise relatively prime. The answer
is no, as we will now show. The proof depends on results of Theorem 1.
n o
Theorem 2 Any nite system of congruences Sn ai; ; ai; ; : : : ; ai; mod bi ,
i =1 1 2 bi
where the bi are pairwise relatively prime, and ai;k = ai;m for k 6= m, and
6
bi bi , cannot form a covering set of congruences.
Proof. We use Theorem 1, for the case bi = 1 for 1 i n. For a system
S
of congruences f n=1 ai mod bi g with pairwise relatively prime moduli bi to
i
be a covering set, it is necessary that the system of congruences forms a com-
Q
plete residue system, that is, a union of n=1 bi distinct incongruent residues
Qn i Q
modulo i=1 bi . However, since the complementary Qn consists n n=1 bi , 1
Qn b , the system consists of set b , Q of bi , 1 con-
congruences modulo i=1 i
Q Q iQ i i=1 i
gruences modulo n=1 bi . This means that n=1 bi , n=1 bi , 1 must equal
=1
Qn b , or Qn b ,1 = 0; this is a contradiction, since 2i b b : : : b .
i i
i=1 i i=1 i 1 2 n
That is to say, it takes in nitely many congruences with pairwise relatively prime
moduli to construct a covering set.
The situation is actually much worse; if we construct a system of congruences
n
ai;1 ; ai;2 ; : : : ; ai; bi mod bi ;
i=1
a system of congruences with bi residues for each modulus bi , where bi bi ,
Q
and ai;k = ai;m for k 6= m, then the complementary set consists of n=1 bi , bi
6 Qn b . Here, Qn b , must equal zero, which is
i
congruences modulo i=1 i i=1 i bi
a contradiction because bi , bi 1. Again, it takes in nitely many such
congruences to construct a covering set. In the extreme case, when bi = bi , 1,
we have the system
8
x a1;1 ; a1;2; : : : ; a1;b1 ,1 mod b1
x a2;1 ; a2;2; : : : ; a2;b2 ,1 mod b2
.. 5.2
.
: x an;1 ; an;2; : : : ; an;b ,1 mod bn
n
where the bi are pairwise relatively prime, and each set of congruences modulo bi
contains bi ,1 congruences, i.e., one congruence shy of a complete residue system
Q
for each modulus bi . Q complement of the system consists of n=1 bi ,bi ,1
The
congruence modulo i=1 in b , or 1 congruence modulo Qn b . iHence the sys-
i=1 i
tem 5.2 contains a complete residue system only if we cap it o with the last
4. CHAPTER 5. ON A CONJECTURE OF ERDOS 4
Q
remaining residue modulo n=1 bi . However, by adding to the system the re-
i
maining congruence, we have relaxed our requirement that all the moduli are
pairwise relatively prime.
Upshot If a nite system of distinct congruences 5.2 with pairwise relatively
prime moduli forms a covering set, it must contain a congruence class which
itself forms a complete residue system, or covering set.
Suppose p1 is a prime such that p1 N , and M = p1 p2 pn , where
1 2 n
p1 Qn p2 : : : pn . The total number of divisors of M which are M
is i=1 i + 1; however, to form a covering set we may only use all factors
Q
greater than 1, the total number of useable factors is = ,1 + n=1 i + 1.
i
We now construct a system of congruences
8
x c1 mod d1
x c2 mod d2
.. 5.3
.
: x c mod d
where the di are the various factors of M = p1 p2 pn , and d1
1 2 n d2 : : :
d . Note that d1 = p1 and d = M .
Observation 1 The number of congruences is given by
XM M M M
i=1 di = d1 + d2 + + d = 1 + d1 + d2 + + d,1 = 0 M ;
after remodulizing all congruences of system 5.3 to the modulus M = p1 p2 pn .
1 2 n
Here, 0 denotes the sum of all proper divisors, i.e., all positive divisors less
than M .
Up to this point we have not made any claims about these residues modulo M ,
that is, we have not yet determined how many repetitions exist, and equivalently,
if the total number of distinct residues is su cient to create a complete residue
system modulo M . We can see, however, that the total number of residues must
be at least M to form a complete residue system modulo M . Therefore, M must
be an abundant or perfect number, that is, the sum of all proper divisors
XM
= 1 + d1 + d2 + + d,1 = 0 M M
i=1 di
in order for this system to contain a complete residue system modulo M . We
have proved the following theorem.
5. CHAPTER 5. ON A CONJECTURE OF ERDOS 5
Theorem 3 In order for the proper divisors of a number M to constitute the
moduli of a covering set it is necessary that M be perfect or abundant, i.e.,
0 M M .
We will prove in Theorem 5 that if M is a perfect number, i.e., 0 M = M ,
then a system 5.3 cannot comprise a covering set.
Observation 2 It may not be necessary to use all divisors of an abundant
number M to form a covering set; however, according to our enumeration of
the residues modulo M of system 5.3, we must remove all residues associ-
ated with the divisors that are removed. For example, suppose we don't use
the divisor dk , then we must remove a total of M=dk residues from the set of
0 M residues counted in Observation 1. However, the divisor dk cannot con-
tain the greatest multiple of any one prime appearing in the prime decomposition
M = p1 p2 pn , for in that case, lcmd1 ; d2 ; : : : ; dk,1 ; dk+1 ; : : : ; d 6= M ,
1 2 n
that is to say, we would not have remodulized the system to the modulus M , but
instead to some modulus M .
In the original set
8
x 0; 2; 4; 6; 8; 10 mod 12
x 0; 3; 6; 9 mod 12
x 1; 5; 9 mod 12
x
: x 1; 7 mod 12
11 mod 12
we nd that the integers 0,1,6, and 9 represent 4 repetitions, since 0 12 , 12 =
4. The total number of repetitions that occur in a system 5.3 which forms a
covering set is 0 M , M . The following theorem enumerates the total number
of repetitions that occur in two congruences.
Theorem 4 If two congruences a mod b and a mod b , where gcdb ; b =
1 1 2 2 1 2
1, are remodulized to the modulus pb b , where p 2 Z, then the solution set
1 2
intersection consists of p residues modulo pb1 b2.
Proof. If we obtain a pair of congruences
x a1 mod b1 5.4
x a2 mod b2
where b1 and b2 are relatively prime, then remodulizing each to modulo b1 b2 ,
the intersection by the Chinese Remainder Theorem 3 is determined to be
the unique congruence x a0 mod b1 b2 , where a1 a0 a1 + b1b2 , 1 and
a2 a0 a2 +b2 b1 ,1. If the pair is remodulized, not to the smallest modulus,
6. CHAPTER 5. ON A CONJECTURE OF ERDOS 6
b1 b2 , but instead to some multiple of it, say pb1 b2 , where p is a positive integer,
then
x a1 ; a1 + b1 ; : : : ; a1 + b1 pb2 , 1 mod pb1 b2 5.5
x a2 ; a2 + b2 ; : : : ; a2 + b2 pb1 , 1 mod pb1 b2
We show that the intersection of 5.5 is
a0 ; a0 + b1 b2 ; : : : ; a0 + b1b2 p , 1 mod pb1 b2 ;
which is equivalent to the solution a0 mod b1 b2 of the original pair after a0 mod b1 b2
has been remodulized by the factor p. By writing the rst congruence of 5.5
as
a1 mod b1 = a1 ; a1 + b1 ; : : : a1 + b1 b2 , 1;
a1 + b1 b2 ; a1 + b1 + b1 b2 ; : : : a1 + b1 2b2 , 1;
.. .. ..
. . .
a1 + p , 1b1b2 ; a1 + b1 + p , 1b1b2 ; : : : a1 + b1 pb2 , 1 mod pb1b2
we note that the rst row contains the solution, a0 mod pb1b2 . Moreover, by
adding multiples of b1 b2 to the residue a0 , we nd the subsequent solutions
within the same column; hence there are p solutions modulo pb1 b2 . Con-
structing the second congruence of 5.5 in the same manner, we extract the
same p solutions. Therefore, if a pair of congruences 5.4 is remodulized to the
modulus pb1 b2, then they share exactly p simultaneous residues.
As an example, if we have the pair of congruences
x a1 mod p1
x a2 mod p2
which are remodulized to the modulus M = p1 p2 pnn , then they share
1 2
M=p1p2 = p1 ,1 p2 ,1 p3 pn
1 2 3 n
residues modulo M .
Theorem 5 If M is a perfect number, then a system of congruences whose
moduli consist of all divisors 1 of M cannot form a covering set.
Proof. Suppose M is an even perfect number 6 ; then it is of the form 2k p,
where p is an odd prime of the form 2k+1 , 1. Suppose we form a system
of congruences 5.3 where the di are all divisors of M greater than 1, and
remodulize all congruences modulo di to the modulus M . By Observation 1,
0 M = M ; a complete residue system modulo M must contain M distinct
residues modulo M . Since p is prime, the congruences modulo 2 and modulo p
7. CHAPTER 5. ON A CONJECTURE OF ERDOS 7
share 1 residue modulo 2p, or 2k,1 residues modulo 2k p by Theorem 4. These
represent 2k,1 repetitions, and M , 2k,1 M = 0 M ; hence the total num-
ber of distinct residues is not su cient to form a covering set.
If M is an odd perfect number if any exist, then it must contain more
than 8 distinct prime factors 7 . Since p1 and p2 are two distinct prime factors,
their intersection contains 1 congruence modulo p1 p2 , or M=p1 p2 congruences
modulo M . In that case, M , M=p1p2 M = 0 M , meaning that the total
number of distinct residues modulo M is too small for a system of congruences
5.3 to form a covering set.
Remark 1 Theorems 3 and 5 combined suggest that if for each N 2 there
exists a covering set whose distinct moduli all exceed N , then there would exist
abundant numbers whose least prime factor exceeds N . This is true. In fact,
even more is true.
De nition 3 A number M is said to be abundant of order k 1 if and only if
0 M =M k.
Theorem 6 If K and N are any integers, then there exists an integer M ,
abundant of order K , whose least prime factor exceeds N .
P
Proof. Since the primes are suchP that 1=pi = +1, we may select N
Q
p1 p2 pn such that n=1 1=pi = K . Set M = n=1 pi , then
i i
0 M = 1 + p1 + p2 + + p1 p2 pn,1 and 0 M =M = 1=p1 pn +
1=p2 pn + + 1=p1 + + 1=pn K .
Remark 2 Theorem 6 shows that there are numbers M whose divisors cannot
yet be excluded from forming a covering set whose moduli all exceed N . However,
a settling of this conjecture may well require nding methods that can accurately
account for the total number of repetitions that occur in such systems.
References
1 Harold Davenport, The Higher Arithmetic, Dover Publications, Inc., New
York, 1983. p. 57.
2 Erdos, Paul, On Integers of the Form 2k + p and some Related Problems,
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