Euler proved Fermat's Last Theorem for n=3 in 1770 by establishing a method of infinite descent. However, some key steps in the proof were not fully justified at the time. Euler had relied on previous work from 1759/1763 that proved important properties of numbers of the form x^2 + 3y^2, which provided the missing justification needed for the descent argument. While Euler claimed a proof by 1753, he waited to publish a more polished version incorporating the supplemental work.
In this video we learn how to solve limits that involve trigonometric functions. It is all based on using the fundamental trigonometric limit, which is proved using the squeeze theorem.
For more lessons: http://www.intuitive-calculus.com/solving-limits.html
Watch video: http://www.youtube.com/watch?v=1RqXMJWcRIA
In this video we learn how to solve limits that involve trigonometric functions. It is all based on using the fundamental trigonometric limit, which is proved using the squeeze theorem.
For more lessons: http://www.intuitive-calculus.com/solving-limits.html
Watch video: http://www.youtube.com/watch?v=1RqXMJWcRIA
This learner's module will discuss or talk about the Graph of Quadratic Functions. It will also discuss on how to draw the Graph of Quadratic Functions using the vertex, axis of symmetry, etc.
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Mathematician Recognized for Proof of Fermat’s Last TheoremMark Thek
Mark Thek is the author of the book Quantification of Human Emotion and the president of Esterline Power Systems, a subsidiary of the Esterline Corporation. As an engineer and physicist, Mark Thek is also interested in new developments in the field of mathematics.
This learner's module will discuss or talk about the Graph of Quadratic Functions. It will also discuss on how to draw the Graph of Quadratic Functions using the vertex, axis of symmetry, etc.
Continuity, Removable Discontinuity, Essential Discontinuity. These slides accompany my lectures in differential calculus with BSIE and GenENG students of LPU Batangas
Mathematician Recognized for Proof of Fermat’s Last TheoremMark Thek
Mark Thek is the author of the book Quantification of Human Emotion and the president of Esterline Power Systems, a subsidiary of the Esterline Corporation. As an engineer and physicist, Mark Thek is also interested in new developments in the field of mathematics.
International Journal of Engineering and Science Invention (IJESI)inventionjournals
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Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
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A Strategic Approach: GenAI in EducationPeter Windle
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This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
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1. n=2 n=4 n=3 E-272
Euler’s Proof of
Fermat’s Last Theorem
(for n = 3)
Lee Stemkoski
Adelphi Univeristy
December 5, 2012
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 1 / 33
2. n=2 n=4 n=3 E-272
Outline
1 n=2
2 n=4
3 n=3
4 E-272
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 2 / 33
3. n=2 n=4 n=3 E-272
Fun Algebra to Check Thoroughly
If p is prime and p | ab,
then p | a or p | b. (Euclid’s Lemma)
If gcd(r, s) = 1 and r · s = tn ,
then r = un and s = v n .
The set S = {x2 + ny 2 | x, y ∈ Z}
is closed under multiplication.
(a +nb2 )(c2 +nd2 ) = (ac±nbd)2 +n(ad bc)2
2
(Later in this talk, we will consider n = 3.)
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 3 / 33
4. n=2 n=4 n=3 E-272
A Babylonian Tablet
Figure: Plimpton 322
Also see:
Euclid, Book X, Lemma 1 - Proposition XXIX
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 4 / 33
5. n=2 n=4 n=3 E-272
Proving a Pythagorean Parameterization
Assume a solution exists: x2 + y 2 = z 2 ,
with x, y, z ∈ Z+ , relatively prime.
Some cases to consider:
odd + odd = even
////// +////// = even
even/// even//////////
even + odd = odd
///// +////// =/////
odd/// even//// odd
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 5 / 33
6. n=2 n=4 n=3 E-272
(odd)2 + (odd)2 = (even)2
This equation is impossible!
(2m + 1)2 + (2n + 1)2 = (2p)2
4(m2 + m + n2 + n) + 2 = 4p2
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 6 / 33
7. n=2 n=4 n=3 E-272
(even)2 + (odd)2 = (odd)2
x2 + y 2 = z 2
x2 = z 2 − y 2
x2 = (z + y)(z − y)
z + y = even = 2p
z − y = even = 2q
x = 2r, y = p − q, z = p + q
p and q: relatively prime, opposite parity
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 7 / 33
8. n=2 n=4 n=3 E-272
Finishing it up...
x2 = (z + y)(z − y)
r2 = p · q
gcd(p, q) = 1 implies p = a2 and q = b2
Putting it all together:
√
x = 2r = 2 pq = 2ab
y = p − q = a2 − b 2
z = p + q = a2 + b 2
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 8 / 33
9. n=2 n=4 n=3 E-272
The Pythagorean Parameterization
Theorem
If x2 + y 2 = z 2 , with x, y, z relatively prime,
then there exist integers a and b,
relatively prime and with opposite parity,
such that
x = 2ab
y = a2 − b2
z = a2 + b2
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 9 / 33
10. n=2 n=4 n=3 E-272
Fermat
Figure: Pierre de Fermat, 1601-1665
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 10 / 33
11. n=2 n=4 n=3 E-272
Method of Infinite Descent
Prove: if P (x) is true, then there exists
y < x with P (y) true, where x, y ∈ Z+ .
Obtain an infinite sequence of strictly
decreasing positive integers.
Contradicts the Well Ordering Principle:
S ⊆ Z+ has a smallest element.
Conclude: initial assumption is false.
Useful for showing solutions do not exist.
Fermat’s account of this method: “Relation des
nouvelles d´couvertes en la science des nombres”
e
letter to Pierre de Carcavi, 1659.
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 11 / 33
12. n=2 n=4 n=3 E-272
Wanted: Larger Margins
Fermat’s annotation of Bachet’s translation
of Diophantus’ Arithmetica
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 12 / 33
13. n=2 n=4 n=3 E-272
A proof
Assume a solution exists: x4 + y 4 = z 4 ,
with x, y, z ∈ Z+ , relatively prime.
Let X = x2 , Y = y 2 , Z = z 2 , then:
X2 + Y 2 = Z2
X = 2ab, Y = a2 − b2 , Z = a2 + b2
a and b: relatively prime, opposite parity
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 13 / 33
14. n=2 n=4 n=3 E-272
The descent
Y = a2 − b2 implies b2 + y 2 = a2
b = 2cd, y = c2 − d2 , a = c2 + d2
c and d: relatively prime, opposite parity
X = 2ab implies x2 = 4cd(c2 + d2 )
cd and (c2 + d2 ): relatively prime
cd = e2 and c2 + d2 = f 2
and c = g 2 , d = h2 ... let the descent begin...
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 14 / 33
15. n=2 n=4 n=3 E-272
Recap
Assume there are positive integers such that
X 2 + Y 2 = Z 2 , where X = x2 and Y = y 2
Obtain positive integers such that
c2 + d2 = f 2 , where c = g 2 and d = h2
f is strictly smaller than Z
We obtain an infinite sequence of strictly
decreasing positive integers, which is
impossible; the original assumption was false.
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 15 / 33
16. n=2 n=4 n=3 E-272
Euler
Figure: Leonhard Euler, 1707-1783
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 16 / 33
17. n=2 n=4 n=3 E-272
Correspondence with Goldbach
196 letters from 1729 to 1764
Goldbach motivates Euler to examine
Fermat’s work
1748 - Euler first mentions Fermat’s Last
Theorem
1753 - Euler announces proof for n = 3
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 17 / 33
18. n=2 n=4 n=3 E-272
Euler to Goldbach, 13 February 1748
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 18 / 33
19. n=2 n=4 n=3 E-272
Euler to Goldbach, 04 August 1753
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 19 / 33
20. n=2 n=4 n=3 E-272
Euler’s Algebra (1770)
Assume a solution exists: x3 + y 3 = z 3 ,
with x, y, z ∈ Z+ , relatively prime.
Exactly one of these three numbers are even.
Case 1: x, y are odd and z is even.
Case 2: y, z are odd and x is even.
Proof of Case 1. Assume that x > y.
x + y = even = 2p and x − y = even = 2q
x = p + q and y = p − q
p and q: positive, relatively prime, opposite parity
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 20 / 33
21. n=2 n=4 n=3 E-272
The proof continues
z 3 = x3 + y 3 = (p + q)3 + (p − q)3
= 2p3 + 6pq
= 2p(p2 + 3q 2 )
Note: p2 + 3q 2 is odd.
If g = gcd(2p, p2 + 3q 2 ) > 1 then g is odd;
g | p, so g | 3q 2 ; since g q, we have g = 3.
Therefore, gcd(2p, p2 + 3q 2 ) = 1 or 3.
(Two subcases to consider.)
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 21 / 33
22. n=2 n=4 n=3 E-272
gcd(2p, p2 + 3q 2) = 1
2p(p2 + 3q 2 ) = z 3
By earlier fact: 2p = u3 and (p2 + 3q 2 ) = v 3 .
Know: v ∈ S → v 3 ∈ S.
Also: v 3 ∈ S → v ∈ S
p2 + 3q 2 = v 3 = (a2 + 3b2 )3
Since gcd(p, q) = 1, we have:
p = a3 − 9ab2 , q = 3a2 b − 3b3 , gcd(a, b) = 1.
2p = 2a(a + 3b)(a − 3b) = u3
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 22 / 33
23. n=2 n=4 n=3 E-272
A descent appears!
2p = 2a(a + 3b)(a − 3b) = u3
2a, (a + 3b), (a − 3b) are relatively prime, so:
2a = α3 , (a + 3b) = β 3 , (a − 3b) = γ 3
α3 = β 3 + γ 3
Move terms if necessary so all terms are positive.
α, β, γ < z, since α3 β 3 γ 3 = 2p < 2p(p2 + 3q 2 ) = z 3
...a smaller positive solution to FLT, n = 3.
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 23 / 33
24. n=2 n=4 n=3 E-272
Excerpt #1, Euler’s Algebra
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 24 / 33
25. n=2 n=4 n=3 E-272
Excerpt #2, Euler’s Algebra
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 25 / 33
26. n=2 n=4 n=3 E-272
Commentarii...
Euler proved FLT(3) in 1770.
Euler proved FLT(3) in 1770, but key steps
were unjustified.
Euler proved FLT(3) in 1770, but key steps
were justified in 1759/1763 (E-272).
The results of E-272 are insufficient to prove
FLT(3).
Euler had a proof of FLT(3) by 1753, but
waited to publish a more polished version.
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 26 / 33
27. n=2 n=4 n=3 E-272
Red Flags Revisited
Need to fully justify:
1 If p2 + 3q 2 = v 3
(also gcd(p, q) = 1, and v 3 is odd)
then v = a2 + 3b2 .
2 In this situation, p2 + 3q 2 = (a2 + 3b2 )3
⇒ p = a3 − 9ab2
⇒ q = 3a2 b − 3b2
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 27 / 33
28. n=2 n=4 n=3 E-272
The missing link?
E272: Supplementum quorundam theorematum
arithmeticorum... (1759/1763)
Proves properties of numbers of the form x2 + 3y 2 .
Quick tour:
1 If gcd(a, b) = m, then m2 |(a2 + 3b2 )
2 2
2 If 3|(a2 + 3b2 ), then a +3b = n2 + 3m2 .
3
2 2
3 If 4|(a2 + 3b2 ), then a +3b = n2 + 3m2 .
4
4 If P = p + 3q is prime and P |(a2 + 3b2 ),
2 2
2 2
then a +3b = n2 + 3m2 .
P
Corollary: a = 3mq ± np and b = mp ± nq
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 28 / 33
29. n=2 n=4 n=3 E-272
5 If Pi = (pi )2 + 3(qi )2 is prime and
2 2
Pi |(a2 + 3b2 ), then a 1+3bk = n2 + 3m2 .
P ...P
6 If A = p2 + 3q 2 is prime and A|(a2 + 3b2 ),
then there exists a similar B < A.
7 All odd prime factors of a2 + 3b2 , when
gcd(a, b) = 1, have the form p2 + 3q 2 .
8 Primes of the form p2 + 3q 2 (except 3)
have the form 6n + 1.
9 Primes of the form 6n + 1
have the form p2 + 3q 2 .
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 29 / 33
30. n=2 n=4 n=3 E-272
Red Flag # 1
“If p2 + 3q 2 = v 3
(also gcd(p, q) = 1, and v 3 is odd)
then v = a2 + 3b2 .”
Fully justified by Euler’s Proposition 7.
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 30 / 33
31. n=2 n=4 n=3 E-272
Red Flag # 2
“In this situation, p2 + 3q 2 = (v)3 = (a2 + 3b2 )3
⇒ p = a3 − 9ab2 and q = 3a2 b − 3b2 ”
Can be addressed by Cor. to Euler’s Prop. 4:
Applied to a prime a2 + 3b2 , yields
uniqueness of representation.
(p2 + 3q 2 )(12 + 3 · 02 ) = (a2 + 3b2 )
a = 3 · 0 · q ± 1 · p and b = 0 · p ± 1 · q
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 31 / 33
32. n=2 n=4 n=3 E-272
“In this situation, p2 + 3q 2 = (v)3 = (a2 + 3b2 )3
⇒ p = a3 − 9ab2 and q = 3a2 b − 3b2 ”
Repeat for each prime power factor of v;
only one of the two methods of composition
will preserve gcd(a, b) = 1.
Combine prime power factors; many
representations of v.
Must use the same representation to
calculating v 3 to preserve gcd(p, q) = 1.
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 32 / 33
33. n=2 n=4 n=3 E-272
Did Euler consider this?
Similar work on sums of two squares
Unusual for Euler to not publish refutation
of FLT for n = 3?
E-255, x3 + y 3 = z 3 + v 3 (pres. 1754)
(one year after letter to Goldbach)
E-256, x2 + cy 2 conjectures (pres. 1753/4)
1755 letter to Goldbach: convinced Fermat
was correct, searching for FLT n = 5 proof
E-272, x2 + 3y 2 (pres. 1759)
Lee Stemkoski (Adelphi) Euler on Fermat’s Last Theorem December 5, 2012 33 / 33