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Journal articles on the topic 'Fermat's last theorem'

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Published: 4 June 2021

Last updated: 16 June 2025

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1

KLYKOV, SERGEY P. "ELEMENTARY PROOFS FOR THE FERMAT'S LAST THEOREM IN Z USING ONE TRICK FOR A RESTRICTION IN ZP." Journal of Science and Arts 23, no. 3 (2023): 603–8. http://dx.doi.org/10.46939/j.sci.arts-23.3-a03.

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An elementary and short proof of Fermat's Last Theorem (FLT) is presented, which is understandable even to a student. Perhaps this proof is precisely the lost proof, which could similar to own Fermat's proof. Restricting some coefficients of polynomials by value 0, except for the first term, allows to prove the Fermat's Last Theorem for domain Z, since in this case the canonical representation of p-adic numbers is limited to only one digit in the corresponding p-ary system. It was shown within the framework of elementary algebra, which corresponds to the Pythagorean theorem (PT) that the assumption of the existence of certain “Fermat’s triples” (FT), as integer solutions of Fermat's Last Theorem, can not be possible in Z due to some fatal inconsistencies for the PT and found by means the PT. Some equations in Zp were shown for n=3, 4 and 5

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2

Dowling, Jonathan P. "Fermat's Last Theorem." Mathematics Magazine 59, no. 2 (1986): 76. http://dx.doi.org/10.2307/2690422.

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3

Devlin, Keith. "Fermat's Last Theorem." Math Horizons 1, no. 2 (1994): 4–5. http://dx.doi.org/10.1080/10724117.1994.11974875.

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4

Dowling, Jonathan P. "Fermat's Last Theorem." Mathematics Magazine 59, no. 2 (1986): 76. http://dx.doi.org/10.1080/0025570x.1986.11977225.

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5

Alkis, Mazaris. "AN IMPORTANT CONCLUSION FOR FERMAT'S LAST THEOREM." INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH 10, no. 08 (2022): 2837–39. https://doi.org/10.5281/zenodo.6956329.

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Fermat’s Last Theorem is perhaps the only most famous mathematical problem of all times. Although finally proved, but the Theorem never stopped being a challenge mainly because the first proof didn’t used mathematics known in Fermat’s era. In the present work we arrive at a very important conclusion for the Theorem. If this conclusion is taken into account, the formulation of the Theorem should be different.

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6

Brill, Michael H. "On Fermat's Last Theorem." Mathematics Magazine 58, no. 2 (1985): 96. http://dx.doi.org/10.2307/2689896.

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7

Azuhata, Takashi. "On Fermat's last theorem." Acta Arithmetica 45, no. 1 (1985): 19–27. http://dx.doi.org/10.4064/aa-45-1-19-27.

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8

Nag, Bibek Baran. "On Fermat's Last Theorem." Journal of Advances in Mathematics and Computer Science 34, no. 4 (2019): 1–4. http://dx.doi.org/10.9734/jamcs/2019/v34i230211.

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9

Brill, Michael H. "On Fermat's Last Theorem." Mathematics Magazine 58, no. 2 (1985): 96. http://dx.doi.org/10.1080/0025570x.1985.11977159.

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10

Gevorkyan, Yuriy. "Geometric approach to the proof of Fermat’s last theorem." EUREKA: Physics and Engineering, no. 4 (July 30, 2022): 127–36. http://dx.doi.org/10.21303/2461-4262.2022.002488.

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A geometric approach to the proof of Fermat’s last theorem is proposed. Instead of integers a, b, c, Fermat’s last theorem considers a triangle with side lengths a, b, c. It is proved that in the case of right-angled and obtuse-angled triangles Fermat's equation has no solutions. When considering the case when a, b, c are sides of an acute triangle, it is proved that Fermat's equation has no entire solutions for p>2. The numbers a=k, b=k+m, c=k+n, where k, m, n are natural numbers satisfying the inequalities n>m, n<k+m, exhaust all possible variants of natural numbers a, b, c which are the sides of the triangle. The proof in this case is carried out by introducing a new auxiliary function f(k,p)=kp+(k+m)p–(k+n)p of two variables, which is a polynomial of degree in the variable . The study of this function necessary for the proof of the theorem is carried out. A special case of Fermat’s last theorem is proved, when the variables a, b, c take consecutive integer values. The proof of Fermat’s last theorem was carried out in two stages. Namely, all possible values of natural numbers k, m, n, p were considered, satisfying the following conditions: firstly, the number (np–mp) is odd, and secondly, this number is even, where the number (np–mp) is a free member of the function f(k, p). Another proof of Fermat’s last theorem is proposed, in which all possible relationships between the supposed integer solution of the equation f(k, p)=0 and the number corresponding to this supposed solution are considered. The proof is carried out using the mathematical apparatus of the theory of integers, elements of higher algebra and the foundations of mathematical analysis. These studies are a continuation of the author's works, in which some special cases of Fermat’s last theorem are proved

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11

James, Joseph. "Fermat's Last theorem Algebraic Proof." Journal of Progressive Research in Mathematics 7, no. 4 (2016): 1129–41. https://doi.org/10.5281/zenodo.4032339.

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In 1995, A, Wiles announced, using cyclic groups ( a subject area which was not available at the time of Fermat), a proof of Fermat’s Last Theorem, which is stated as follows: If π is an odd prime and x, y, z, are relatively prime positive integers, then z π 6= x π + y π . In this note, an elegant proof of this result is given. It is proved, using elementary algebra, that if π is an odd prime and x, y, z are positive integers satisfying z π = x π +y π , then z, x, are each divisible by π.

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12

Kleiner, Israel. "From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem." Elemente der Mathematik 55, no. 1 (2000): 19–37. http://dx.doi.org/10.1007/pl00000079.

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13

Granville, Andrew, and Alf van der Poorten. "Notes on Fermat's Last Theorem." American Mathematical Monthly 106, no. 2 (1999): 177. http://dx.doi.org/10.2307/2589072.

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14

Cox, David A. "Introduction to Fermat's Last Theorem." American Mathematical Monthly 101, no. 1 (1994): 3. http://dx.doi.org/10.2307/2325116.

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15

Patyal, Jagjit Singh. "Results Beyond Fermat's Last Theorem." International Journal of Mathematics Trends and Technology 68, no. 8 (2022): 116–28. http://dx.doi.org/10.14445/22315373/ijmtt-v68i8p511.

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16

Eynden, Charles Vanden. "Fermat's Last Theorem: 1637—1988." Mathematics Teacher 82, no. 8 (1989): 637–40. http://dx.doi.org/10.5951/mt.82.8.0637.

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Around 1637 the French jurist and amateur mathematician Pierre de Fermat wrote in the margin of his copy of Diophantus's Arithmetic that he had a “truly marvelous” proof that the equation xn + yn = zn has no solution in positive integers if n > 2. Unfortunately the margin was too narrow to contain it. In 1988 the world thought that the Japanese mathematician Yoichi Miyaoka, working at the Max Planck Institute in Bonn, West Germany, might have discovered a proof of this theorem. Such a proof would be of considerable interest because no evidence has been found that Fermat ever wrote one down, and no one has been able to find one in the 350 years since. In fact Miyaoka's announcement turned out to be premature, and a few weeks later articles reported holes in his argument that could not be repaired.

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17

Cipra, B. "Fermat's Last Theorem Finally Yields." Science 261, no. 5117 (1993): 32–33. http://dx.doi.org/10.1126/science.261.5117.32.

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18

MERMIN, N. DAVID. "Physics and Fermat's last theorem." Nature 330, no. 6149 (1987): 615–16. http://dx.doi.org/10.1038/330615b0.

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19

Cox, David A. "Introduction to Fermat's Last Theorem." American Mathematical Monthly 101, no. 1 (1994): 3–14. http://dx.doi.org/10.1080/00029890.1994.11996897.

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20

Gevorkyan, Yuriy. "Geometric approach to the proof of Fermat's last theorem." EUREKA: Physics and Engineering, no. 4 (July 30, 2022): 127–36. https://doi.org/10.21303/2461-4262.2022.002488.

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Abstract:

A geometric approach to the proof of Fermat's last theorem is proposed. Instead of integers a, b, c, Fermat's last theorem considers a triangle with side lengths a, b, c. It is proved that in the case of right-angled and obtuse-angled triangles Fermat's equation has no solutions. When considering the case when a, b, c are sides of an acute triangle, it is proved that Fermat's equation has no entire solutions for p>2. The numbers a=k, b=k+m, c=k+n, where k, m, n are natural numbers satisfying the inequalities n>m, n

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21

P. N Seetharaman. "A Proof for Fermat's Last Theorem using an Auxiliary Fermat's Equation." Indian Journal of Advanced Mathematics 4, no. 2 (2024): 19–24. https://doi.org/10.54105/ijam.a1182.04021024.

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Fermat’s Last Theorem states that there exists no three positive integers x, y and z satisfying the equation xn + yn = zn, where n is any integer > 2. Fermat and Euler had already proved the theorem for the exponents n = 4 and n = 3 in the equations x4 + y4 = z4 and x3 + y3 = z3 respectively. Hence taking into account of the same, it is enough to prove the theorem for the exponent n = p, where p is any prime > 3. In this proof, we have hypothesized that r, s and t are positive integers in the equation rp + sp = tp where p is any prime >3 and prove the theorem by the method of contradiction. To support the proof in the above equation we have used an Auxiliary equation x3 + y3 = z3. The two equations are linked by means of transformation equations. Solving the transformation equations we prove the theorem.

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22

JOSEPH, JAMES E. "2016 ALGEBRAIC PROOF FERMAT'S LAST THEOREM (2-18)." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 1 (2016): 5825–26. http://dx.doi.org/10.24297/jam.v12i1.606.

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In 1995, A, Wiles [2], [3], announced, using cyclic groups ( a subject area which was not available at the time of Fermat), a proof of Fermat's Last Theorem, which is stated as fol-lows: If is an odd prime and x; y; z; are relatively prime positive integers, then z 6= x + y: In this note, a new elegant proof of this result is presented. It is proved, using elementary algebra, that if is an odd prime and x; y; z; are positive integers satisfying z = x + y; then z; y; x; are each divisible by :

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23

P., N. Seetharaman. "A Proof for Fermat's Last Theorem using an Auxiliary Fermat's Equation." Indian Journal of Advanced Mathematics (IJAM) 4, no. 2 (2024): 19–24. https://doi.org/10.54105/ijam.A1182.04021024.

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<strong>Abstract:</strong> Fermat’s Last Theorem states that there exists no three positive integers x, y and z satisfying the equation x n + y n = z n , where n is any integer > 2. Fermat and Euler had already proved the theorem for the exponents n = 4 and n = 3 in the equations x 4 + y 4 = z 4 and x 3 + y 3 = z 3 respectively. Hence taking into account of the same, it is enough to prove the theorem for the exponent n = p, where p is any prime > 3. In this proof, we have hypothesized that r, s and t are positive integers in the equation r p + s p = t p where p is any prime >3 and prove the theorem by the method of contradiction. To support the proof in the above equation we have used an Auxiliary equation x 3 + y 3 = z 3 . The two equations are linked by means of transformation equations. Solving the transformation equations we prove the theorem.

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24

Mednis, P. "ALGEBRAIC PROOF OF FERMAT'S LAST THEOREM." POLISH JOURNAL OF SCIENCE, no. 57 (December 18, 2022): 23–27. https://doi.org/10.5281/zenodo.7455439.

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In this paper is suggested the sufficiently simple solution the widely known Fermat’s last theorem. The proof is based on Ruffini’s – Abel’s theorem concerning the solution of degree algebraic equations in an integer number.

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25

P. N Seetharaman. "A Comprehensible Proof for Fermat's Last Theorem." Indian Journal of Advanced Mathematics 4, no. 1 (2024): 29–34. https://doi.org/10.54105/ijam.a1181.04010424.

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Fermat’s Last Theorem states that it is impossible to find positive integers A, B and C satisfying the equation An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3. We hypothesize that all r, s and t are non-zero integers in the equation r p + sp = tp and establish contradiction. Just for supporting the proof in the above equation, we have another equation x 3 + y3 = z3 Without loss of generality, we assert that both x and y as non-zero integers; z3 a non-zero integer; z and z2 irrational. We create transformed equations to the above two equations through parameters, into which we have incorporated the Ramanujan - Nagell equation. Solving the transformed equations we prove the theorem.

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26

Quine, W. V. "Fermat's Last Theorem in Combinatorial Form." American Mathematical Monthly 95, no. 7 (1988): 636. http://dx.doi.org/10.2307/2323308.

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27

Whitt, Lee. "Medical Cozenage on Fermat's Last Theorem." College Mathematics Journal 16, no. 1 (1985): 55. http://dx.doi.org/10.2307/2686630.

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28

Abbott, Steve, Gary Cornell, Joseph H. Silverman, and Glenn Stevens. "Modular Forms and Fermat's Last Theorem." Mathematical Gazette 84, no. 501 (2000): 574. http://dx.doi.org/10.2307/3620827.

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29

Sun, Zhi-Wei. "Fibonacci numbers and Fermat's last theorem." Acta Arithmetica 60, no. 4 (1992): 371–88. http://dx.doi.org/10.4064/aa-60-4-371-388.

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30

Peterson, I. "Fermat's Last Theorem: A Promising Approach." Science News 133, no. 12 (1988): 180. http://dx.doi.org/10.2307/3972496.

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31

Bennett, Curtis D., A. M. W. Glass, and Gabor J. Szekely. "Fermat's Last Theorem for Rational Exponents." American Mathematical Monthly 111, no. 4 (2004): 322. http://dx.doi.org/10.2307/4145241.

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32

Peterson, I. "Fermat's Famous Theorem: Proved at Last?" Science News 146, no. 19 (1994): 295. http://dx.doi.org/10.2307/3978539.

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33

NISHINO, HITOSHI. "SUPERSYMMETRY BREAKINGS AND FERMAT'S LAST THEOREM." Modern Physics Letters A 10, no. 02 (1995): 149–57. http://dx.doi.org/10.1142/s0217732395000168.

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A mechanism of supersymmetry breaking in two or four dimensions is given, in which the breaking is related to the Fermat's last theorem. It is shown that supersymmetry is exact at some irrational number points in parameter space, while it is broken at all rational number points except for the origin. Accordingly, supersymmetry is exact almost everywhere, as well as broken almost everywhere on the real axis in the parameter space at the same time. This is the first explicit mechanism of supersymmetry breaking with an arbitrarily small change of parameters around any exact supersymmetric model, which is possibly useful for realistically small nonperturbative supersymmetry breakings in superstring model building. Our superpotential can be added as a "hidden" sector to other useful supersymmetric models.

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34

Bennett, Curtis D., A. M. W. Glass, and Gábor J. Székely. "Fermat's Last Theorem for Rational Exponents." American Mathematical Monthly 111, no. 4 (2004): 322–29. http://dx.doi.org/10.1080/00029890.2004.11920080.

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35

Whitt, Lee. "Medical Cozenage on Fermat's Last Theorem." College Mathematics Journal 16, no. 1 (1985): 55–56. http://dx.doi.org/10.1080/07468342.1985.11972851.

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36

Quine, W. V. "Fermat's Last Theorem in Combinatorial Form." American Mathematical Monthly 95, no. 7 (1988): 636. http://dx.doi.org/10.1080/00029890.1988.11972061.

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37

Kolyvagin, V. A. "Modular hypothesis and Fermat's last theorem." Mathematical Notes 55, no. 2 (1994): 157–58. http://dx.doi.org/10.1007/bf02113295.

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38

Joseph, James E. "Algebraic Proof II- Fermat's Last Theorem." International Journal of Algebra and Statistics 4, no. 1 (2015): 42. http://dx.doi.org/10.20454/ijas.2015.967.

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39

Chiaho, Wong. "The proof of Fermat's Last Theorem." Applied Mathematics and Mechanics 17, no. 11 (1996): 1031–38. http://dx.doi.org/10.1007/bf00119950.

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40

Bosco, Adriko. "The Beal's Conjecture and Fermat's Last Theorem." International Journal of Science and Research (IJSR) 10, no. 11 (2021): 435–41. https://doi.org/10.21275/mr211031162015.

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41

Sóstenes, Rônmel da Cruz, Miguel da Silva Cleomacio, and Jacinto do Nascimento Júnior Agrinaldo. "A heuristic solution to Fermat's last theorem." Revista Brasileira do Ensino Médio 6 (March 6, 2023): 1–10. https://doi.org/10.5281/zenodo.7702972.

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For centuries, notable mathematicians spent time to prove or refute Fermat's last theorem. As a result, many mathematical proofs have been generated for different values of <em>n</em>. The English mathematician Andrew Wiles, after many attempts, and with the help of his former student Richard Taylor, finally came up with a proof of Fermat's last theorem, which was published in 1995 in the prestigious journal <em>Annals of Mathematics</em>. Because centuries had passed without a proof of the theorem, many mathematicians suspected that Fermat had not developed a real proof. The demonstration presented by Andrew Wiles involved very complex mathematical operations that didn’t exist in Fermat's time. Due to the high complexity of the demonstration of Fermat's last theorem, it is highly challenging to try to find solutions where it is mathematically valid. Therefore, and within this context, this paper aimed to present a heuristic solution of Fermat's last theorem using a specifically developed algebraic method. The purpose was not, under any circumstances, to propose a demonstration of the theorem, but to perform mathematical operations that aimed to facilitate its understanding, and consequently, its application. The outcomes obtained showed that it was possible to obtain valid solutions. This serves as a motivating element for teaching mathematics in undergraduate and even high school.

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42

James, Joseph. "UNIQUE FACTORIZATION FERMAT'S LAST THEOREM BEAL'S CONJECTURE." Journal of Progressive Research in Mathematics 10, no. 1 (2016): 1434–39. https://doi.org/10.5281/zenodo.3976651.

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In this paper the following statememt of Fermat\rq{}s Last Theorem is proved.  If  $x, y, z$ are positive integers$\pi$ is an odd prime and  $z^\pi=x^\pi+y^\pi, x, y, z$ are all even. Also, in this paper, is proved (Beal\rq{}s conjecture): The equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive integers $x, y, z, $ with $\xi, \mu, \nu$ primes at least $3.

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43

P.N Seetharaman. "An Alternative Elementary Proof for Fermat's Last Theorem." International Journal of Basic Sciences and Applied Computing 11, no. 8 (2025): 11–16. https://doi.org/10.35940/ijbsac.h0534.11080425.

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Fermat’s Last Theorem states that the equation x n + y n = z n has no solution for x, y and z as positive integers, where n is any positive integer > 2. Taking the proofs of Fermat and Euler for the exponents n = 4 and n = 3, it would suffice to prove the theorem for the exponent n = p, where p is any prime > 3. We hypothesize that r, s and t are positive integers satisfying the equation r p + s p = t p and establish a contradiction in this proof. We include another Auxiliary equation x 3 + y 3 = z 3 and connect these two equations by using transformation equations. On solving the transformation equation we get rst = 0, thus proving that only a trivial solution exists in the main equation r p + s p = t p.

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44

Shotsberger, Paul G. "Kepler and Wiles: Models of Perseverance." Mathematics Teacher 93, no. 8 (2000): 680–81. http://dx.doi.org/10.5951/mt.93.8.0680.

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If you have seen the videotape The Proof or read the companion book by Simon Singh (1997) called Fermat's Enigma, you know that the story of Andrew Wiles's journey toward the proof of Fermat's last theorem is a remarkable tale of hope, disappointment, persistence, and ultimate triumph. Discovered by Pierre Fermat around 1637, the theorem is simple to state: “The conjecture that xn + yn = zn, where n > 2, has no solution with x, y, and z positive integers” (James and James 1992, p. 163). Yet the proof of this well-known theorem eluded generations of mathematicians. Wiles spent nearly a decade attempting to prove the theorem before he finally succeeded in 1994. The videotape and the book each offer an all-too-rare glimpse into the private struggles that preceded the more formal presentation of a finished product.

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45

Smith, Stuart T. "Fermat's last theorem and Bezout's theorem in GCD domains." Journal of Pure and Applied Algebra 79, no. 1 (1992): 63–85. http://dx.doi.org/10.1016/0022-4049(92)90127-2.

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46

P., N. Seetharaman. "A Comprehensible Proof for Fermat's Last Theorem." Indian Journal of Advanced Mathematics (IJAM) 4, no. 1 (2024): 29–34. https://doi.org/10.54105/ijam.A1181.04010424.

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Abstract:

<strong>Abstract:</strong> Fermat’s Last Theorem states that it is impossible to find positive integers A, B and C satisfying the equation An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3. We hypothesize that all r, s and t are non-zero integers in the equation r p + sp = tp and establish contradiction. Just for supporting the proof in the above equation, we have another equation x 3 + y3 = z3 Without loss of generality, we assert that both x and y as non-zero integers; z3 a non-zero integer; z and z2 irrational. We create transformed equations to the above two equations through parameters, into which we have incorporated the Ramanujan - Nagell equation. Solving the transformed equations we prove the theorem.

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47

James, E. Joseph. "An algebraic proof of Fermat's last theorem." Journal of Progressive Research in Mathematics 4, no. 4 (2015): 414–17. https://doi.org/10.5281/zenodo.3980405.

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In 1995, A, Wiles announced, using cyclic groups, a proof of Fermat's Last Theorem, which is stated as follows: If is an odd prime and x; y; z are relatively prime positive integers, then z 6= x +y: In this note, a proof of this theorem is offered, using elementary Algebra. It is proved that if is an odd prime and x; y; z are positive integers satisfying z = x +y; then x; y; and z are each divisible by 

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48

Zuehlke, John A. "Fermat's Last Theorem for Gaussian Integer Exponents." American Mathematical Monthly 106, no. 1 (1999): 49. http://dx.doi.org/10.2307/2589586.

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49

Wiles, Andrew. "Modular Elliptic Curves and Fermat's Last Theorem." Annals of Mathematics 141, no. 3 (1995): 443. http://dx.doi.org/10.2307/2118559.

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50

McCleary, John. "How Not to Prove Fermat's Last Theorem." American Mathematical Monthly 96, no. 5 (1989): 410. http://dx.doi.org/10.2307/2325146.

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