An Elementary Chapter in Number Theory: Proof of Fermat's Last Theorem
Abstract
Abstract. Pierre de Fermat first stated around 1637 that for any integer n > 2, the equation an + bn = cn has no positive integer solutions, and he said the theorem in the margin of a copy of Arithmetica. His proof is available only for the equation a 4 + b 4 = c 4 for the exponent n = 4. Subsequently, Euler proved the theorem in the equation a 3 + b 3 = c 3 for the exponent n = 3. Taking the above two proofs of Fermat and Euler, it would suffice to prove the theorem for n = p, where p is any prime > 3. In this proof, we hypothesize all r, s and t as positive integerssatisfying the equation rp + sp = tp and establish a contradiction. We use another auxiliary equation, x 3 + y 3 = z 3 , and combine the two equations using transformation equations. Solving the transformation equations, we establish a contradiction, thereby proving the theorem.
Full text
Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 63 Retrieval Number:100.1/ijam.B121705021025 DOI: 10.54105/ijam.B1217.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. An Elementary Chapter in Number Theory: Proof of Fermat's Last Theorem P. N. Seetharaman Abstract. Pierre de Fermat first stated around 1637 that for any integer n > 2, the equation an + bn = cn has no positive integer solutions, and he said the theorem in the margin of a copy of Arithmetica. His proof is available only for the equation a 4 + b 4 = c 4 for the exponent n = 4. Subsequently, Euler proved the theorem in the equation a 3 + b 3 = c 3 for the exponent n = 3. Taking the above two proofs of Fermat and Euler, it would suffice to prove the theorem for n = p, where p is any prime > 3. In this proof, we hypothesize all r, s and t as positive integers satisfying the equation rp + sp = tp and establish a contradiction. We use another auxiliary equation, x 3 + y 3 = z3, and combine the two equations using transformation equations. Solving the transformation equations, we establish a contradiction, thereby proving the theorem. Keywords: Transformation Equations. Mathematics Subject Classification: 2010: 11A–XX. I. INTRODUCTION Pierre-de-Fermat, a French mathematician around 1637, wrote in the margin of a copy of Arithmetica that it is impossible to find positive integers A, B and C satisfying the equation A n + B n = Cn, where n is an integer greater than 2. He stated that he himself had found a marvellous proof for the equation, but the margin was too narrow to contain it. His proof for the theorem is available only for n = 4, using the infinite descent method. Subsequently, Euler proved the theorem for n = 3 [1]. Dirichlet, Legendre, and Lame proved the theorem for the exponents n = 5 and n = 7. Around 1820, Sophie Germain proved the theorem for some specific cases. Kummer proved the theorem for regular primes. He invented ideal number theory, and number theory advanced significantly into newer areas. Mathematicians observed a close connection between Fermat’s Last Theorem and Elliptic Curves [2]. After 358 years, in 1995, Prof. Andrew Wiles proved the theorem completely [3]. Many mathematicians and number theorists have contributed to and analysed the theorem [4]. In this proof, we are trying for an alternative elementary proof of Fermat’s Last Theorem. Manuscript received on 11 September 2025 | First Revised Manuscript received on 16 September 2025 | Second Revised Manuscript received on 23 September 2025 | Manuscript Accepted on 15 October 2025 | Manuscript published on 30 October 2025. *Correspondence Author(s) P.N. Seetharaman*, Retired Executive Engineer, Energy Conservation Cell, Tamil Nadu State Electricity Board, Anna Salai, Chennai (Tamil Nadu), India. Email ID: [email protected], ORCIID ID: 0000 - 0002-4615-1280 © The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/ II. ASSUMPTIONS A. We hypothesize that r, s and t are positive integers satisfying the equation r p +s p =t p Here, p is any prime >3. Clearly, gcd (r, s, t) = 1, and we establish a centre contradiction in this proof. B. We include the auxiliary equationx 3 +y 3 =z3in this proof, in which we can have both x and y to be positive integers; z 3 will be a positive integer; both z and z 2 irrational (as proved already by Euler and others) gcd (x, y, z3) = 1 and √𝑟𝑡 will be irrational. Since both x and z 3 cannot simultaneously be squares. C. Let F = (Ryz3rs), where R = y. D. We can have x, y and z 3 such that each has some other odd prime factors coprime to r, s and t. Proof. By random trials, we have created the following equations. (𝑎√𝑡 𝑝+ 𝑏√𝐹1 3 ⁄ )2+ (𝑐√ 𝑥+ 𝑑√𝑅1 3 ⁄ )2 = (𝑒√𝑅2 3 ⁄ + 𝑓√𝑟𝑠𝑡 )2 and (𝑎√𝑧 3− 𝑏√𝑠𝑝 )2+ (𝑐√𝐹2 3 ⁄ − 𝑑√𝐸2 3 ⁄ )2= (𝑒√𝑦𝑟 𝑝− 𝑓√𝐸1 3 ⁄ )2 (1) is the transformation equations of x 3 + y 3 = z 3 and r p + s p = tp , respectively, through the parameters called a, b, c, d, e and f. Here F = (Ryz3rs). From equation (1), we get 1/ 3 3 p a t b F x+= (2) 𝑎√𝑧 3− 𝑏√𝑠 𝑝= √𝑟 𝑝 (3) 𝑐√ 𝑥+ 𝑑√𝑅1 3 ⁄ = √𝑦 3 (4) 𝑐√𝐹2 3 ⁄ − 𝑑√𝐹2 3 ⁄ = √𝑠 𝑝 (5) 𝑒√𝑅2 3 ⁄ + 𝑓√ 𝑟𝑠𝑡= √𝑧 3 (6) and 𝑒√𝑦𝑟 𝑝− 𝑓√𝐹1 3 ⁄ = √𝑡 𝑝 (7) Solving simultaneously (2) and (3), (4) and (5), (6) and (7), we get
An Elementary Chapter in Number Theory: Proof of Fermat's Last Theorem 64 Retrieval Number:100.1/ijam.B121705021025 DOI: 10.54105/ijam.B1217.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. 𝑎=(√𝑥3𝑠𝑝+√𝐹1 3 ⁄𝑟𝑝) (√𝑠𝑝𝑡𝑝+√𝑧3𝐹1 3 ⁄)⁄ 𝑏=(√𝑥3𝑧3−√𝑟𝑝𝑡𝑝) (√𝑠𝑝𝑡𝑝+√𝐹1 3 ⁄𝑧3)⁄ 𝑐=(√𝐹2 3 ⁄𝑦3+√𝑅1 3 ⁄𝑠𝑝) (√𝐹2 3 ⁄𝑥+√𝐹2 3 ⁄𝑅1 3 ⁄) ⁄ 𝑑=(√𝐹2 3 ⁄𝑦3−√𝑥𝑠𝑝) (√𝐹2 3 ⁄𝑥+√𝐹2 3 ⁄𝑅1 3 ⁄) ⁄ 𝑒=(√𝐹1 3 ⁄𝑧3+√𝑟𝑠𝑡𝑝+1) (√𝐹1 3 ⁄𝑅2 3 ⁄+√𝑦𝑟𝑝+1𝑠𝑡)⁄ And 𝑓=(√𝑦𝑧3𝑟𝑝−√𝑅2 3 ⁄𝑡𝑝) (√𝐹1 3 ⁄𝑅2 3 ⁄+√𝑦𝑟𝑝+1𝑠𝑡)⁄ From (2) & (7), we get √𝑡𝑝×√𝑡𝑝 =(√𝑥3−𝑏√𝐹1 3 ⁄)(𝑒√𝑦𝑟𝑝−𝑓√𝐹1 3 ⁄)(𝑎) ⁄ e., 𝑡𝑝={(𝑒)√𝑥3𝑦𝑟𝑝−(𝑓)√𝐹1 3 ⁄𝑥3−(𝑏𝑒)√𝐹1 3 ⁄𝑦𝑟𝑝+(𝑏𝑓)(𝐹1 3 ⁄)} (𝑎) ⁄ From (3) & (7), we have √𝑟𝑝×√𝑟𝑝=(𝑎√𝑧3−𝑏√𝑠𝑝)(√𝑡𝑝+𝑓√𝐹1 3 ⁄)(𝑒√𝑦) ⁄ i.e., 𝑟𝑝={(𝑎)√𝑧3𝑡𝑝+(𝑎𝑓)√𝐹1 3 ⁄𝑧3−(𝑏)√𝑠𝑝𝑡𝑝−(𝑏𝑓)√𝐹1 3 ⁄𝑠𝑝} (𝑒√𝑦) ⁄ From (3) & (5), we get √𝑠𝑝×√𝑠𝑝=(𝑎√𝑧3−√𝑟𝑝)(𝑐√𝐹2 3 ⁄−𝑑√𝐹2 3 ⁄)(𝑏) ⁄ i.e., 𝑠𝑝={(𝑎𝑐)√𝐹2 3 ⁄𝑧3−(𝑎𝑑)√𝐹2 3 ⁄𝑧3−(𝑐)√𝐹2 3 ⁄𝑟𝑝+(𝑑)√𝐹2 3 ⁄𝑟𝑝}(𝑏) ⁄ Substituting the above equivalent values of tp, rp and sp in Fermat’s equation, rp + sp = tp, after multiplying both sides by {𝑎𝑏𝑒√𝑦}, we get {𝑏𝑒}√𝑦{(𝑒)√𝑥3𝑦𝑟𝑝−(𝑓)√𝐹1 3 ⁄𝑥3−(𝑏𝑒)√𝐹1 3 ⁄𝑦𝑟𝑝+(𝑏𝑓)(𝐹1 3 ⁄)} ( ) ( ) ( ) ( ) 3 1/ 3 3 1/ 3 () p p p p ab a t z af F z b s t bf F s= + − − ( ) ( ) ( ) ( ) ( ) 2/ 3 3 2/ 3 3 2/ 3 2/ 3pp ae y ac F z ad F z c F r d F r+ − − + (8) Our aim is to compute all rational terms in equation (8) after multiplying both sides by {(√𝑠𝑝𝑡𝑝+√𝐹1 3 ⁄𝑧3)3(√𝐹2 3 ⁄𝑥+√𝐹2 3 ⁄𝑅1 3 ⁄)(√𝐹1 3 ⁄𝑅2 3 ⁄+√𝑦𝑟𝑝+1𝑠𝑡)2} To be free from denominators on the parameters a, b, c, d, e and f and again multiplying both sides by t for getting some rational terms. I term in LHS of equation (8), after multiplying by the respective terms and substituting for {be2} =(𝑦√𝑥3𝑟𝑝){(𝑠𝑝𝑡𝑝)+(𝐹1 3 ⁄𝑧3)+2√𝑠𝑝𝑡𝑝√𝐹1 3 ⁄𝑧3}(√𝐹2 3 ⁄𝑥+√𝐹2 3 ⁄𝑅1 3 ⁄) ×√𝑡(√𝑥3𝑧3−√𝑟𝑝𝑡𝑝){(𝐹1 3 ⁄𝑧3)+(𝑟𝑠𝑡𝑝+1)+2√𝐹1 3 ⁄𝑧3𝑟𝑠𝑡𝑝+1} On multiplying by {(𝑦√𝑥3𝑟𝑝)(𝐹1 3 ⁄𝑧3)(𝐹1 3 ⁄√𝑥)√𝑡(−√𝑟𝑝𝑡𝑝)(𝐹1 3 ⁄𝑧3)} We get {−(𝐹𝑥2𝑦𝑧6𝑟𝑝)√𝑡𝑝+1} II term in LHS of equation (8), after multiplying by the respective terms and substituting for {b(ef)} =(−√𝐹1 3 ⁄𝑥3𝑦){(𝑠𝑝𝑡𝑝)+(𝐹1 3 ⁄𝑧3)+(2√𝑠𝑝𝑡𝑝√𝐹1 3 ⁄𝑧3)}(√𝐹2 3 ⁄𝑥+√𝐹2 3 ⁄𝑅1 3 ⁄) ×√𝑡(√𝑥3𝑧3−√𝑟𝑝𝑡𝑝)(√𝐹1 3 ⁄𝑧3+√𝑟𝑠𝑡𝑝+1)(√𝑦𝑧3𝑟𝑝−√𝑅2 3 ⁄𝑡𝑝) (i) On multiplying by
Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 65 Retrieval Number:100.1/ijam.B121705021025 DOI: 10.54105/ijam.B1217.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. {(−√𝐹1 3 ⁄𝑥3𝑦)(𝑧3√𝐹2 3 ⁄)√𝐹2 3 ⁄𝑥√𝑡(−√𝑟𝑝𝑡𝑝)√𝐹1 3 ⁄𝑧3√𝑦𝑧3𝑟𝑝} We get {(𝐹𝑥2𝑦𝑧6𝑟𝑝)√𝑡𝑝+1} (This term algebraically gets cancelled with the I term in LHS above) (ii) Also, on multiplying by {(−√𝐹1 3 ⁄𝑥3𝑦)(𝑠𝑝𝑡𝑝)√𝐹2 3 ⁄𝑅1 3 ⁄√𝑡√𝑥3𝑧3√𝑟𝑠𝑡𝑝+1(−√𝑅2 3 ⁄𝑡𝑝)} We get {(𝑥3𝑠𝑝𝑡2𝑝+1)√𝐹𝑅𝑦𝑧3𝑟𝑠} which is rational, since F = (Ryz3rs) (iii) Again, on multiplying by {(−√𝐹1 3 ⁄𝑥3𝑦)(𝑧3√𝐹2 3 ⁄)√𝐹2 3 ⁄𝑅1 3 ⁄√𝑡√𝑥3𝑧3√𝐹1 3 ⁄𝑧3(−√𝑅2 3 ⁄𝑡𝑝)} We get {(𝐹𝑥3𝑧6)√𝑡𝑝+1√𝑅𝑦} which is rational, since R = y. III term in LHS of equation (8), after multiplying by the respective terms and substituting for {b2e2} =(−𝑦√𝐹1 3 ⁄𝑟𝑝)(√𝑠𝑝𝑡𝑝+√𝐹1 3 ⁄𝑧3)(√𝐹2 3 ⁄𝑥+√𝐹2 3 ⁄𝑅1 3 ⁄)√𝑡 ×{(𝑥3𝑧3)+(𝑟𝑝𝑡𝑝)−2√𝑥3𝑧3𝑟𝑝𝑡𝑝}{(𝐹1 3 ⁄𝑧3)+(𝑟𝑠𝑡𝑝+1)+2√𝐹1 3 ⁄𝑧3𝑟𝑠𝑡𝑝+1} (i) On multiplying by (−𝑦√𝐹1 3 ⁄𝑟𝑝)√𝑠𝑝𝑡𝑝√𝐹2 3 ⁄𝑥√𝑡(𝑥3𝑧3+𝑟𝑝𝑡𝑝)(𝑟𝑠𝑡𝑝+1) We get {−(𝑦𝑟𝑠𝑡𝑝+1)√𝑡𝑝+1(𝑥3𝑧3+𝑟𝑝𝑡𝑝)√𝐹𝑥𝑟𝑝𝑠𝑝} Which will be irrational, since √𝐹𝑥𝑟𝑝𝑠𝑝=√𝑅𝑥𝑦𝑧3√(𝑟𝑠)𝑝+1, which will be irrational, since R = y and √𝑥𝑧3 will be irrational. (ii) Also, on multiplying by {(−𝑦√𝐹1 3 ⁄𝑟𝑝)√𝐹1 3 ⁄𝑧3√𝐹2 3 ⁄𝑥√𝑡(−2√𝑥3𝑧3𝑟𝑝𝑡𝑝)(𝑧3√𝐹2 3 ⁄)} We get {(2𝐹𝑥2𝑦𝑧6𝑟𝑝)√𝑡𝑝+1} (This term algebraically gets cancelled with the IV term in the LHS below) IV term in LHS of equation (8), after multiplying by the respective terms and substituting for {b2(ef)} =(𝐹1 3 ⁄√𝑦)(√𝑠𝑝𝑡𝑝+√𝐹1 3 ⁄𝑧3)(√𝐹2 3 ⁄𝑥+√𝐹2 3 ⁄𝑅1 3 ⁄)√𝑡 ×{(𝑥3𝑧3)+(𝑟𝑝𝑡𝑝)−2√𝑥3𝑧3𝑟𝑝𝑡𝑝}(√𝐹1 3 ⁄𝑧3+√𝑟𝑠𝑡𝑝+1)(√𝑦𝑧3𝑟𝑝−√𝑅2 3 ⁄𝑡𝑝) (i) On multiplying by {(𝐹1 3 ⁄√𝑦)√𝐹1 3 ⁄𝑧3√𝐹2 3 ⁄𝑥√𝑡(−2√𝑥3𝑧3𝑟𝑝𝑡𝑝)√𝐹1 3 ⁄𝑧3√𝑦𝑧3𝑟𝑝} We get {−(2𝐹𝑥2𝑦𝑧6𝑟𝑝)√𝑡𝑝+1} (This term algebraically gets cancelled with the III term in LHS worked out above) (ii) Also, on multiplying by {(𝐹1 3 ⁄√𝑦)√𝐹1 3 ⁄𝑧3√𝐹2 3 ⁄𝑅1 3 ⁄√𝑡(−2√𝑥3𝑧3𝑟𝑝𝑡𝑝)√𝐹1 3 ⁄𝑧3√𝑅2 3 ⁄𝑡𝑝} We get {−(2𝐹𝑧3𝑡𝑝)√𝑅𝑥3𝑦𝑧3𝑟𝑝𝑡} Which will be irrational, since R = y if r and t are coprimes to x and z3.
An Elementary Chapter in Number Theory: Proof of Fermat's Last Theorem 66 Retrieval Number:100.1/ijam.B121705021025 DOI: 10.54105/ijam.B1217.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. (iii) Again, on multiplying by {(𝐹1 3 ⁄√𝑦)√𝐹1 3 ⁄𝑧3√𝐹2 3 ⁄𝑅1 3 ⁄√𝑡(𝑥3𝑧3+𝑟𝑝𝑡𝑝)√𝐹1 3 ⁄𝑧3(−√𝑅2 3 ⁄𝑡𝑝)} We get {−(𝐹𝑧3√𝑅𝑦)√𝑡𝑝+1(𝑥3𝑧3+𝑟𝑝𝑡𝑝)} Which will be rational, since we have defined R = y. I term in RHS of equation (8), after multiplying by the respective terms and substituting for {a2b} =√𝑧3𝑡𝑝(√𝐹2 3 ⁄𝑥+√𝐹2 3 ⁄𝑅1 3 ⁄){(𝐹1 3 ⁄𝑅2 3 ⁄)+(𝑦𝑟𝑝+1𝑠𝑡)+2√𝐹1 3 ⁄𝑅2 3 ⁄√𝑦𝑟𝑝+1𝑠𝑡}√𝑡 ×((𝑥3𝑠𝑝)+(𝐹1 3 ⁄𝑟𝑝)+2√𝐹1 3 ⁄𝑥3𝑟𝑝𝑠𝑝)(√𝑥3𝑧3−√𝑟𝑝𝑡𝑝) On multiplying by {√𝑧3𝑡𝑝√𝐹2 3 ⁄𝑅1 3 ⁄(2√𝐹1 3 ⁄𝑅2 3 ⁄√𝑦𝑟𝑝+1𝑠𝑡)√𝑡(𝑥3𝑠𝑝)(−√𝑟𝑝𝑡𝑝)} we get {−(2𝑥3𝑟𝑝𝑠𝑝𝑡𝑝+1)√𝐹𝑅𝑦𝑧3𝑟𝑠} Which is rational, since we have defined F = (Ryz3rs) II term in RHS of equation (8), after multiplying by the respective terms and substituting for {(a2b)f} =√𝐹1 3 ⁄𝑧3(√𝐹2 3 ⁄𝑥+√𝐹2 3 ⁄𝑅1 3 ⁄)(√𝐹1 3 ⁄𝑅2 3 ⁄+√𝑦𝑟𝑝+1𝑠𝑡)√𝑡 ×((𝑥3𝑠𝑝)+(𝐹1 3 ⁄𝑟𝑝)+2√𝐹1 3 ⁄𝑥3𝑟𝑝𝑠𝑝)(√𝑥3𝑧3−√𝑟𝑝𝑡𝑝)(√𝑦𝑧3𝑟𝑝−√𝑅2 3 ⁄𝑡𝑝) (i) On multiplying by {√𝐹1 3 ⁄𝑧3√𝐹2 3 ⁄𝑅1 3 ⁄√𝑦𝑟𝑝+1𝑠𝑡√𝑡(𝑥3𝑠𝑝)(−√𝑟𝑝𝑡𝑝)(−√𝑅2 3 ⁄𝑡𝑝)} we get {(𝑥3𝑟𝑝𝑠𝑝𝑡𝑝+1)√𝐹𝑅𝑦𝑧3𝑟𝑠} Which is rational. (ii) Also, on multiplying by {√𝐹1 3 ⁄𝑧3√𝐹2 3 ⁄𝑅1 3 ⁄√𝐹1 3 ⁄𝑅2 3 ⁄√𝑡(𝑟𝑝√𝐹2 3 ⁄)(−√𝑟𝑝𝑡𝑝)√𝑦𝑧3𝑟𝑝} we get {−(𝐹𝑧3𝑟2𝑝)√𝑡𝑝+1√𝑅𝑦} Which is rational, since R = y. III term in RHS of equation (8), after multiplying by the respective terms and substituting for {ab2} =(−√𝑠𝑝𝑡𝑝)(√𝐹2 3 ⁄𝑥+√𝐹2 3 ⁄𝑅1 3 ⁄){(𝐹1 3 ⁄𝑅2 3 ⁄)+(𝑦𝑟𝑝+1𝑠𝑡)+2√𝐹1 3 ⁄𝑅2 3 ⁄√𝑦𝑟𝑝+1𝑠𝑡} () ( ) ( ) 3 1/3 3 3 3 3 2 p p p p p p t x s F r x z r t x z r t + + − On multiplying by {(−√𝑠𝑝𝑡𝑝)√𝐹2 3 ⁄𝑅1 3 ⁄(2√𝐹1 3 ⁄𝑅2 3 ⁄√𝑦𝑟𝑝+1𝑠𝑡)√𝑡√𝑥3𝑠𝑝(−2√𝑥3𝑧3𝑟𝑝𝑡𝑝)} we get {(4𝑥3𝑟𝑝𝑠𝑝𝑡𝑝+1)√𝐹𝑅𝑦𝑧3𝑟𝑠} Which is rational. IV term in RHS of equation (8), after multiplying by the respective terms and substituting for {(ab2)f} =(−√𝐹1 3 ⁄𝑠𝑝)(√𝐹2 3 ⁄𝑥+√𝐹2 3 ⁄𝑅1 3 ⁄)(√𝐹1 3 ⁄𝑅2 3 ⁄+√𝑦𝑟𝑝+1𝑠𝑡)√𝑡 ×(√𝑥3𝑠𝑝+√𝐹1 3 ⁄𝑟𝑝){(𝑥3𝑧3)+(𝑟𝑝𝑡𝑝)−2√𝑥3𝑧3𝑟𝑝𝑡𝑝}(√𝑦𝑧3𝑟𝑝−√𝑅2 3 ⁄𝑡𝑝) On multiplying by {(−√𝐹1 3 ⁄𝑠𝑝)√𝐹2 3 ⁄𝑅1 3 ⁄√𝑦𝑟𝑝+1𝑠𝑡√𝑡√𝑥3𝑠𝑝(−2√𝑥3𝑧3𝑟𝑝𝑡𝑝)(−√𝑅2 3 ⁄𝑡𝑝)} we get
Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 67 Retrieval Number:100.1/ijam.B121705021025 DOI: 10.54105/ijam.B1217.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. {−(2𝑥3𝑟𝑝𝑠𝑝𝑡𝑝+1)√𝐹𝑅𝑦𝑧3𝑟𝑠} Which is rational. V term in RHS of equation (8), after multiplying by the respective terms and substituting for {a2ce} =√𝐹2 3 ⁄𝑦𝑧3(√𝑠𝑝𝑡𝑝+√𝐹1 3 ⁄𝑧3)(√𝐹1 3 ⁄𝑅2 3 ⁄+√𝑦𝑟𝑝+1𝑠𝑡)√𝑡 ×{(𝑥3𝑠𝑝)+(𝐹1 3 ⁄𝑟𝑝)+2√𝐹1 3 ⁄𝑥3𝑟𝑝𝑠𝑝}(√𝐹2 3 ⁄𝑦3+√𝑅1 3 ⁄𝑠𝑝)(√𝐹1 3 ⁄𝑧3+√𝑟𝑠𝑡𝑝+1) (i) On multiplying by {√𝐹2 3 ⁄𝑦𝑧3√𝑠𝑝𝑡𝑝√𝐹1 3 ⁄𝑅2 3 ⁄√𝑡(𝑥3𝑠𝑝)√𝑅1 3 ⁄𝑠𝑝√𝑟𝑠𝑡𝑝+1} we get {(𝑥3𝑠2𝑝𝑡𝑝+1)√𝐹𝑅𝑦𝑧3𝑟𝑠} Which is rational. (ii) Also, on multiplying by {√𝐹2 3 ⁄𝑦𝑧3√𝑠𝑝𝑡𝑝√𝐹1 3 ⁄𝑅2 3 ⁄√𝑡(𝑟𝑝√𝐹2 3 ⁄)√𝑅1 3 ⁄𝑠𝑝√𝐹1 3 ⁄𝑧3} We get {(𝐹𝑧3𝑟𝑝𝑠𝑝)√𝑡𝑝+1√𝑅𝑦} Which will be rational. VI term in RHS of equation (8), after multiplying by the respective terms and substituting for {a2de} =(−√𝐹2 3 ⁄𝑦𝑧3)(√𝑠𝑝𝑡𝑝+√𝐹1 3 ⁄𝑧3)(√𝐹1 3 ⁄𝑅2 3 ⁄+√𝑦𝑟𝑝+1𝑠𝑡)√𝑡 ×{(𝑥3𝑠𝑝)+(𝐹1 3 ⁄𝑟𝑝)+2√𝐹1 3 ⁄𝑥3𝑟𝑝𝑠𝑝}(√𝐹2 3 ⁄𝑦3−√𝑥𝑠𝑝)(√𝐹1 3 ⁄𝑧3+√𝑟𝑠𝑡𝑝+1) (i) On multiplying by {(−√𝐹2 3 ⁄𝑦𝑧3)√𝑠𝑝𝑡𝑝√𝑦𝑟𝑝+1𝑠𝑡√𝑡(𝑥3𝑠𝑝)(−√𝑥𝑠𝑝)√𝐹1 3 ⁄𝑧3} we get {(𝑥3𝑦𝑧3𝑠2𝑝)√(𝑟𝑡)𝑝+1√𝐹𝑥𝑠𝑡} (ii) Also, on multiplying by {(−√𝐹2 3 ⁄𝑦𝑧3)√𝐹1 3 ⁄𝑧3√𝑦𝑟𝑝+1𝑠𝑡√𝑡(𝑥3𝑠𝑝)(−√𝑥𝑠𝑝)√𝑟𝑠𝑡𝑝+1} we get {(𝑥3𝑦𝑧3𝑠𝑝𝑡)√(𝑟𝑠𝑡)𝑝+1√𝐹𝑥𝑟𝑠} The above two terms will be irrational, since F = (Ryz3rs) and R = y and √𝑥𝑧3 will irrational. VII term in RHS of equation (8), after multiplying by the respective terms and substituting for {ace} =(−√𝐹2 3 ⁄𝑦𝑟𝑝){(𝑠𝑝𝑡𝑝)+(𝐹1 3 ⁄𝑧3)+2√𝐹1 3 ⁄𝑧3𝑠𝑝𝑡𝑝}(√𝐹1 3 ⁄𝑅2 3 ⁄+√𝑦𝑟𝑝+1𝑠𝑡)√𝑡 ×(√𝑥3𝑠𝑝+√𝐹1 3 ⁄𝑟𝑝)(√𝐹2 3 ⁄𝑦3+√𝑅1 3 ⁄𝑠𝑝)(√𝐹1 3 ⁄𝑧3+√𝑟𝑠𝑡𝑝+1) On multiplying by {(−√𝐹2 3 ⁄𝑦𝑟𝑝)(𝑠𝑝𝑡𝑝)√𝐹1 3 ⁄𝑅2 3 ⁄√𝑡√𝑥3𝑠𝑝√𝑅1 3 ⁄𝑠𝑝√𝑟𝑠𝑡𝑝+1} we get {−(√𝐹𝑅𝑥3𝑦𝑠𝑡)(𝑠2𝑝𝑡𝑝)√(𝑟𝑡)𝑝+1} Which will be irrational, since F = (Ryz3rs), R = y and 33 x z rt will be irrational VIII term in RHS of equation (8), after multiplying by the respective terms and substituting for {ade}
An Elementary Chapter in Number Theory: Proof of Fermat's Last Theorem 68 Retrieval Number:100.1/ijam.B121705021025 DOI: 10.54105/ijam.B1217.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. =√𝐹2 3 ⁄𝑦𝑟𝑝{(𝑠𝑝𝑡𝑝)+(𝐹1 3 ⁄𝑧3)+2√𝐹1 3 ⁄𝑧3𝑠𝑝𝑡𝑝}(√𝐹1 3 ⁄𝑅2 3 ⁄+√𝑦𝑟𝑝+1𝑠𝑡)√𝑡 ×(√𝑥3𝑠𝑝+√𝐹1 3 ⁄𝑟𝑝)(√𝐹2 3 ⁄𝑦3−√𝑥𝑠𝑝)(√𝐹1 3 ⁄𝑧3+√𝑟𝑠𝑡𝑝+1) (i) On multiplying by {√𝐹2 3 ⁄𝑦𝑟𝑝(2√𝐹1 3 ⁄𝑧3𝑠𝑝𝑡𝑝)√𝑦𝑟𝑝+1𝑠𝑡√𝑡√𝑥3𝑠𝑝√𝐹2 3 ⁄𝑦3√𝐹1 3 ⁄𝑧3} we get {(2𝐹𝑦2𝑧3)√(𝑟𝑠𝑡)𝑝+1√𝑥3𝑦𝑟𝑝𝑠𝑝𝑡} This will be irrational if x and y are coprime to r, s, and t. (ii) Also, on multiplying by {√𝐹2 3 ⁄𝑦𝑟𝑝(𝑠𝑝𝑡𝑝)√𝑦𝑟𝑝+1𝑠𝑡√𝑡√𝐹1 3 ⁄𝑟𝑝(−√𝑥𝑠𝑝)√𝑟𝑠𝑡𝑝+1} we get {−(𝑦𝑟𝑝𝑠𝑝𝑡𝑝+1)√(𝑟𝑠𝑡)𝑝+1√𝐹𝑥𝑟𝑠} Which will be irrational, since we have defined F = (Ryz3rs) and √𝐹𝑥𝑟𝑠=√(𝑅𝑦𝑧3𝑟𝑠)(𝑥𝑟𝑠)={√𝑅𝑦(𝑟𝑠)√𝑥𝑧3} Where R = y; √𝑥𝑧3 will be irrational since gcd(x, z3) = 1 and both x and z3 cannot simultaneously be squares. Sum of all rational terms in the LHS of equation (8) ={(𝑥3𝑠𝑝𝑡2𝑝+1)√𝐹𝑅𝑦𝑧3𝑟𝑠} (vide II term) +{(𝐹𝑥3𝑧6)√𝑡𝑝+1√𝑅𝑦} −{(𝐹𝑧3√𝑅𝑦)√𝑡𝑝+1(𝑥3𝑧3+𝑟𝑝𝑡𝑝)} (vide IV term) ={(𝑥3𝑠𝑝𝑡2𝑝+1)√𝐹𝑅𝑦𝑧3𝑟𝑠} −{(𝐹𝑧3𝑟𝑝𝑡𝑝)√𝑡𝑝+1√𝑅𝑦} Sum of all rational terms in the RHS of equation (8) ={(𝑥3𝑟𝑝𝑠𝑝𝑡𝑝+1)√𝐹𝑅𝑦𝑧3𝑟𝑠} (Adding I to IV terms) +{(𝑥3𝑠2𝑝𝑡𝑝+1)√𝐹𝑅𝑦𝑧3𝑟𝑠} (vide V term) +{(𝐹𝑧3𝑟𝑝𝑠𝑝)√𝑡𝑝+1√𝑅𝑦} −{(𝐹𝑧3𝑟2𝑝)√𝑡𝑝+1√𝑅𝑦} (vide II term) ={(𝑥3𝑠𝑝𝑡2𝑝+1)√𝐹𝑅𝑦𝑧3𝑠} (∵ 𝑟𝑝+𝑠𝑝 =𝑡𝑝) +{(𝐹𝑧3𝑟𝑝)√𝑡𝑝+1√𝑅𝑦} (𝑠𝑝−𝑟𝑝) Equating the rational term on both sides, we get {(𝐹𝑧3𝑟𝑝√𝑅𝑦√𝑡𝑝+1)(𝑡𝑝+𝑠𝑝−𝑟𝑝)}=0 (2𝐹𝑧3𝑟𝑝𝑠𝑝)√𝑅𝑦√𝑡𝑝+1 =0 That is, either r = 0 or s = 0 or t = 0. ( Q F = Ryz3rs) This contradicts our hypothesis that all r, s and t are nonzero integers in the equation rp + sp = tp and proves that only a trivial solution exists. III. CONCLUSIONS Equation (8) was derived from the two transformation equations by substituting the equivalent values of rp, sp & tp in Fermat’s equation rp + sp = tp. The central hypothesis we made in the proof, namely that r, s, and t are non-zero integers, has been shattered by the result r = 0; thus, we are proving the theorem. DECLARATION STATEMENT Some of the references cited are older, noted explicitly as [1], [2], [3] and [4]. However, these works remain significant for the current study, as they are pioneering in their fields. I must verify the accuracy of the following information as the article's author. ▪ Conflicts of Interest/ Competing Interests: Based on my understanding, this article has no conflicts of interest. ▪ Funding Support: This article has not been funded by any organizations or agencies. This independence ensures that the research is conducted with objectivity and without any external influence. ▪ Ethical Approval and Consent to Participate: The content of this article does not necessitate ethical approval or consent to participate with supporting documentation. ▪ Data Access Statement and Material Availability: The adequate resources of this article are publicly accessible. ▪ Author's Contributions: The authorship of this article is contributed solely by the author. REFERENCES 1. Hardy G. H. and Wright E. M., An introduction to the theory of numbers, 6th ed. Oxford University Press, 2008, pp. 261-586. DOI: https://dx.doi.org/10.1080/00107510903084414, works remain significant, see declaration 2. Lawrence C. Washington, Elliptic Curves, Number Theory and Cryptography, 2nd ed. 2003, pp. 445-448. DOI: https://doi.org/10.1201/9781420071474, works remain significant, see declaration 3. Andrew Wiles, Modular Elliptic Curves and Fermat's Last Theorem, Annals of Mathematics, 1995; 141(3); pp.443-551.
Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 69 Retrieval Number:100.1/ijam.B121705021025 DOI: 10.54105/ijam.B1217.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. DOI: https://doi.org/10.2307/2118559, works remain significant, see declaration 4. 13 Lectures on Fermat's Last Theorem by Paulo Ribenboim, Publisher: Springer, New York, originally published in 1979, pages 159. DOI: https://doi.org/10.1007/978-1-4684-9342-9., works remain significant, see declaration AUTHOR’S PROFILE P.N. Seetharaman, B.Sc. (Mathematics); B.E. (Electrical Engineering), is a retired Executive Engineer from the Tamil Nadu Electricity Board. He had served in Mettur Tunnel Hydro Power Station for ten years, and finally worked in the Research and Development wing, Energy Conservation Cell, at Chennai. He retired from service in 2002. After retirement, he studied Number Theory, especially Fermat's Last Theorem and worked on finding an elementary proof for the Theorem. Disclaimer/Publisher’s Note: The statements, opinions, and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of the Lattice Science Publication (LSP)/journal and/or the editor(s). The Lattice Science Publication (LSP)/ journal and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions, or products referred to in the content.