An Elementary Procedure in the Proof of Fermat's Last Theorem

Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 95 Retrieval Number:100.1/ijam.A122706010426 DOI: 10.54105/ijam.A1227.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. An Elementary Procedure in the 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 wrote the theorem in the margin of a copy of Arithmetica. His proof is available only for the equation a4 + b4 = c4 for the exponent n = 4. Subsequently, Euler proved the theorem in the equation a3 + b3 = c3 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, x3 + y3 = 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 An + Bn = Cn, where n is an integer greater than 2. He stated that he himself had found a marvelous 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 has developed leaps and bounds 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 30 September 2025 | Revised Manuscript received on 07 October 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: 00000002-4615-1280 © 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 rp +sp =tp Here, p is any prime >3. Clearly, gcd (r,s,t) = 1, and we establish a centre contradiction in this proof. Any two of the variables r, s and t cannot simultaneously be squares. B. We include the auxiliary equationx3 +y3 =z3in this proof, in which we can have both x and y to be positive integers; z3 will be a positive integer; both z and z2 irrational. As gcd(xyz3) = 1. xy will be foolish, since both x and y cannot simultaneously be squares. C. We have defined F, R as positive odd primes each coprime to each x, y, z3, r, s, & t and E = (xyz3rt)3. D. We can have r, s and t some other odd prime factors coprime to x, y and z3. Proof. By random experiment, we have created the following equations. ()()() 2 2 2 3 1/ 3 1/ 3 1/ 3 5/3 a z b F c E d R e xr f E+ + + = + and ()()() 2 2 2 5/3p p p a t b ys c r d t e F f s− + − = − (1) as the transformation equations of x3 + y3 = z3 and rp + sp = tp respectively, through the parameters called a, b, c, d, e and f. Here, F and R are distinct odd primes, each coprime to x, y, z3, r, s, and t, and E = (xyz3rt)3. From equation (1), we get 3 1/ 3 3 a z b F x+= (2) pp a t b ys r−= (3) 1/ 3 1/ 3 3 c E d R y+= (4) pp c r d t s−= (5) 5/ 3 3 e xr f E z+= (6) and 5/ 3 pp e F f s t−= (7) Solving simultaneously (2) and (3), (4) and (5), (6) and (7), we get An Elementary Procedure in the Proof of Fermat's Last Theorem 96 Retrieval Number:100.1/ijam.A122706010426 DOI: 10.54105/ijam.A1227.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. ()() 3 1/ 3 3 1/ 3pp a x ys F r yz s F t= + + ()() 3 3 3 1/ 3p p p b x t r z yz s F t= − + ()() 3 1/ 3 1/ 3 1/ 3pp c y t R s E t R r= + + ()() 3 1/ 3 1/ 3 1/ 3p p p d y r E s E t R r= − + () ( ) 5/ 3 3 5/ 3p p p e z s E t xrs FE  = + +   And () ( ) 5/ 3 5/ 3 3 pp f F z xrt xrs FE  = − +   From (3) & (7), we get ()() ( ) 5/ 3p p p p t t r b ys e F f s a = + − ( ) ( ) ( ) ( )   ( ) 5/ 3 5/ 3 1 i.e., p p p p p t e F r f r s be F ys bf ys a + = − + − From (3) & (5), we have ()() ( ) p p p p r r a t b ys s d t c = − + ( ) ( ) ( ) ( )   ( ) 11 i.e., p p p p p r a s t ad t b ys bd yst c ++ = + − − From (5) & (7), we get ()() ( ) 5/ 3p p p p s s c r d t e F t f = − − ( ) ( ) ( ) ( )   ( ) 5/ 3 5/ 3 1 i.e., p p p p p s ce F r c r t de F t d t f + = − − + Substituting the above equivalent values of tp, rp and sp in the Fermat’s equation rp + sp = tp after multiplying both sides by   acf , we get   ( ) ( ) ( ) ( )   5/ 3 5/3 1p p p p cf e F r f r s be F ys bf ys + − + − ( ) ( ) ( ) ( )   11 () p p p p af a s t ad t b ys bd yst ++ = + − − ( ) ( ) ( ) ( ) ( )   5/ 3 5/ 3 1p p p p ac ce F r c r t de F t d t + + − − + (8) Our aim is to compute all rational terms in equation (8) after multiplying both sides by ()() ( ) 2 22 5/ 3 3 1/ 3 1/ 3 1/ 3p p p yz s F t E t R r xrs FE    + + +      for freeing from denominators on the parameters a, b, c, d, e and f and again multiplying both sides by 3 xz st for getting some rational terms. I term in LHS of equation (8), after multiplying by the respective terms and substituting for {c(ef )} ( ) ( )  () 5/ 3 3 1/ 3 1/ 3 3 1/ 3 1/ 3 2 p p p p F r yz s F t F yz st E t R r= + + + ()()() 3 3 1/ 3 3 5/ 3 5/ 3 3p p p p xz st y t R s z s E t F z xrt + + − (i) On multiplying by ()()   5/ 3 1/ 3 3 1/ 3 3 3 5/ 3 2 p p p p F r F yz st E t xz st y t E t xrt− We get ( ) ( )   1 2 3 1 2p p FExy z st rt + + − (ii) Also, on multiplying by ( )   5/ 3 1/ 3 1/ 3 3 3 3 5/ 3 3p p p F r F t E t xz st y t z s F z We get Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 97 Retrieval Number:100.1/ijam.A122706010426 DOI: 10.54105/ijam.A1227.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. ( )   2 3 1 1 1/3 3 3p p p F z t s E xy z r t ++ Which is rational. II term in LHS of equation (8), after multiplying by the respective terms and substituting for {cf 2} () ( ) ( ) ()   () 3 1/ 3 1/ 3 3 1/ 3 1/ 3 3 2 p p p p p r s yz s F t F yz st E t R r xz st= − + + + () ( ) ( )   3 1/ 3 5/ 3 3 5/ 3 3 2 p p p y t R s F z xrt F xz rt + + − On multiplying by () ( ) ( )     3 2 3 1/ 3 3 3p p p p r s xyz rst F t z E t xz st y t−+ We get () ( ) ( )   1 1/ 3 3 3 3 2 3p p p p t s E xy z r t xyz rst F t z + −+  Which will be rational since we have defined E1/3 = (xyz3rt). III term in LHS of equation (8), after multiplying by the respective terms and substituting for {bc(ef)} ()()() 5/ 3 3 1/ 3 1/ 3 1/ 3 3 3 3p p p p F ys yz s F t E t R r xz st x t r z= + + − ()()() 3 1/ 3 3 5/ 3 5/3 3p p p p y t R s z s E t F z xrt + + − On multiplying by ()()   5/ 3 1/ 3 1/ 3 3 3 3 5/ 3p p p p F ys F t E t xz st r z y t E t xrt−− We get ( ) ( )   1 2 3 1 p p FExy z st rt + + IV term in LHS of equation (8), after multiplying by the respective terms and substituting for {bcf2)} ()()() 1 3 1/3 1/ 3 1/ 3 3p p p ys yz s F t E t R r xz st + = − + + ()() ( ) ( )   3 3 3 1/ 3 5/ 3 3 5/ 3 3 2 p p p p p x t r z y t R s F z xrt F xz rt − + + − On multiplying by () ( )   1 3 1/ 3 3 3 3p p p ys yz s E t xz st x t y t xrt + − We get ( ) ( )   1 3 2 3 1 1/ 3 p p x y z rst st E y + + − Which will be irrational since we have defined E1/3 = (xyz3rt). I term in RHS of equation (8), after multiplying by the respective terms and substituting for {a2f} ( ) ( )   1/ 3 1/ 3 1/ 3 1/ 3 3 2 p p p p s t E t R r E R r t xz st= + + ( ) ( ) ( )  () 5/ 3 3 1/ 3 1/ 3 3 5/ 3 3 2 p p p p xrs FE x ys F r F x yr s F z xrt   + + + −   On multiplying by ( ) () 1/ 3 3 1/ 3 3 5/ 3 3 2 p p p p s t E t xrs xz st F x yr s F z we get ( ) ( )   1 1/ 3 2 3 1 2p p FE x z s t rt xy + + Which is irrational. II term in RHS of equation (8), after multiplying by the respective terms and substituting for {(a2df} An Elementary Procedure in the Proof of Fermat's Last Theorem 98 Retrieval Number:100.1/ijam.A122706010426 DOI: 10.54105/ijam.A1227.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. () ( ) 5/ 3 1 1/ 3 1/ 3 3p p p t E t R r xrs FE xz st + = + +   ( ) ( )  ()() 3 1/ 3 1/3 3 3 1/ 3 5/ 3 3 2 p p p p p x ys F r F x yr s y r E s F z xrt + + − − (i) On multiplying by ( ) ( )   5/ 3 1 1/ 3 3 1/ 3 3 5/ 3 3p p p t E t FE xz st F r y r F z + we get ( )   2 3 1 3p p p F Er z t t xy r s + (ii) Also, on multiplying by ()()   1 1/ 3 3 1/ 3 3 1/ 3 5/ 3 3 2 p p p p t E t xrs xz st F x yr s E s F z +− we get ( ) ( )   1 1/ 3 2 3 1 2p p FE x z s t rt xy + + − Which will be irrational, since both x and y cannot simultaneously be squares, since gcd(x, y) = 1 in the equation x3 + y3 = z3. III term in RHS of equation (8), after multiplying by the respective terms and substituting for {(ab)f} () ( ) ( ) ( )   5/ 3 1 1/ 3 1/ 3 1/ 3 1/ 3 3 2 p p p ys xrs FE E t R r E R r xz st + = − + + +   ()()() 3 1/ 3 3 3 5/ 3 3p p p p x ys F r x t r z F z xrt + − − On multiplying by () ( )   1 1/ 3 3 1/ 3 3 5/ 3 3p p p p ys xrs E t xz st F r x t F z + − we get ( ) ( )   1 1/ 3 2 3 1 p p FE x z s t rt xy + + Which is irrational. IV term in RHS of equation (8), after multiplying by the respective terms and substituting for {(ab)df} ( ) () ( ) 5/ 3 1/ 3 1/ 3 3pp yst E t R r xrs FE xz st  = − + +   ()()()() 3 1/ 3 3 3 3 1/ 3 5/ 3 3p p p p p p x ys F r x t r z y r E s F z xrt + − − − (i) On multiplying by ( ) ( ) ()()   5/ 3 1/ 3 3 1/ 3 3 3p p p p yst E t FE xz st F r r z y r xrt− − − we get ( ) ( )   1 23 p p FExy z r st rt + − (ii) Also, on multiplying by ( ) ()()   1/ 3 3 3 3 3p p p p yst E t xrs xz st x ys r z y r xrt− − − we get ( ) ( )   1 3 2 3 1 1/ 3 p p x y z r st st E y + + Which is irrational. V term in RHS of equation (8), after multiplying by the respective terms and substituting for {ac2e} () ( ) 5/ 3 5/ 3 3 1/ 3 3p p p F r yz s F t xrs FE xz st  = + +   Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 99 Retrieval Number:100.1/ijam.A122706010426 DOI: 10.54105/ijam.A1227.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. () ( ) ( )  () 3 1/ 3 3 1/ 3 1/ 3 3 3 5/ 3 2 p p p p p x ys F r y t R s R y s t z s E t + + + + On multiplying by ( )   5/ 3 1/ 3 3 3 3 3p p p p F r F t xrs xz st x ys y t z s we get ( ) ( )   1 2 3 3 1 p p Fx y z s t rt xy + + Which is irrational. VI term in RHS of equation (8), after multiplying by the respective terms and substituting for {ac2} ()() ( ) ( ) ( )   5/ 3 5 / 3 3 1/ 3 3 2 p p p p p r t yz s F r xrs FE FE xrs xz st= − + + + () ( ) ( )   3 1/ 3 3 1/ 3 1/ 3 3 2 p p p x ys F r y t R s R y s t − + + On multiplying by () ( ) ( )   3 3 3 3p p p r t yz s xrs xz st x ys y t− we get ( )   3 4 3 1 1p p p x y z rs t r s ++ − Which will be irrational, since both r & s cannot simultaneously be squares. VII term in RHS of equation (8), after multiplying by the respective terms and substituting for {a(cd)e} =(−√𝐹5/3𝑡)(√𝑦𝑧3𝑠+√𝐹1/3𝑡𝑝)(√𝑥𝑟𝑠𝑝+√(𝐹𝐸)5/3)√𝑥𝑧3𝑠𝑡 ()()()() 3 1/ 3 3 1/ 3 3 1/ 3 3 5/ 3p p p p p p x ys F r y t R s y r E s z s E t + + − + On multiplying by {(−√𝐹5/3𝑡)√𝑦𝑧3𝑠√𝑥𝑟𝑠𝑝√𝑥𝑧3𝑠𝑡√𝐹1/3𝑟𝑝√𝑦3𝑡(−√𝐸1/3𝑠𝑝)√𝐸5/3𝑡𝑝} we get ( ) ( )   1 2 3 1 p p FExy z s t rt + + Which is rational. VIII term in RHS of equation (8), after multiplying by the respective terms and substituting for {a(cd)} () ( ) ( ) ( )   5/ 3 5/ 3 1 3 1/ 3 3 2 p p p p t yz s F t xrs FE FE xrs xz st + = + + + ()()() 3 1/ 3 3 1/ 3 3 1/ 3p p p p x ys F r y t R s y r E s + + − (i) On multiplying by ( ) ()   1 3 3 3 3 1/ 3p p p t yz s xrs xz st x ys y t E s +− we get ( ) ( )   1 3 2 3 1 1/ 3 p p x y z rs t st E y + + − Which will be irrational, since we have defined E1/3 = (xyz3rt). (ii) Also, on multiplying by ( ) () 5/ 3 1 3 3 1/ 3 3 1/ 3 2 p p p p t yz s FE xrs xz st F r y t E s +   −     we get ( ) ( )   1 2 3 1 2p p FExy z s t rt + + − Which will be rational. Sum of all rational terms in the LHS of equation (8) ( )   3 1 1 1/ 3 3p p p xyz rst s E xyz r t ++ =− (combining I & II terms) An Elementary Procedure in the Proof of Fermat's Last Theorem 100 Retrieval Number:100.1/ijam.A122706010426 DOI: 10.54105/ijam.A1227.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. ( ) ( )   1 2 3 1 p p FExy z st rt + + − (combining I & III terms) Sum of all rational terms in the RHS of equation (8) ( ) ( )   1 23 p p FExy z r st rt + =− (vide IV term) ( ) ( )   1 2 3 1 p p FExy z s t rt + + + (vide VII term) ( ) ( )   1 2 3 1 2p p FExy z s t rt + + − ( ) ( ) ( ) 1 23 ppp FExy z st rt r s + = − + ( ) ( ) ( ) 1 2 3 1 p p p p p FExy z st rt r s t + + = − + = Equating the rational term on both sides, we get ( )( ) 3 1 1 1/ 3 3 0 p p p xyz rst s E xyz r t ++ −= Dividing both sides by ( ) 3 xyz− and substituting for E1/3 = xyz3rt We get ( ) ( )   1 1 3 1 0 p p p rst s xyz t r + + + = That is, either r = 0 or s = 0 or t = 0. This contradicts our hypothesis that all r, s and t are nonzero integers in the Fermat equation rp + sp = tp, which 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 rst = 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. 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. 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, initially published in 1979, page 159. DOI: https://doi.org/10.1007/978-1-4684-9342-9, works remain significant, see declaration SUPPLEMENTARY DATA In the II term in the RHS of equation (8) (iii) Again on multiplying by ( ) ()   1 1/ 3 3 3 3p p p p t E t xrs xz st x ys y r xrt +− We get the rational term. ( )   4 2 1 1 1/ 3 3p p p x y rst s E xyz r t ++ − This algebraically cancels with the rational term worked out below under the IV term in the RHS of equation (8). In the IV term in the RHS of equation (8) (iii) Again on multiplying by ( ) ()   1/ 3 3 3 3 3p p p p yst E t xrs xz st x ys x t y r xrt−− We get ( )   4 2 1 1 1/ 3 3p p p x y rst s E xyz r t ++ In the VII term in the RHS of equation (8) (ii) On multiplying by () ( ) ()   5/ 3 5/ 3 1/ 3 3 1/ 3 3 1/ 3 5/ 3p p p p F t F t FE xz st F r y t E s E t−− we get the rational term [E=(xyz3rt)3] This term gets cancelled algebraically with the rational term worked out under the VIII term in the RHS below. In the VIII term in the RHS of equation (8) ( )   2 5/3 1 1 1/ 3 3 3p p p F E t s E xy z r t ++ Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 101 Retrieval Number:100.1/ijam.A122706010426 DOI: 10.54105/ijam.A1227.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. (iii) Again on multiplying by ( ) ()   5/ 3 1 1/ 3 3 1/ 3 3 1/ 3p p p p t F t FE xz st F r y t E s +− We get the rational term. ( )   2 5/ 3 1 1 1/ 3 3 3p p p F E t s E xy z r t ++ − 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 at the Mettur Tunnel Hydro Power Station for 10 years and later worked in the Research and Development wing of the Energy Conservation Cell in 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.