An Elementary Proof for Fermat’s Last Theorem using Three Distinct Odd Primes F, E and R
P. N. Seetharaman
P.N Seetharaman, Retired Executive Engineer, Energy Conservation Cell), Tamil Nadu State Electricity Board, Anna Salai, Chennai (Tamil Nadu), India.
Manuscript received on 09 February 2025 | First Revised Manuscript received on 13 February 2025 | Second Revised Manuscript received on 20 March 2025 | Manuscript Accepted on 15 April 2025 | Manuscript published on 30 April 2025 | PP: 22-26 | Volume-5 Issue-1, April 2025 | Retrieval Number: 100.1/ijam.A119105010425 | DOI: 10.54105/ijam.A1191.05010425
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Abstract: In number theory, Fermat’s Last Theorem states that no three positive integers a, b and c satisfy the equation a n + b n = c n where n is any integer > 2. Fermat and Euler had already proved that there are no integral solutions to the equations x 3 + y3 = z3 and x4 + y4 = z4 . Hence, it would suffice to prove the theorem for the index n = p, where p is any prime greater than 3. In this proof, we have hypothesised that r, s, and t are positive integers in the equation rp + sp = tp, where p is any prime greater than 3, and we prove the theorem using the method of contradiction. We have used auxiliary equations x³ + y³ = z³, along with the primary equation rp + sp = tp, which are connected using a transformation equation through the parameters. Solving the transformation equations, we get the result rst = 0, showing that only a trivial solution exists in the central equation.
Keywords: Transformation Equations Two Fermat’s Equations. 2010 Mathematics Subject Classification 2010: 11A–XX.
Scope of the Article: Applied Mathematics