A Proof for Fermat’s Last Theorem using an Auxiliary Fermat’s Equation
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 01 October 2024 | Revised Manuscript received on 12 October 2024 | Manuscript Accepted on 15 October 2024 | Manuscript published on 30 October 2024 | PP: 19-24 | Volume-4 Issue-2, October 2024 | Retrieval Number: 100.1/ijam.A118205010425 | DOI: 10.54105/ijam.A1182.04021024
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© 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/)
Abstract: 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.
Keywords: Transformation Equations. Mathematics Subject Classification 2010: 11A–XX.
Scope of the Article: Applied Mathematics