Published April 30, 2025 | Version CC-BY-NC-ND 4.0
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An Elementary Proof for Fermat's Last Theorem using a Transformation Equation to Fermat's Equation

Creators

  • 1. Retired Executive Engineer, Energy Conservation Cell), Tamil Nadu State Electricity Board, Anna Salai, Chennai (Tamil Nadu), India.

Description

Abstract: Fermat’s Last Theorem states that there are no positive integers x, y and z satisfying the equation x n + y n = z n , where n is any integer > 2. Around 1637 Fermat proved that there are non-zero solutions to the above equation with n = 4. In the 18th century Euler treated the case n = 3, thereby reducing the proof for the case of a prime exponent ≥ 5 in this proof we hypothesize that r, s and t are positive integers satisfying the equation rp + sp = tp , where p is any prime >3 and establish a contradiction. We use an Auxiliary equation x 3 + y3 = z3 and create transformation equations. Solving the transformation equations we prove that only a trivial solution exists in the main equation r p + sp = tp .

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Additional details

Identifiers

DOI
10.54105/ijam.A1192.05010425
EISSN
2582-8932

Dates

Accepted
2025-04-15
Manuscript received on 17 February 2025 | First Revised Manuscript received on 19 February 2025 | Second Revised Manuscript received on 21 March 2025 | Manuscript Accepted on 15 April 2025 | Manuscript published on 30 April 2025.

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