Year: 2025
Type: article
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 = c 4 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, x 3 + y3 = z3 , and combine the two equations using transformation equations. Solving the transformation equations, we establish a contradiction, thereby proving the theorem.
Authors: Retired Executive Engineer, Energy Conservation Cell, Tamil Nadu State Electricity Board, Anna Salai, Chennai (Tamil Nadu), India., P.N. Seetharaman
Institution: Government of Tamil Nadu
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4415653539