An Elementary Chapter in Number Theory: Proof of Fermat’s Last Theorem

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 said 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 integerssatisfying 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.