Fermat’s Last Theorem states that the equation a^n+b^n=c^n does not have any solutions for any positive integers a, b, and c except n=1 and n=2. For n=1 it is just a simple addition of two integers. For n=2 it is the Pythagorean Theorem which was proved by the School of Pythagoras. For n=0 a^0+b^0=1+1=2 which is not equal to c^0=1. All other positive integers were proven impossible solutions by Andrew Wiles in 1994.
Many false proofs of the “theorem” were submitted over the years. But many helpful ideas were discovered toward the proof of Fermat’s Last Theorem. One of them was the fact that a, b, and c must be relatively prime, meaning they do not have any common divisors except one and themselves. Wieferich considered exponents that do not divide a, b, or c (n does not divide a, b, or c evenly). The female mathematician Sophie Germain proved the theorem holds using Wieferich’s idea for any prime n that is also prime when multiplied by two with one added (2n+1). The famous mathematician Legendre proved that the theorem holds also using Wieferich’s idea for any prime n that is also prime for forms 4n+1, 8n+1, 10n+1, 14n+1, and 16n+1. It was also proven that only irregular primes can be a solution to the theorem. It was proven by computers before modern computers that irregular primes up to four million are not a solution.
Fermat’s Last Theorem was not a proven theorem; it was just a conjecture because Pierre Fermat just wrote he had found a proof of the equation in the margin of his copy of the book “Arithmetica” by the ancient Greek Diophantus in 1637. The equation a^n+b^n=c^n is one of the Diophantine equations discovered by Diophantus.
Fermat’s proof of n=4 as an impossible solution is the only part of the proof that was found. This left only the prime numbers as possible solutions. The prime numbers 3, 5, and 7 were the only ones proven during the next two centuries. Euler proved that n=3 is an impossible solution to the equation a^n+b^n=c^n. Both Euler and Fermat used the method of infinite descent for their proofs. The case for n=5 was proven independently by Legendre and Dirichlet in 1825. The case for n=7 was proven in 1839 by Gabriel Lame.
Yves Hellegouarch came up with an elliptic curve that cold be used to prove Fermat’s Last Theorem. The equation he discovered is y^2=x(x-a^n)(x+b^n). Gerhard Frey made a suggestion in 1984 that the modularity theorem (then called the Taniyama Shimura Conjecture) for Hellegouarch’s elliptic curves could be used to prove Fermat’s Last Theorem. Andrew Wiles, with the help of Richard Taylor, made a proof of the “theorem” in 1994 using the suggested modularity Theorem. He used Ribet’s proof of the Epsilon Conjecture in 1986.