Proof of Fermat's Last Theorem for specific exponents

Fermat's Last Theorem is a theorem in number theory, originally stated by Pierre de Fermat in 1637 and proved by Andrew Wiles in 1995. The statement of the theorem involves an integer exponent n larger than 2. In the centuries following the initial statement of the result and its general proof, various proofs were devised for particular values of the exponent n. Several of these proofs are described below, including Fermat's proof in the case n = 4, which is an early example of the method of infinite descent.

Fermat's Last Theorem states that no three positive integers (a, b, c) can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than two. (For n equal to 1, the equation is a linear equation and has a solution for every possible a, b. For n equal to 2, the equation has infinitely many solutions, the Pythagorean triples.)

A solution (a, b, c) for a given n leads to a solution for all the factors of n: if h is a factor of n then there is an integer g such that n = gh. Then (a^g, b^g, c^g) is a solution for the exponent h:

Therefore, to prove that Fermat's equation has no solutions for n > 2, it suffices to prove that it has no solutions for n = 4 and for all odd primes p.

For any such odd exponent p, every positive-integer solution of the equation a^p + b^p = c^p corresponds to a general integer solution to the equation a^p + b^p + c^p = 0. For example, if (3, 5, 8) solves the first equation, then (3, 5, −8) solves the second. Conversely, any solution of the second equation corresponds to a solution to the first. The second equation is sometimes useful because it makes the symmetry between the three variables a, b and c more apparent.

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