A Framework For Fermat's Marvellous Proof

A Framework For Fermat's Marvellous Proof
Pythagorean triples are integer solutions to the equation:

a²+b²=c²

If u and v are positive integers, with v>u and a=v²-u², b=2uv, and c=v²+u² provides integer solutions to a²+b²=c²

It is easily possible to demonstrate that this solution can be generalized in the following manner.

a²+b²=c² can be rewritten as:

b²=c²-a²

and further to

b²=(c+a)(c-a)

Consider then integer solutions to the more general equation (where n is a positive integer):

bⁿ=(c+a)^n-1(c-a)

If u and v are positive integers, with v>u and b=2uv^n-1, c=vⁿ+uⁿ, and a=vⁿ-uⁿ provides integer solutions to bⁿ=(c+a)^n-1(c-a)

Proof:

bⁿ=(c+a)^n-1(c-a)
(2uv^n-1)ⁿ=((vⁿ+uⁿ)+(vⁿ-uⁿ))^n-1((vⁿ+uⁿ)-(vⁿ-uⁿ))
2ⁿuⁿv^n²-n=(2vⁿ)^n-1(2uⁿ)
2ⁿuⁿv^n²-n=2^n-1v^n²-n2ⁿuⁿ
2ⁿuⁿv^n²-n=2ⁿuⁿv^n²-n

The more general equation, bⁿ=(c+a)(c-a)^n-1, can also be demonstrated in a similar manner if the conditions are the same as above except that b=2u^n-1v.

Now, Fermat's Last Theorem can be written as:

bⁿ=cⁿ-aⁿ

and further to

bⁿ=(a⁰c^n-1+ac^n-2...+a^n-2c+a^n-1c⁰)(c-a)

For all n > 2, c^n-1 and a^n-1, for any combination of v and u, will produce non-cancellable intermediate terms and, thus, no integer solutions for this form.