I am writing a paper on this subject as I have developed Sums of Power for arithmetic progression. Setting n=2, the generalized equation reduces to polynomials of stepping down 2nd power. It looks like Kummer's cyclotomic expression but a slightly different form. This form had been used by Euler for p=3 and Sophie Germain for p=5, they got it by substitution of variables, it was not known during their time that there is a generalized equation that can describe the same form for any p. 

There are two simplified forms for these equations:

Consider Fermat's Last Theorem Equation as follows:

Therefore:

For odd p

For even p

Where:

and

Some of the equations:

p=2

p=3

p=4

p=5

Maybe it could offer an alternative proof for Fermat's Last Theorem. I had tried rational root theorem and substitution of solution of these forms:

or

Still got stumbled upon few steps. Maybe Galois theory might be used to explain why

can never be rational in the form of root w and variable s.

Last edited by Stangerzv (2012-04-23 09:58:59)