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:
For odd p
For even p
Some of the equations:
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 whycan never be rational in the form of root w and variable s.
Last edited by Stangerzv (2012-04-22 11:58:59)