1. Introduction

In the early time fermat, a French number theorist, had stated that the equation has no solution in integers, if n>2 [1]. The problem to find the exact value of z is enhanced as the number of terms increases rather than power two. Let us have n terms each of which has a common power p in the series then the series is called a p-series which is applied by the following theorem for p>2 which is also an answer for the claim of fermat.

Theorem 1. Let such that then

Where p=2t+1 for all p odd Natural numbers(N), p=2t+2 for all p even natural numbers.

Proof. Let such that if where then multiplying up to n times we get , therefor we can write this expression to p power as If the binomial theorem for expansion .

Can be written as

Where

,

Putting and dividing both sides by we get

(1)

The other way of writing is which can be written to p power as

If binomial theorem for expansion can be written as

Where

,

Putting in and dividing both sides by we get

(2)

Therefor, the summation of EQ(1) and EQ(2) can be put as follow

Where p=2t+1 for all p odd natural numbers, whereas p=2t+2 for all p even natural numbers

Let put and if be divided by such that (Real numbers) then we get the following result

This result led us to a conclusion that every couple terms which have common power can undergo this formula.