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Extension for perfect prime. Consider this equation

When p=7,

No solution up to n=10,000

Extension for perfect prime. Consider this equation

When p=6,

No solution up to n=10,000

An approach to proof Fermat's Last Theorem.

Consider when n=3, and i=3, yields:

Let

andthen;

let

Then

this equation is solvable.

Now let n=2

Let

andthen

There is no whole number solution according to Euler.

List of the smallest prime for n=1 and p=[0-50]

Ps=-7,-5, 19, 73, 2179, 6553,..

New Formulation for alternating sums of power

Let the p-th power of an alternating arithmetic series as follows

The General Equations are given as follows:

For odd power:

For even power:

Where

Extension for perfect prime. Consider this equation

When p=5,

No solution up to n=10,000

Extension for perfect prime. Consider this equation

Where

Consider when p=4, n=8

Smallest solution

So far there is no solution for p=3 up to a=40,000

Extension for perfect prime. Consider this equation

Where,

is the n-th primeConsider when p=2, n=49;

Smallest solution

Smallest solution for Pt=19

Smallest solution for Pt=18

Smallest solution for Pt=17

Smallest solution for Pt=16

Smallest solution for Pt=15

Smallest solution for Pt=14

Smallest solution for Pt=13

Smallest solution for Pt=12

Smallest solution for Pt=11

Smallest solution for Pt=10

Next Solution

Next Solution

Smallest solution for Pt=9

Next Solution

Smallest solution for Pt=8