Products of Odd Numbers or Prime Number Can Generate the Three Members’ Families of Fermat Last Theorem and the Theorem Is Valid for Summation of Squares of More Than Two Natural Numbers

Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that A n + B n = C n , in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A 2 + B 2 + C 2 + D 2 + so on = 2 n A where all are natural numbers.


Introduction
The Pythagorean equation, x 2 + y 2 = z 2 , has an infinite number of positive integer solutions for x, y, and z; these solutions are known as Pythagorean triples (with the simplest example 3 2 + 4 2 = 5 2 ).Around 1637, Fermat wrote in the margin of a book that the more general equation x n + y n = z n , had no solutions in positive integers if n is an integer greater than 2. In theory, this statement is known as Fermat's Last Theorem (it is also called as Fermat's conjecture before 1995).The cases n = 1 and n = 2 have been known from Pythagoras time having infinite solutions [1]  1637.It was written in the margin of a copy of Arithmetica.Fermat claimed that he had a proof and due length of the calculation, he was unable to fit in the margin of the copy.However, after his death no document was found to substantiate his claim.Consequently, the proposition became as a conjecture rather than a theorem.After 358 years of effort by mathematicians, the first successful proof was completed in 1994 by Andrew Wiles and formally published in 1995.
Here are some simple calculations to find A, B, C from product of any two odd numbers.
As theorem Let it is assumed that A is product of two odd numbers X and Y.
where X > Y (Y may be 1 whether A is prime or not).
Thus, any product two natural odd numbers can generate Fermat triplet (FT).
When A is prime number then A can be presented as 1 × A.
So, all-natural number (NN) those may be prime or compound odd members can generate Fermat triplet (FT).
For simplest example: So, members are ( ) Here are examples with prime numbers (A) which is always odd (Table 1).
For the cases of non-prime numbers and product of odd numbers, Table 2 shows the illustration to generate F Ts.
Any set of three (A, B, C) when multiplied with square of any number, then it generates another FT.
Fermat conjecture cannot be limited to three number (A, B, C) it can be expanded to any numbers of family.Illustration is given below we like to extend the relation to  where all are natural numbers.
As per Fermat triplet (if there is no common factor) A or B will be even and other one will be odd so that C will be odd number.A B C + = as per Equation (1).
Equation (1) will be incorporated by another term say D generating Equation

C E D E D E D
C 2 may be expressed as where F and G both must be odd as C is odd.
Then as per Equation ( 7) it can be expressed as and Thus, Equation ( 5) This expression generate Fermat quartet.
This principle may be used any number of terms for the Fermat expression.
As example Henceforth  On application shake this relation can generate a new class mathematical formalism for application of Fermat theorem.Most of the time the application of basic mathematics come late and it is expected that this extension will introduce a new class of cryptography which is under study.

Conclusions
Any products of odd numbers or prime number (p) (which is product of p and 1) can generate family of Fermat last theorem.Fermat last theorem was expressed with three natural numbers.
It has been shown that n [2] [3].The proposition was first stated as a theorem by Pierre de Fermat around How to cite this paper: Pramanik, S., Das, D.K. and Pramanik, P. (2023) Products of Odd Numbers or Prime Number Can Generate the Three Members' Families of Fermat Last Theorem and the Theorem Is Valid for Summation of Squares of More Than Two Natural Numbers.Advances in Pure Mathematics, 13, 635-641.https://doi.org/10.4236/apm.2023.1310043 term is not acceptable as both factors have same values because (D − C) and (D + C) should have different values, so (D − C) = 5 and (D + C) = 125.S. Pramanik et al.
not acceptable as both the terms are equal.Let second one is chosen (25 × 169).So, (F − E) = 25 and (F + E) it can generate Fermat family of any number, so possible n = 2 and A, B, C etc and are natural numbers.
B, C etc are natural numbers.It is an extension of Fermat relation.
To get B and C as integers X and Y should be odd numbers as (2 This implies S. Pramanik et al.DOI: 10.4236/apm.2023.1310043637 Advances in Pure Mathematics

Table 1 .
Generation of FT from prime numbers.

Table 2 .
Generation of FT from products of odd numbers.
where all are integers.This is a family of 4 numbers Fermat Quartet (A, B, D, E)