The Pythagorean equation, x² + y² = z², has an infinite number of positive integer solutions for x, y, and z; these solutions are known as Pythagorean triples (with the simplest example 3² + 4² = 5²). Around 1637, Fermat wrote in the margin of a book that the more general equation xⁿ + yⁿ = zⁿ, 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] [2] [3] .

The proposition was first stated as a theorem by Pierre de Fermat around 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. It was described as a “stunning advance in mathematics” in the citation for Wiles’s Abel Prize award in 2016 [2] . It also proved many parts of the Taniyama-Shimura conjecture. Afterward, it was defined as the modularity theorem, It opened up new approaches to numerous other problems and developed powerful technique known as modularity lifting in mathematics. It is among the most remarkable theorems in the history of mathematics Fermat himself proved the special case n = 4. Hilbert D (1897) elaborated some of the studies [3] . Alternative proofs of the case n = 4 were developed later by Frénicle de Bessy (1676) [4] , Leonhard Euler (1738) [5] , Kausler (1802) [6] , Peter Barlow (1811) [7] , Adrien-Marie Legendre (1830) [8] , Joseph Bertrand (1851) [9] , Victor Lebesgue (1853, 1859, 1862) [10] , Tafelmacher (1893) [11] , Gambioli (1901) [12] , Bang (1905) [13] , Sommer (1907) [14] , Nutzhorn (1912) [15] , Robert Carmichael (1913) [16] , Hancock (1931) [17] , Grant and Perella (1999) [18] , and Barbara (2007) [19] . The conjecture was proved for only the primes 3, 5, and 7 during 1637 to 1839. In addition, some innovative proof was provided by Sophie Germain and that was very relevant to entire class of primes [20] . In mid-19^th century Ernst Kummer extended the analysis. He proved the theorem for all regular primes, leaving irregular primes which were analyzed separately [21] [22] . Based on Kummer’s work and using advanced computer, many mathematicians were able to extend the proof covering prime exponents up to four million [23] . It was understood that a proof for all exponents was unsolvable.

British mathematician Andrew Wiles showed that it is a special case of the modularity theorem for elliptic curves expressed in 170 pages in the year 1995.

For x = 2, it is preferred to define A, B, C as a member of Fermat triplet (FT). Here are some simple calculations to find A, B, C from product of any two odd numbers.

As theorem

A 2 + B 2 = C 2 (1)

→ Math_5# (2)

→ Math_7# (3)

Let it is assumed that A is product of two odd numbers X and Y. then A² = X² ∙ Y² as

A = X × Y (4)

where X > Y (Y may be 1 whether A is prime or not).

This implies

C + B = X 2 and C − B = Y 2

Thus

C = ( X 2 + Y 2 ) / 2 and B = ( X 2 − Y 2 ) / 2 (5)

To get B and C as integers X and Y should be odd numbers as ( X 2 + Y 2 ) and ( X 2 − Y 2 ) are divided by 2 to get B and C as integers.

On substitution of values of C and B in Equation (3)

We get,

[ ( X 2 + Y 2 ) / 2 ] 2 − [ ( X 2 − Y 2 ) / 2 ] 2

1 / 4 [ ( X 4 + Y 4 + 2 X 2 Y 2 ) − ( X 4 + Y 4 − 2 X 2 Y 2 ) ]

X 2 Y 2 = A 2 as per Equation (4).

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:

If NN = 3 then NN = 3 × 1.

So, members are ( 3 2 + 1 2 ) / 2 = 5 and ( 3 2 − 1 2 ) / 2 = 4 .

Thus 3 2 + 4 2 = 5 2 (FT or PT).

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

A 2 + B 2 = C 2

we like to extend the relation to

A 2 + B 2 + D 2 = E 2

where all are integers. This is a family of 4 numbers Fermat Quartet (A, B, D, E) 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 2 + B 2 = C 2 as per Equation (1).

Equation (1) will be incorporated by another term say D generating Equation (5)

Thus

A 2 + B 2 + D 2 = E 2 Math_25##Math_26# (6)

And

C 2 = E 2 − D 2 = ( E − D ) ( E + D ) (7)

C² may be expressed as

C 2 = F × G (8)

where F and G both must be odd as C is odd.

Then as per Equation (7) it can be expressed as F = E − D and G = E + D .

So, E = ( G + F ) / 2 and D = ( G − F ) / 2 .

Thus, Equation (5)

A 2 + B 2 + D 2 = E 2 or A 2 + B 2 + [ ( G − H ) / 2 ] 2 = [ ( G + H ) / 2 ] 2 .

This expression generate Fermat quartet.

This principle may be used any number of terms for the Fermat expression.

As example

7 2 + 24 2 = 25 2 (Three members family) (9)

Let me introduce the family of 4 members.

Thus 7 2 + 24 2 + C 2 = D 2 using Equation (9).

We get 25 2 = D 2 − C 2 or 625 = ( D − C ) ( D + C ) , 625 = 5 × 125 = 25 × 25 . Last 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.

Thus D = ( 5 + 125 ) / 2 = 65 and C = ( 125 − 5 ) / 2 = 60 .

Henceforth 7 2 + 24 2 + C 2 = D 2 .

So, 7 2 + 24 2 + 60 2 = 65 2 (Fermat quartet) as mentioned in Equation (5).

Again, for example of five members family (Fermat Quintet, it is expressed as

7 2 + 24 2 + 60 2 + E 2 = F 2

Or 7 2 + 24 2 + 60 2 = F 2 − E 2 = ( F − E ) ( F + E ) .

Or 4225 = ( F − E ) ( F + E ) .

Now 4225 = 5 × 845 = 25 × 169 = 13 × 325 = 65 × 65 .

Last factor is not acceptable as both the terms are equal.

Let second one is chosen (25 × 169).

So, (F − E) = 25 and (F + E) = 169.

Thus F = ( 25 + 169 ) / 2 = 97 and E = ( 169 − 25 ) / 2 = 72 .

Thus

7 2 + 24 2 + 60 2 + 72 2 = 97 2 (Fermat quintet)

If first factor is chosen as 5 × 845.

Then F = ( 845 + 5 ) / 2 = 425 and E = ( 845 − 5 ) / 2 = 420 .

Thus

7 2 + 24 2 + 60 2 + 420 2 = 425 2 (Fermat quintet)

With this principle it can generate Fermat family of any number, so

A n + B n + C n + D n + ⋯ = X n

is possible n = 2 and A, B, C etc and are natural numbers.

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.

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 A n + B n + C n + D n + ⋯ = X n is possible for n = 2 where A, B, C etc are natural numbers. It is an extension of Fermat relation.

This article is our tribute to Pierre de Fermat, a great mathematician of world. I express my thank to Dr Arindam Pramanik for preparing the manuscript and to Indian Statistical Institute, Kolkata, India for academic help We also recognize GLA Univesrsity, Mathura, India for technical help.

The authors declare no conflicts of interest regarding the publication of 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