Fermat’s path to his last theorem

This explores some ideas I had about Fermat, how he may have thought of his famous last theorem, and I add some related conjectures.

Since Fermat was investigating polygonal numbers it was apparent that just as some squares were the sum of two other squares, so too some polygonal numbers are the sum of two other polygonal numbers of the same type. For example there are an infinite number of solutions where 2 triangular numbers sum to a third triangular number. These can be found from the following formula:

Equation (1)

If s is the number of sides in a polygon, the formula for the n^th s-gonal number is ${(s-2)n^2-(s-4)n}\over 2$ .

So generally for each value of s, there are an infinite number of solutions for each value of n. So there are an infinite number of solutions X_n1 +Y_n2 =Z_n3 where each has the same value of s. So then for r = 2 there are an infinite number of solutions where two triangular numbers add to a third.

Fermat also looked at higher powers and so it would have been natural to him to ask whether solutions like this occurred there. For example whether two tetrahedral numbers can add to a third tetrahedral number, and whether there are an infinite number of solutions. As it turns out solutions like these can be found relatively easily but he might have seen that the sum of two cubes is not findable as a cube by trial and error. So in time there would arise a conjecture, that there were solutions in all such figurate numbers except cubes and their higher powers. His last theorem would then be about this exception to the other solutions.

A figurate number is a number that can be represented as a regular and discrete geometric pattern (e.g. dots). If the pattern is polytopic, the figurate is labeled a polytopic number, and may be a polygonal number or a polyhedral number.

The first few triangular numbers can be built from rows of 1, 2, 3, 4, 5, and 6 items:

Equation (2)

The n-th regular r-topic number is given by the formula:

$P_r(n) = {{n + r - 1} \choose r} = {n^{(r)} \over {r!}} \quad \mbox{for} \ n \ge 1$

$r$ $!$ is the factorial of $r$ , $n \choose r$ is a binomial coefficient, and $n (r)$ is the rising factorial.

Polytopic numbers for r = 2, 3, and 4 are:

P₂(n) = 1/2 n(n + 1) (triangular numbers)
P₃(n) = 1/6 n(n + 1)(n + 2) (tetrahedral numbers)
P₄(n) = 1/24 n(n + 1)(n + 2)(n + 3) (pentatopic numbers)

On investigating higher powers Fermat likely noticed more solutions for polyhedral numbers A+B=C. For example:

Equation (3)

P₃ (n₁) + P₃ (n₂) = P₃ (n₃)

Is derived from equation (2).

This would describe a solution where 2 tetrahedral numbers add to a third tetrahedral number.

Fermat then may have proved his last theorem by examining solutions for equation (3) and showing why there are solutions for tetrahedrons but no solutions for cubes. It’s likely he would have initially thought there would be solutions for two cubes adding to a third because while there are solutions for two triangular numbers adding to a third, there are also solutions for two squares adding to a third. So because there are solutions for two tetrahedral numbers adding to a third, one might expect there to also be solutions for two cubes adding to a third.

So I would propose then a conjecture that Fermat may have thought of, which may lead to his original proof if he had one. This may be an old conjecture but I could not find it, and so I am proposing it here. Referring to this equation:

Equation (4)

P_r (n₁) + P_r (n₂) = P_r (n₃)

For any value of r greater than 3 in (4) are there solutions?
If there are solutions for a given r, are there an infinite number of solutions, and is there an identity or equation which gives some or all these solutions?
If there is an identity, is there a more general identity which gives solutions for more than one value of r, or even for all r?
Would such an identity or related proofs point to a proof for Fermat’s last theorem?
Does the same apply for all polyhedral numbers? For example two pentagonal numbers can add to a third pentagonal number infinitely often. There are also solutions in the third dimension like adding tetrahedral numbers. Are there solutions in all higher dimensions than 3, and if so questions 1-4 would apply.