Quadrados magicos

The number 26, between 25 and 27
Resolution of the diophantine equation y 3 − x2 = 2
Axel Gougam & Julien Baglio
May 28, 2006

Abstract
The goal of this article is to demonstrate a funny property of the number 26 : it is the only integer which is at a discrete distance of one from a square and a cube : 25 = 52 ≤ 26 ≤
27 = 33 . This problem is related to Pierre de Fermat, a french mathematician of the XVIIe century, who stated that there should exist five integers which verify the property exposed above. It has been proved after that he was somewhat wrong for there is a unique solution to the problem. We here divide our work in two parts. First we expose the basic ideas which come to mind and lead to the solution which is then exposed in the second part.

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Introduction to the problem, basic ideas

The problem discussed in this article can be formally exposed as :
If we take an integer p > 0 which verify p − 1 = k 2 and p + 1 = k 3 , then it implies that p = 26, k = 5 and k = 3.
If we subtract the first equality with the second one, we obtain −2 = k 2 − k 3 , that is to say (k, k ) is a solution of the diophantine equation y 3 − x2 = 2 with (x, y ) ∈ N2 . In fact, the problem is to resolve this equation. Thus, we will prove that
Theorem 1 (Particular case of the Catalan problem) The unique solution of the diophantine equation in N2 y 3 − x2 = 2
(1)
is x = 5 and y = 3
The first idea is to study the parity of the solutions. It leads to
Lemma 1 If (x, y ) is a solution of the equation (1), then both x and y are odd numbers

1

We have, taken in Z/2Z, xn = x for any integer x, n. That leads to y 3 = y , x2 = x and y − x = 0. The integers y and x have the same parity. In order to identify it, we take the equation in Z/4Z : for any integer class a we have a3 = a according to Euler’s theorem which extends Fermat’s theorem about rests in division by prime integer. Thus we have y = x2 + 2.
If we suppose that 2|x then x2 = 0 and y 3 = y =