Are there any numbers greater than 26 that are adjacent on either side to a square and a cube?

Question

Formally: For how many integers $n,m$ do we have $n^3-m^2=\pm 2$ ?

I've found two numbers this is true for:

• $0$ [ with $(-1)^3$ and $(\pm1)^2$ ]
• $26$ [with $3^3$ and $5^2$]

I've computed up to 10,000 using a program, and these are the only ones I've found. Are there any other number pairs this is true for? If not, is it possible to prove these are the only pairs?

Answer 1

Sage's built in computation confirms that the solutions you have found are the only solutions.

The commands

E = EllipticCurve([0,2])
E.integral_points(both_signs=True)

yield the results [(-1 : -1 : 1), (-1 : 1 : 1)], corresponding to the full set of solutions of the equation $$ y^2 = x^3 + 0x + 2, $$ namely $\{(-1, -1),(-1,1)\}$. Similarly, the commands

E = EllipticCurve([0,-2])
E.integral_points(both_signs=True)

yield the results [(3 : -5 : 1), (3 : 5 : 1)], corresponding to the full set of solutions of the equation $$ y^2 = x^3 + 0x - 2, $$ namely $\{(3,-5), (3,5)\}$.

Note that Sage can be used for free here. Here's the documentation for Sage's elliptic curve tools.