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?

The equations $x^3 - y^2 = 2$ and $x^3 - y^2 = -2$ are elliptic curves. It is known that these curves will contain finitely many points with integer coordinates, but I'm not sure how these integral points can be systematically counted. — Ben Grossmann, Jan 12 '24 at 23:00
Related: https://math.stackexchange.com/questions/462524/step-in-a-solution-of-y2-x3-2 and https://math.stackexchange.com/questions/1279001/solving-the-mordell-equation-y2-x3-%e2%88%92-2-what-would-be-a-general-strategy and https://math.stackexchange.com/questions/571473/solving-the-diophantine-equation-y2-x3-2 and others. — Gerry Myerson, Jan 13 '24 at 01:40
$y^2=x^3+k$ is called a Mordell equation. See https://personal.math.ubc.ca/~bennett/BeGh-LMSJCM-2015.pdf and https://kconrad.math.uconn.edu/blurbs/gradnumthy/mordelleqn1.pdf and http://alpha.math.uga.edu/~pete/4400MordellEquation.pdf — Gerry Myerson, Jan 13 '24 at 01:44

Ben Grossmann · Answer 1 · 2024-01-12T23:29:28.610

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.

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

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