# [seqfan] Fixing A175155 Numbers n satisfying n^2 + 1 = x^2 y^3

Georgi Guninski guninski at guninski.com
Fri Nov 16 16:56:36 CET 2012

```A175155 Numbers n satisfying n^2 + 1 = x^2 y^3

I am not sure this is entirely correct:
>This sequence is infinite. The fundamental solution of n^2 + 1 = x^2 y^3 is (n,x,y) = (682,61,5), that mean the Pellian equation n^2 - 125x^2 = -1 has the solution (n,x) = (682,61) =(n(1),x(1)). Then, this Pellian equation admit an infinity solutions (n(2k+1),x(2k+1))

This indeed is a family of solutions giving the smallest one,
but there are infinitely many other solutions arising from
x^2 - k^3 y^2 = -1

In particular n=1459639851109444 is missing from the sequence.
n^2 + 1 = 17^3 * 79153^2 * 263090369^2

I suggest:
1. Adding the missing term and other low hanging fruit from pell eqs
2. Indicating that terms might be missing (the sequence contains a
22 digit number and I suppose it is infeasible to find all terms up to
it)

Should I submit another sequence that currently numerically coincides with A175155?

Solution x to x^2 - 125 y^2 = -1

```