Assume wlog that deg(P) <= deg(Q). If deg(P) = 1, then you're essentially solving Q(x) = a (mod b). If deg(P) = 2, then that's exactly the problem of finding integer points on a conic (if deg(Q) = 2), elliptic (if 3 <= deg(Q) <= 4), or hyperelliptic (if deg(Q) >= 5) curve. This arXiv paper seems as though it might be helpful: https://arxiv.org/abs/0801.4459 If deg(P) = 3 and deg(Q) = 3, then it's a cubic curve, and you can apply some birational map to reduce it to one of the above cases. Beyond that, I have no idea. -- APG.

Sent: Thursday, April 25, 2019 at 11:28 PM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Square hex numbers?

This suggests the question:

----- P and Q be integer polynomials of degrees >= 2.

Find the solution set {(N,K) in Z^2 | P(N) = Q(K)}. -----

What's known about the general case when

max(deg(P), deg(Q)} = 2, 3, 4, or >= 5

??? I would try googling this but I'm not sure how to search for it.

—Dan

Adam Goucher wrote: -----

Your equation is of the form:

(quadratic in n) = (quadratic in K)

so you can complete the square on both sides and ..... -----

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