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Define,

$$P_1 := a^2+b^2\\ P_2 := a^2+c^2\\ P_3 := b^2+c^2$$

Let,

$$a,b,c = 2xy,\;x(y^2-1),\;y(x^2-1)$$

and $P_1,P_2$ become squares. If we wish to make $P_3$ a square as well, then,

$$P_3:=x^2(y^2-1)^2+y^2(x^2-1)^2=z^2\tag1$$

or a quartic polynomial to be made a square (hence may be amenable to being treated as an elliptic curve). Euler gave a rational parameterization to $(1)$.

However, if we seek integer solutions, then I find there are only six $x,y$ with $1<x<y<500$, namely,

$$7,\,10\\11,\,17\\17,\,65\\22,\,24\\27,\,66\\186,\,288$$

Question:

  1. Are there more with bound $B>500$?
  2. Are there, in fact, infinitely many integer solutions to $(1)$?
Tito Piezas III
  • 54,565

1 Answers1

2

I tried to pursue the perspective of $(1)$ being a family of elliptic curves for a fixed $y \ge 2$. However, one quickly comes across curves, where the rank is not so easily computed (or maybe even computable with current methods).

Instead I used Michael Stoll's ratpoints to find integral points and in the range $500 \le y \le 3000$ I found the following $(x,y,z)$

$$ (1272,\,901,\,1786468933)\\(142,\,1376,\,270286962)\\(21659,\,2024\,\ 953619976155) $$

which atleast answers the first question.

The second question I cannot answer. Empirically, the integral points are very scarce, but my feeling is, that it is a very bad heuristic for concluding anything.

Jesper Petersen
  • 1,128
  • +1 Thanks. I had in fact found $x,y = 142,1346$ by assuming $x,y = 2u, 2v$. If integral points are sparse, perhaps a Pell equation is involved. For example, one can show $x^2(x-1)^2+y^2(y-1)^2 = w^2$ discussed here has an infinite number of integer solutions using an elliptic curve. But it also has subsets that can be solved by a Pell equation. – Tito Piezas III Feb 28 '16 at 19:39