I am asking whether the existence of a parametric identity involving a set of quadruple signed integers $\{i, j, k, l\}$, which is true for infinitely many sets of quadruples for each integer value of $n, (n>0)$, implies that there is a relationship between some or all of the variables in the set $\{i, j, k, l\}$ and/or relationship between some (or all) of the parameters of the $\{i, j, k, l\}$ with $n$? And furthermore, I am asking whether such relationship would allow rewriting the parametric identity to hold true for subsets of the $\{i, j, k, l\}$ set that contain fewer than four parameters and perhaps involve direct relationship with $n$?
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I wonder whether this is the kind of thing you have in mind:
$n=(i^2-j^2)-(k^2-l^2)$ has infinitely many integer solutions for each integer $n>0$.
I don't think this implies any relation among some or all of $i,j,k,l$ (unless you allow relations that also involve $n$).
But one does have $2r=(r+s+1)^2-(r+s)^2-((s+1)^2-s^2)$ which uses only two parameters, and has infinitely many solutions for each $r$.
Gerry Myerson
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Thanks Gerry - I meant to allow relations that also involve n. I edited my question to make it clear. Sorry for confusion and thanks for answering. – Alex Jul 02 '23 at 12:08
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OK, then I'm not sure what you mean by a parametric identity involving $i,j,k,l$, true for infinitely many quadruples for each $n$ – that sounds to me like a relation among the parameters and $n$. Maybe you can give an example? – Gerry Myerson Jul 02 '23 at 12:15
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Proposition 1. For every n ∈ N, it is possible to find infinitely many quadruples (i, j, k, l) ∈ Z that fulfil $(-1)^n\cdot(\pi - \text{A002485}(n)/\text{A002486}(n)) =(|i|\cdot2^j)^{-1} \int_0^1 \big(x^l(1-x)^m(k+(i+k)x^2)\big)/(1+x^2); dx$ – Alex Jul 02 '23 at 14:18
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Where $m=2(j+2)$ – Alex Jul 02 '23 at 22:49