## What do solutions of the Jones-Sato-Wada-Wiens polynomial look like?

As soon as one comes across the story of the set of primes being the positive values taken by the Jones-Sato-Wada-Wiens polynomial of degree 25 in 26 variables (see also the senior thesis of Cabusora, or, for french-speaking readers, this RMS paper by Collet and Vidiani), one wants to start playing with it, and see some solutions.

But the aforementioned sources don’t mention any, and neither do more recent books like Denef’s on Hilbert’s tenth problem, the one by de Koninck and Luca on analytic number theory, the one by Crandall and Pomerance on a computational perspective on primes, the one by Everest and Ward on introductory number theory.

Fortunately, there is an Msc thesis by Gupta which does take the trouble to find the values of all 26 variables for the prime 2, and it is informative. Using his notations, these values are:

$k=0$, $g=0$, $f=17$, $n=2$, $p=3$, $q=16$, $z=9$, $w=1$, $h=2$, $j=5$, $e=32$ (parameter of a Pell equation), $a=7901690358098896161685556879749949186326380713409290912$, $o=8340353015645794683299462704812268882126086134656108363777$ (these two being the smallest solution of that Pell equation), $y=2a$, $x=2a^2-1$, $m=a$, $\ell=1$, $i=0$, $v=2a-3$, $b=0$$s=1$$t=0$, $r\simeq 10^{10^{52}}$, the final $u, c, d$ being larger still.

The basic idea is that the built-in Pell equations inside the polynomial quickly lead to very large values indeed. No hope to play with 26 small integers, say less than $10^6$.