1) the representability of diophantine systems as polynomials

2) the formulation of primes as such a system (eg. here)

If you inspect the polynomial closely, you notice that it has the form

(k+2) * (1 – P(a,b,c…)^2 – P'(a,b,c…)^2 …), so it will be positive only when all the polynomials P are zero. They essentially select the prime values of k+2. More on Wikipedia.

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