After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial.

In particular, it's the case for the set of prime numbers.

In 1976, Jones-Sata-Wada-Wiens published (see here) the following polynomial $P$ of degree $25$ in $26$ variables $a$, $b$, $c$,..., $z$, admitting the property that $P(\mathbb{N}^{26})\cap \mathbb{N} = \mathbb{P}$, the set of prime numbers !

$P(a,b,...,z) = (k+2)[1 – (wz+h+j–q)^{2}$

$ – [(gk+2g+k+1)(h+j) + h – z]^{2}$

$ -(2n+p+ q+z–e)^{2} $

$– [16(k+1)^{3}(k+2)(n+1)^{2} + 1 – f^{2}]^{2}$

$– [e^{3}(e+2)(a+1)^{2} + 1 –o^{2}]^{2}$

$ - [(a^{2}–1)y^{2} + 1 –x^{2}]^{2}$

$– [16r^{2}y^{4}(a^{2}–1) + 1 – u^{2}]^{2}$

$– [((a+u^{2}(u^{2}–a))^{2}–1)(n+4dy)^{2} + 1 –(x+cu)^{2}]^{2}$

$–[n+l+ v–y]^{2}– [(a^{2}–1)l^{2} + 1 – m^{2}]^{2} $

$– [ai+k+1–l–i]^{2} $

$– [p + l(a–n–1) + b(2an+2a–n^{2}–2n–2) – m]^{2} $

$ – [q+ y(a–p–1) +s(2ap + 2a – p^{2} – 2p – 2)– x]^{2}$

$– [z + pl(a–p) + t(2ap – p^{2} – 1) – pm]^{2}]$

In "The Book of Prime Number Records" Paulo Ribenboim reports:

"It should be noted that this polynomial also takes on negative values, and that a prime number may appear repeatedly as a value of the polynomial."

I'm agree that this polynomial encodes an algorithm, but it's also a concrete polynomial!

We certainly can't generate large primes with a direct use of it in a reasonable time, but if we analyse it with the tools of differential calculus, maybe we can find some generic extremal points admitting neighborhood areas of positive range, and then generate easily computable sequence of prime numbers, as for the Catalan-Mersenne conjecture:

$2$, $2^{2}-1$, $2^{2^{2}-1}-1$, $2^{2^{2^{2}-1}-1}-1$, $2^{2^{2^{2^{2}-1}-1}-1}-1$... a sequence of prime numbers ?

I don't know if it's possible, but it's worth a try... now there exists very powerfull supercomputers:

$10^{6}$ times more powerfull than in 1989 when Paulo write its book...

In my opinion this formula can't be useful too for a global understanding of the prime numbers, in particular, we certainly can't use it to upgrade our statistic understanding of primes and prove the Riemann hypothesis. My point is closer to Green-Tao theorem than RH.

Main question: Does differential calculus can be helpful to get results on primes by analyzing $ P $?

For those who already know: What's the minimal size of a solution for $P$ to get a fixed prime ?