For clarity, I have reedited this question, by using some comments below, so I apologize if its contents is redundant with some comments...

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 $P(\mathbb{N})\cap \mathbb{N} = \mathbb{P}$, 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 that: "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, so it can be analyse by the tools of differential calculus to identify its extremal points and the areas with positive range... I don't know if it's easy to compute, but now there exists very powerfull computers and supercomputers: $10^{6}$ times more powerfull than in 1989 when Paulo write its book...

I don't know, perhaps that by this process we can access to sequences of arbitrary big prime numbers, written with easily computable formulas... 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 ?

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

So does anyone can explain to me if all this investigation I'm talking about would be useful to obtain some results on prime numbers ?