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 !

\begin{align} P(a,b,...,z)=&\phantom-\,(k+2)\Big(1\\ &-(wz+h+j-q)^2\\ &-\left[(gk+2g+k+1)(h+j)+h-z\right]^2\\ &-[2n+p+q+z-e]^2\\ &-\left[16(k+1)^{3}(k+2)(n+1)^2+1-f^2\right]^2\\ &-\left[e^{3}(e+2)(a+1)^2+1-o^2\right]^2\\ &-\left[(a^2-1)y^2+1-x^2\right]^2\\ &-\left[16r^2y^{4}\left(a^2-1\right)+1-u^2\right]^2\\ &-\left[\left(\left(a+u^2\left(u^2-a\right)\right)^2-1\right)(n+4dy)^2+1-(x+cu)^2\right]^2\\ &-\left[n+l+v-y\right]^2-\left[\left(a^2-1\right)l^2+1-m^2\right]^2\\ &-\left[ai+k+1-l-i\right]^2\\ &-\left[p+l(a-n-1)+b\left(2an+2a-n^2-2n-2\right)-m\right]^2\\ &-\left[q+y(a-p-1)+s\left(2ap+2a-p^2-2p-2\right)-x\right]^2\\ &-\left[z+pl(a-p)+t\left(2ap-p^2-1\right)-pm\right]^2\\ &\Big) \end{align}

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 ?

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    $\begingroup$ I think rather not. There is even a story related to this, in that some mathematician when being told about these types of results for primes in an informal way said something like this is great we will learn a lot about the prime numbers from it. Then, however, they were informaed that essatially same thing can be done with many other types of sets than the primes (the precise technical term for these sets escapes me at the moment, but they are not overly special sets) so not much hope to learn much on the primes specifically was the conclusion. $\endgroup$ – user9072 Jun 6 '13 at 15:46
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    $\begingroup$ The term quid is looking for is r.e. (recursively enumerable), also called c.e. (computably enumerable). These are the sets of numbers that can be the output of a computer program running forever. $\endgroup$ – Andrés E. Caicedo Jun 6 '13 at 15:54
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    $\begingroup$ @Sébastien: of course, everything is potentially useful, but maybe your question is not sufficiently focused to understand what you would like to do with this polynomial. By the way, I took the liberty of editing out the tags "differential geometry" and "calculus of variations", which do not seem to be related to the question. $\endgroup$ – alvarezpaiva Jun 6 '13 at 17:07
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    $\begingroup$ As Paulo Ribenboim pointed out in The Book of Prime Number Records, "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." It's my understanding that what this sort of polynomial does is encode an algorithm for listing the set. So there's no reason to expect it to give any particular insight not already implicit in the algorithm. $\endgroup$ – Nathan Jun 6 '13 at 17:22
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    $\begingroup$ The first of your bonus questions seems to me to be a bit lazy, surely unintentionally. The proof of the Jones-Sata-Wada-Wiens theorem tells you how to find the variables. $\endgroup$ – Andrés E. Caicedo Jun 8 '13 at 22:47

Added to address additional/clarifies questions.

a. As mentioned below there are various 'formulas' that will yield arbitrarily large primes, however, they are not efficient in a certain way. Personally, I doubt one can get one (or at least a 'better' one) from considering this polynomial. Now, perhaps, I am wrong. I would be curious to learn if this were the case.

b. For Green-Tao and related results the following as some 'warning': The sixty-first Putnam competition (2000) had the following question (paraphrasing): the values the polynomial $Q=X^2+Y^2$ takes on $\mathbb{Z}^2$, contains infinitely many triples of consecutive integers. This is not hard to show. Yet, then in the book "The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary" by Kedlaya, Poonen, Vakil there is mentioned the related question (p.279), (then) stated as open problem (again paraphrasing):

For $Q=X^2+Y^2$ the set $Q(\mathbb{Z}^2)$ contains arbitrarily long arithmetic progressions.

So if one cannot do this (directly) for this polynomial I am rather doubtful one can expect much for the one given here. One should note that, now, this problem is solved, but only (at least this is the 'official' solution on Kedlaya's webpage) via the result of Green and Tao [the set contains all primes congruent $1$ mod $4$, this is a set of positive relative density in the primes and thus it contains arbitrarily long arithmetic progressions].

As remarked in comments it is always hard to say that something is definitely not useful in particular if the use refers to a vague notion as understanding the primes better.

Yet, I will try to demonstrate by an analogy, somewhat close to the problem at hand, at least in my opinion, what is about the intuition.

Recall that Lagrange's Four Squares Theorem asserts that every non-negative number is the sum of four squares of integers. Or in other words, for $L= X_1^2 + X_2^2 + X_3^2 + X_4^2$ one has $L(\mathbb{Z}^4)=\mathbb{N}$ (where we choose the convention that $\mathbb{N}$ includes $0$).

Now, this is on the one hand an interesting result in its own right, and on the other hand the fact that one can characterize the non-negative integers among all integers in such a way/by such a formula is sometimes also used. In fact, it is used in considerations close to the one at hand see its mention in the context of Hilbert's tenth problem.

So this is fine and interesting. What however does not seem like a very feasible idea is to try to understand the non-negative integers better by analysing the polynomial $X_1^2 + X_2^2 + X_3^2 + X_4^2$.

And, to some extent the situation for the primes and this polynomial seems comparable. Yes, one has this characterization of primes, this is very interesting but not so interesting to understand the primes (in the sense, say, some analytic number theorist would like to understand them, frequency, gaps, etc.): the description is not very convenient to work with, the primes do not come out in a systematic way, there are various polynomials having the same property (so why this and not another), the fact that such a polynomial exists is nothing very specific to the primes but also true for all recursively enumerable sets, and so on.

There are also various other prime generating formulas that have some interest, yet also they are not used typically to study primes.

A reason why the idea might seem tempting could be that one somehow thinks that polynomials are relatively easy to understand and handle. However, a key point of all these investigations was, very informally, to show that polynomials are in fact not simple, but can be used to "encode" complex things and thus there can be no algorithm to solve general Diophantine equations.

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  • $\begingroup$ Thank you very much quid for your answer, I pressed the upper arrow of it. I agree, 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 RH... But, as I have just add on the content of the question, perhaps that the use of this polynomial combined with differential calculus allow access to sequences of arbitrary big prime numbers, written with easily computable formulas... My point is closer to Green-Tao theorem that RH... $\endgroup$ – Sebastien Palcoux Jun 7 '13 at 9:46

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