How likely is it that someone finds a record prime with those equations?

Question

An amazing result in number theory is that the equations shown here

http://mathworld.wolfram.com/PrimeDiophantineEquations.html

open a way to find prime numbers. It is probably very difficult to find a solution for huge $k$, but on the other hand, I see no reason, why it will not be found at some time.

How likely is that with those equations, someone finds a prime larger than any known prime ?

Has this been estimated, or has it just been stated that it will be "very difficult" to find a solution ? Theoretically, even numbers much too large for the usual primality tests could be proven prime with those equations, so it might be worth to think about this possibility.

Answer 1

That system of equations, even if algorithmically possible to solve, is so much non-feasible and also non-linear and have a constraint on the solutions (that solutions are positive integers) and clearly is mainly of theoretical interest.

If you would plug some very large possible prime $k+2$ into that system and would try to do the test (that is, would try to solve the system non-algorithmically) for those $26$ variables in some range to see do you have a solution in that range, even if the range is just $\{1,2,..,100\}$ the number of possible choices for all the variables in that small range is $100^{26}$.

Yes, there is some dependence of some equations on some other equations, so, some choices would not need to be checked but still, without "very fast" algorithmic method the system is practically, not non-solvable, but, not-recommended-to-solve.

Mostly the interesting part is the presence of "iff" (that is "if and only if") in the statement, meaning that primes are actually characterized by that system and its solutions, but, in order to actually find some methods for determining are some numbers of some special forms primes we would need different approaches.

Answer 2

There is zero chance.

To get a feeling for why that is, try to find a solution of the system with $k+2=23$, say. The system is more like a "toy" system that shows that the defining property of being a prime can be encoded in this way. It is in no way "practical".

To compare it with something remotely similar: Conway's Game of Life is Turing-complete (as is Minecraft or Magic The Gathering). Hence it is possible to encode good old trial division of a large prime candidate as some initial position for the Game. That still doesn't make this a good idea if one wants to attempt to beat more elaborate algorithms on a standard computer ...

Answer 3

Let me give a more "mathematical reason" as to why this equation is of absolutely no use in actually finding large primes. The details are all in the original paper available here, specifically Theorem 2.12.

The way this equation works is that if there is a solution $a,\dots,z$ for a given $k$, then $g$ is a number satisfying $(g+1)(k+1)=k!+1$ (existence of such $g$ is equivalent to $k+1$ being a prime, by Wilson's theorem). If we were to find a new largest prime this way, considering we would need to have $k>2^{82589933}$, we would need to have $g>(2^{80000000})!$, which is way too large to even store, let alone do computations with it and verify a solution, ever in the lifetime of our universe.

As other comments and answers have stated, this system of equations is nothing more than a theoretical curiosity. It never was intended to be, and as you can see it never will be, a feasible way to find primes.