# The prime number sieve

I've always been fascinated by the prime number sieve. While it is easy to understand the algorithm, it may be impossible to predict the next prime as a function of all those prior in terms of some sort of mathematical calculation. Recently I've been trying to come up with some way to mathematically model the sieve. The best I can come up with so far is simply an external direct product of cyclic groups... $$G_n = \mathbb{Z}_{p_1} \times \ldots \times \mathbb{Z}_{p_n} ~= \mathbb{Z}_{p_1 \cdots p_n}$$ Here, $G_n$ is the external direct product of cyclic groups for the first $n$ primes. However, if we were to try to use this structure to predict the value of $p_{n+1}$, then I would have no idea how to proceed. The elements of most interest in $G_n$ to me appear to be those that may be described as not existing in the kernel of any projective homomorphism. The subset of all such elements, however, does not form any kind of sub-structure.

I guess the only question I can come up with is...is this approach completely dumb, or might there be any merit to it? I must fully concede right now that I consider myself to be a complete mathematical moron. Still, I can't seem to keep myself away from thinking about this stuff, because it's interesting, even if completely inaccessible to me.

• The prime number theorem allows to estimate with high accuracy the number of primes below a given number. But whether the prime numbers actually have a structure (and not only observed structures usually breaking down if more primes are involved) is an open question. Some useful theorems about primes are known, for example the Dirichlet theorem stating that every arithmetic progression $an+b$ with $\gcd(a,b)=1$ contains infinite many primes. But it is even unknown whether $x^2+1$ produces infinite many primes, if $x$ is an integer, whether there are infinite many twin primes , is open as well. – Peter Apr 6 '17 at 18:41
• Finally, some big open problems like the Goldbach conjecture show that it will be a long way to really understand the structure of the primes. – Peter Apr 6 '17 at 18:42
• Thanks for the feedback. For fun, I'm considering counting the said elements of G_n using the inclusion/exclusion principle. So while there may be some mystery to their structure, if any, we may at least be able to count them. – Spencer Parkin Apr 6 '17 at 19:06
• I think fundamentally the problem with primes is the way they are defined. A number is a prime if its only divisors are 1 and itself. Predicting the next prime would imply that the numbers between current prime and next are not prime which would require a sieve like method. – sku Apr 7 '17 at 19:34