http://polymathprojects.org/2009/08/09/research-thread-ii-deterministic-way-to-find-primes/

All new research comments should now be made at that thread. Participants are of course welcome to summarise, review, or otherwise carry over existing comments from this thread at the next one; this would be a good time to recap any progress that has not been sufficiently emphasised already.

Assuming that factoring is “easy”, I want to show that either we can find large primes quickly (say sub-exponential in k), or else such number theory conjectures as the “No Siegel Zeros” conjecture are true (and perhaps strong forms of the “Chebotarev’s density theorem”, and many other such conjectures, can be assumed to hold by similar ideas). The hope would be that having such a theorem be true can be used to boost the effectiveness of other potential methods to find large prime numbers (for one thing, if “No Siegel Zeros” holds, then we know that primes are “well-distributed in arithmetic progressions”, a powerful ingredient in potential sieve method approaches to locating primes).

My first suggestion is that if there are infinitely many (sufficiently bad) Seigel zeros, then we know, say, that there are infinitely many moduli M for which all the Jacobi symbols

(M/q) = -1, for primes $q < B := \exp(\sqrt{\log M})$, gcd(q,M)=1.

This means that the Jacobi symbol can be used to “approximate the Mobius function mu'', and since good bounds on “character sums'' are much easier to come by than analogous sums involving mu, it is can be used to improve the quality of results produced by sieve methods to find primes; indeed, Roger Heath-Brown used this idea to show that if there just enough Seigel zeros, then the Twin Prime Conjecture holds. Perhaps there are other important instances of the Hardy-Littlewood conjecture that can be attacked in a similar way.

My second suggestion is incomplete, but more direct: suppose that Siegel zeros exist (there are infinitely many of them, say). Then consider the form

$$
f(x,y) = x^2 – M y^2,
$$
where M is the modulus of a Siegel zero; and consider, for example, f(1,1) = 1 – M. This number cannot be divisible by any small prime q < B, else (M/q)= (1/q) = +1. So, all prime factors of f(1,1) lie in $[B,M^3]$, and therefore if factoring is “easy'', we will have found a large prime!

But how do we locate such a modulus M, assuming they exist? I'm not sure, but it seems like maybe we could use the fact that we have lots of different values of the form f to play around with — we would just need to choose xy coprime to M. Of course we could also try to build a sieve somehow, as numbers M satisfying (M/q)=-1 for primes q < B are quite rare!

The basic issue here is that at the k^th step of this algorithm, one has only excluded poly(k) primes from consideration. In the worst-case scenario, the next number generated by this algorithm could be generated by the first prime that has not already been excluded, which would be only of polynomial size in k rather than exponential. Admittedly, this is an unlikely scenario, but the trick is to figure out how to rule it out completely.

Currently, all of our algorithms, after k steps, are only capable of producing a prime of size as low as k^2 (modulo lower order terms) in the worst case (and this requires GRH). Even getting a prime of size or greater in k steps unconditionally would be a breakthrough, I think.

The paper is “On strings of consecutive integers with no large prime factors”, J. Austral. Math. Soc. Ser. A 64 (1998), no. 2, 266–276 (the length of the string grows like )…
See here.

Last night I made a pen and paper algolrithm I am talking about. I picked Mersenne100 because it’s the first 3 digit and 100 is a nice decimal number. Here are the associated primes:

101, 8101, 268501.

I also cheated to check that there really are primes in googles. All the talk so far strengthen and supports my algol. Picking Mersenne 100 is by whim, it could have been any other Mersennes but inline with your project slowing building up all the smaller Mersennes will help up the test later for sieving out potential composite. This Mersenne algol is similar to euclid prime production and has the non-trivial geometric series Tim Gower pointed out on Paul Erdos results. Mersennes has the other property other have not pointed out. It has the property of Fibonacci sequennce, I really don’t know the impact of Fibonacci sequence. Together the near geometric progression and Fibonacci like sequencing this has a good mix of properties your looking for.

Besides it is backed up by powerful theorems and conjectures. Fermat’s Little Theorem, Euler function, Carmichael function, Poclington’s theorem, Legendre’s theorem on Mersenne, Wilson’s primes, Wagstaff primes, Riesel composite numbers, Euclid prime production, Gause logarithmic prime connection, works of Maurer highlights aspects of this, Carl Pomerance is studying a logarithnmic function that has game theory aspect of it, I believe it has connection with my J function, Cramer’s conjecture and Bertrands Postulate I see them as two side of a coin in coining to my algol.

1) Multiply the first ln(k) primes together and call this A.
2) Factor A. If there is a k-digit prime divisor your done. Otherwise multiply A by all of the prime factors of A+1.
3) Repeat until you find a k-digit prime.

The general idea is that if A doesn’t have a large prime factor, it must have a lot of small prime factors. Since every prime divisor of A+1 is distinct from the prime divisors of A, the list of primes you multiply together to get (the next) A should grow very quickly. I imagine this gives some kind of gain over a brute force search?