All Mersenne prime exponents below 100m tested at least once
When I was at EFF, I helped run the Cooperative Computing Awards project (<a href="https:&#x2F;&#x2F;www.eff.org&#x2F;awards&#x2F;coop&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.eff.org&#x2F;awards&#x2F;coop&#x2F;</a>), which was funded decades ago by an individual donor and provides cash rewards for finding prime numbers of certain sizes.<p>Because of EFF&#x27;s advocacy for cryptography, people always thought that there was a more direct connection between the Mersenne prime search and cryptographic techniques, but the connection is at best distant and tenuous. The record primes that GIMPS finds and that can receive EFF&#x27;s awards are<p>* much (much much) bigger than the primes that we use for cryptography<p>* too big to do almost any kind of practical computation with<p>* not secret, so couldn&#x27;t be used as part of a secret key<p>The distribution or explicit values of Mersenne primes (and, correspondingly, perfect numbers) also don&#x27;t really help in our assessment of the difficulty of the RSA problem or the security of public-key cryptography, because they aren&#x27;t closely related to any of the fastest factorization algorithms that apply to arbitrary numbers.<p>(Mersenne primes are much easier to deterministically test for primality than numbers in general, which is why the record primes that GIMPS finds are Mersenne primes. The runtime complexity of primality-testing algorithms for special-form numbers is sometimes much less than algorithms for an arbitrary number, and this is a great example.)<p>I think the main contribution of projects like GIMPS to cryptography and computer security lies in getting people more interested in number theory and mathematics in general. Hopefully that will lead to faster improvements in humanity&#x27;s knowledge of mathematics in ways that lead to more secure public-key algorithms or to better public assessments of the algorithms that we have now. (Although a lot of the important current research on public-key cryptography is not about factorization but rather about elliptic-key systems and post-quantum systems.)<p>I always felt I had a hard time explaining to people that the awards existed in order to publicize <i>the idea of cooperative distributed computing projects</i>, which were quite novel when EFF&#x27;s awards were first created, and which were a demonstration of the power of the Internet to help people who&#x27;ve never even met each other work together. (The awards, contrary to the hopes and dreams of math enthusiasts and cranks around the world, didn&#x27;t anticipate progress through new <i>research</i> or <i>ideas</i> so much through more <i>volunteer computer time</i>.)<p>Probably today we have ample other demonstrations of that power, and that concept is no longer in much doubt, and the GIMPS project probably no longer even registers for most people as a particularly impressive example. But it is impressive in continuing to set multiple world records on one of the simplest, purest, most basic mathematical problems.