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In the last few weeks, the Great Internet Mersenne Prime Search (GIMPS) announced the discovery of two new Mersenne primes, both over ten million digits in length, including one discovered by the computing team right here at UCLA (see this page for more information). [I was not involved in this computing effort.] As for the question “Why do we want to find such big primes anyway?”, see this page, though this is not the focus of my post today.
The GIMPS approach to finding Mersenne primes relies of course on modern computing power, parallelisation, and efficient programming, but the number-theoretic heart of it – aside from some basic optimisation tricks such as fast multiplication and preliminary sieving to eliminate some obviously non-prime Mersenne number candidates – is the Lucas-Lehmer primality test for Mersenne numbers, which is much faster for this special type of number than any known general-purpose (deterministic) primality test (such as, say, the AKS test). This test is easy enough to describe, and I will do so later in this post, and also has some short elementary proofs of correctness; but the proofs are sometimes presented in a way that involves pulling a lot of rabbits out of hats, giving the argument a magical feel rather than a natural one. In this post, I will try to explain the basic ideas that make the primality test work, seeking a proof which is perhaps less elementary and a little longer than some of the proofs in the literature, but is perhaps a bit better motivated.
I’ve just uploaded to the arXiv the short note “A remark on primality testing and the binary expansion“, submitted to the Journal of the Australian Mathematical Society. In this note I establish the following result: for any sufficiently large integer n, there exists an n-bit prime p, such that the numbers for are all composite. In particular, if one flips any one of the bits in the binary expansion of the prime p, one obtains a composite number. As a consequence, one obtains the (rather plausible) consequence that in order to (deterministically) test whether an n-bit integer is prime or not, one needs (in the worst-case) to read all n bits of the prime. (This question was posed to me by my colleague here at UCLA, Yiannis Moschovakis.)
Primes p of the form mentioned in the above result are somewhat rare at first; the first some prime is . But in fact, the argument in my note shows that the set of such primes actually has positive relative density inside the set of all primes. (Amusingly, this means that one can apply a theorem of Ben Green and myself and conclude that there are arbitrarily long arithmetic progressions of such primes, although I doubt that there is any particular significance or application to this conclusion.)
The same remark applies to other bases; thus, for instance, there exist infinitely many prime numbers with the property that if one changes any one of the base 10 digits of that number, one obtains a composite number.
(Presumably the first such number can be located by computer search, though I did not attempt to do so.) [Update, Feb 25: see comments.]