Every Prime has a Succinct Certificate

To prove that a number n is composite, it suffices to exhibit the working for the multiplication of a pair of factors. This working, represented as a, string, is of length bounded by a polynomial in log n. We show that the same property holds for the primes. It is noteworthy that almost no other set is known to have the property thatshort proofs for membership or nonmembership exist for all candidates without being known to have the property that such proofs are easy to come by. It remains an open problem whether a prime n can be recognized in only log n operations of a Turing machine for any fixed The proof system used for certifying primes is as follows. tp-1)/q (mod p) and ql(P 1). THEOREM 1. p is a theorem p is a prime. THEOREM 2. p is a theorem p has a proof of [4 log p lines. 1. Proofs. We know of no efficient method that will reliably tell whether a given number is prime or composite. By "efficient", we mean a method for which the time is at most a polynomial in the length of the number written in positional notation. Thus the cost of testing primes and composites is very high. In contrast, the cost of selling composites (persuading a potential customer that you have one) is very low-in every case, one multiplication suffices. The only catch is that the salesman may need to work overtime to prepare his short sales pitch; the effort is nevertheless rewarded when there are many customers. At a meeting of the American Mathematical Society in 1903, Frank Cole used this property of composites to add dramatic impact to the presentation of his paper. His result was that 267 1 was composite, contradicting a two-centuries-old conjecture of Mersenne. Although it had taken Cole "three years of Sundays" to find the factors, once he had done so he could, in a few minutes and without uttering a word, convince a large audience of his result simply by writing down the arithmetic for evaluating 267 and 193707721 x 761838257287. We now show that the primes are to a lesser extent similarly blessed; one may certify p with a proof of at most [4 log2 p] lines, in a system each of whose inference rules are readily applied in time O(log 3 p). The method is based on the Lucas-Lehmer heuristic …