A SEARCH FOR WIEFERICH AND WILSON PRIMES

An odd prime p is called a Wieferich prime if 2(p-1) equivalent to 1 ( mod p(2)); alternatively, a Wilson prime if (p-1)! equivalent to-1 (mo d p(2)). To date, the only known Wieferich primes are p = 1093 and 351 1, while the only known Wilson primes are p = 5, 13, and 563. We repor t that there exist no new Wieferich primes p < 4 x 10(12), and no new Wilson primes p < 5 x 10(8). It is elementary that both defining congr uences above hold merely (mod p), and it is sometimes estimated on heu ristic grounds that the ''probability'' that p is Wieferich (independe ntly: that p is Wilson) is about 1/p. We provide some statistical data relevant to occurrences of small values of the pertinent Fermat and W ilson quotients (mod p).