A Very High-Speed Digital Number Sieve

such that N is a solution to s of the congruences. (It is, of course, clear that no Ar can be a solution to more than s of the congruences (1).) Examples : On the one extreme there is the Sieve of Eratosthenes for finding all primes p in the range A = B12 ^ p < B, where ti = m, — 1 for all i. (Here to, are all the primes < Bl .) On the other extreme there is the Chinese remainder type of problem, where ¿, = 1 for all i, and there is only one solution among IJ¿=i to¿ numbers. In between these two extremes, there is the important quadratic sieve, where roughly U = mi/2 for all i. It is used in problems involving quadratic residues, Diophantine equations of second degree and other quadratic type problems. About thirty years ago, Lehmer [1], [2] constructed a novel special-purpose device for sifting. It used the first 30 primes as moduli. Its processing rate was