Symmetric and asymmetric primes

Abstract

In a well-known proof of the quadratic reciprocity law, one counts the lattice points inside the rectangle with sides parallel to the axes and opposite vertices at the origin and (p/2, q/2), where p and q are distinct odd primes. In particular, the Legendre symbols (p/q) and (q/p) depend, respectively, on the number of lattice points in the rectangle above and below the main diagonal. Say p, q form a symmetric pair if the number of lattice points above the main diagonal is equal to the number of lattice points below. Say a prime p is symmetric if it belongs to some symmetric pair, and otherwise call it asymmetric. We first characterize symmetric pairs p, q with the condition (p - 1,q - 1) = |p - q|. In particular, twin primes form a symmetric pair. Of the first 100,000 odd primes, about 5/6 of them are symmetric. However, we are able to prove that, asymptotically, almost all primes are asymmetric. © 1996 Academic Press, Inc.