Symmetric and Asymmetric Primes

✍ Scribed by Peter Fletcher; William Lindgren; Carl Pomerance

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 314 KB
• Volume: 58
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1996.0066

✦ Synopsis

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 ( pÂ2, qÂ2), where p and q are distinct odd primes. In particular, the Legendre symbols ( pÂq) 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 5Â6 of them are symmetric. However, we are able to prove that, asymptotically, almost all primes are asymmetric.