Symmetric primes

Abstract: Two odd primes $p,q$ are said to form a symmetric pair if $|p-q|=\gcd(p-1,q-1)$, and we say a prime is symmetric if it belongs to some symmetric pair. The concept comes from a standard proof of quadratic reciprocity where one counts lattice points in the $p/2\times q/2$ rectangle nestled in the first quadrant, both above and below the diagonal: $p$ and $q$ are a symmetric pair if and only if these counts agree. Over 20 years ago, Fletcher, Lindgren, and I showed that most primes are {\it not} symmetric, though the numerical evidence for this is very weak (only about $1/6$ of the primes to $10^6$ are asymmetric). In a new paper with Banks and Pollack we get a conjecturally tight upper bound for the distribution of symmetric primes and we prove that there are infinitely many of them.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

The conference website is www.theoryofnumbers.com/cant/ Lectures will be broadcast on Zoom. The Zoom login will be emailed daily to everyone who has registered on eventbrite. To join the meeting, you may need to download the free software from www.zoom.us.

The conference program, list of speakers, and abstracts are posted on the external website.