Minimizing the number of carries in addition

TL;DR

This paper proves that using symmetric digit sets minimizes the probability of carries in addition for all odd primes, confirming a conjecture that was previously only asymptotically verified.

Contribution

It proves the conjecture that symmetric digit choices minimize carries in addition for all odd primes, extending prior asymptotic results.

Findings

Symmetric digit sets reduce carry probability in addition.

The minimal carry probability is achieved by digits symmetric around zero.

The conjecture holds for all odd primes, not just large primes.

Abstract

When numbers are added in base $b$ in the usual way, carries occur. If two random, independent 1-digit numbers are added, then the probability of a carry is $\frac{b-1}{2b}$. Other choices of digits lead to less carries. In particular, if for odd $b$ we use the digits $\{-(b-1)/2, -(b-3)/2,...,...(b-1)/2\}$ then the probability of carry is only $\frac{b^2-1}{4b^2}$. Diaconis, Shao and Soundararajan conjectured that this is the best choice of digits, and proved that this is asymptotically the case when $b=p$ is a large prime. In this note we prove this conjecture for all odd primes $p$.