When Does the Dice Sum Become Prime?

TL;DR

This paper develops a high-precision computational method to determine the expected number of die rolls until the sum hits a set A, especially focusing on prime numbers, with applications in probability and number theory.

Contribution

It introduces a dynamic programming approach with rigorous error bounds for accurate computation of expected rolls, achieving unprecedented precision for prime-related sums.

Findings

Expected number of rolls until sum in A computed to over 1000 decimal places.

Sharper estimates for expectations and moments than previous studies.

Exponential decay of survival probability enables high-accuracy truncation.

Abstract

Given a (possibly infinite) subset $A$ of the natural numbers, we ask how many times a fair six-sided die must be rolled until the rolled numbers add up to an element of $A$. Using a one-dimensional dynamic programming recursion together with truncation and rigorous error bounds, we compute the expected number of rolls efficiently and with very high accuracy. When $A$ is the set of prime numbers, the irregular distribution of primes makes it difficult to obtain explicit error estimates. Nevertheless, the density of primes implies that the associated survival probability decays exponentially fast, which enables highly accurate truncation estimates. As a result, our calculations yield significantly sharper estimates for this expectation and its higher moments than the original results of Conroy, Alon, and Malinovsky. In particular, we determine the expectation to more than $1000$ decimal places.