Hitting a prime by rolling a die with infinitely many faces

TL;DR

This paper extends a previous result on the expected number of rolls needed to reach a prime sum with a fair die, showing that for large face counts, the expectation grows roughly logarithmically with the number of faces.

Contribution

It proves that the expected number of rolls to hit a prime sum scales approximately as the logarithm of the number of die faces for large M.

Findings

Expected rolls grow as log M for large M

Expected value approaches a logarithmic function of die faces

Generalizes previous six-sided die result

Abstract

Alon and Malinovsky recently proved that it takes on average $2.42849\ldots$ rolls of fair six-sided dice until the first time the total sum of all rolls arrives at a prime. Naturally, one may extend the scenario to dice with a different number of faces. In this paper, we prove that the expected stopping round in the game of Alon and Malinovsky is approximately $\log M$ when the number $M$ of die faces is sufficiently large.