Hitting a Prime by Rolling a Die with Infinitely Many Faces

Abstract

Alon and Malinovsky recently proved that it takes on average $2.42849\dots$ 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 article, we prove that the expected stopping round in the game of Alon and Malinovsky is approximately $\hspace{0.17em}\log \hspace{0.17em}M$ when the number M of die faces is sufficiently large.

KEYWORDS:

• Asymptotic relation
• Dice roll
• Expected stopping round
• Prime number

Acknowledgments

I would like to acknowledge my gratitude to the referees and the associate editor for many helpful suggestions that were extremely helpful in improving the exposition of the article. This work was supported by a Killam Postdoctoral Fellowship from the Killam Trusts.

Disclosure Statement

The author declares that there are no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.

Notes

1 Albert Einstein, however, in a letter to Max Born on Dec 4th, 1926, proclaimed that “God does not play dice with the universe.” See Born and Einstein (1971, pp. 90–91, Letter 52).

2 See https://sites.math.rutgers.edu/∼zeilberg/tokhniot/oHIT1a.txt, compiled by Martinez and Zeilberger (2023).

3 See https://sites.math.rutgers.edu/∼zeilberg/tokhniot/oHIT2a1.txt, compiled by Martinez and Zeilberger (2023).

4 See the GitHub page https://github.com/shanechern/Hitting-a-prime for related codes, which are written in Mathematica. It is worth noting that the Maple codes of Martinez and Zeilberger (2023) are built upon generatingfunctionology, while the original Matlab codes of Alon and Malinovsky (2023) are based on a dynamic programming algorithm, which is mathematically equivalent to the one of Martinez and Zeilberger (2023).