Title: Hitting a prime by rolling a die with infinitely many faces Show Chinese title

Title: 通过掷一个无限面的骰子击中素数

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. Show Chinese abstract

Abstract: Alon 和 Malinovsky 最近证明，平均需要$2.42849\ldots$次公平的六面骰子掷出，直到所有掷出的总和第一次达到一个质数。 自然地，人们可以将这种情形扩展到具有不同面数的骰子。 在本文中，我们证明，当骰子的面数$M$足够大时，Alon 和 Malinovsky 游戏中的期望停止轮数大约为$\log M$。