Written: Feb. 12, 2024

In this paper [arxiv version] by Noga Alon and Yaakov Malinovsky, extended in this paper, by Lucy Martinez and Doron Zeilberger, found (and proved) the expected number of rolls it takes, rolling a standard die (and more general dice) until the running total is a prime (and other interesting kinds of numbers) for the FIRST time. But how long would it take (on average) until you see a prime k times?, for any k. We use computation to give exact values (up to many decimal places) for these numbers for k from 1 to 30 (and for k=40,60,80,100) and use human ingenuity to prove that it is asymptotiaclly k*ln(k)*(1+o(1))

Added March 23, 2025: Read this nice post .

Added May 19, 2025: Read the slides of Thotsaporn Aek Thanatipanoda's great talk at the MUIC mathematics seminar (May 21, 2025) (see abstract).

Maple package

• PRIMESk.txt, a Maple package that computes not only the expectation, but also the variance and higher moments for reaching a prime (and many other kinds of numbers) k times

Sample Input and Output for PRIMESk.txt

• If you want to see statistical information about reaching a prime for the k-th prime with at most 400 rolls, rolling a fair standard die, starting at 0, for k from 1 to 30, the
  input file yields the output file

• If you want to see statistical information about the number of rolls until reaching a product of two distinct primes for the k-th time with at most 200 rolls, rolling a fair standard die, starting at 0, for k from 1 to 30, the
  input file yields the output file

• If you want to see statistical information about the number of rolls until reaching a product of three distinct primes for the k-th time with at most 200 rolls, rolling a fair standard die, starting at 0, for k from 1 to 30, the
  input file yields the output file

• If you want to see statistical information about the number of rolls until reaching a prime for the k-th time with at most 500 rolls, rolling a fair standard die, starting at 0, for k=20, k=40 and k=60,
  input file yields the output file

• If you want to see statistical information about the number of rolls until reaching a prime for the k-th time with at most 1000 rolls, rolling a fair standard die, for k=80 times
  input file yields the output file

• If you want to see statistical information about the number of rolls until reaching a prime for the k-th time with at most 1200 rolls, rolling a fair standard die, for k=100 times
  input file yields the output file

• If you want to see statistical information about the number of rolls until reaching a perfect square for the k-th time, starting at 2, with at most 1000 rolls, rolling a fair standard die, for k=1 to k=10
  input file yields the output file

Plots of the Probability Density Functions (pdf) of the Random Variable "Reaching a prime for the k-th time" by rolling a fair die for k=20,40,60,80,100

• Here is a plot of the pdf of the r.v.
  number of rolls until you see 20 primes

• Here is a plot of the pdf of the r.v.
  number of rolls until you see 40 primes

• Here is a plot of the pdf of the r.v.
  number of rolls until you see 60 primes

• Here is a plot of the pdf of the r.v.
  number of rolls until you see 80 primes

• Here is a plot of the pdf of the r.v.
  number of rolls until you see 100 primes

Noga Alon's Home Page

Yaakov Malinovsky's Home Page

Lucy Martinez's Home Page

Doron Zeilberger's Home Page