Patterns in Final Digits of Consecutive Primes

Directed by Robert Lemke Oliver

Project Description

Apart from the prime numbers 2 and 5, every prime number ends in a digit that’s one of 1, 3, 7, and 9.  It turns out that there are infinitely many primes that have each possible last digit, and what’s more, that each last digit is essentially equally likely.  Thus, we can loosely think of the primes as being like playing cards drawn from an infinite deck, with the last digit corresponding to one of the four suits.  Surprisingly, however, if we look at two consecutive prime numbers, it turns out that the sixteen possible patterns are not all equally likely and exhibit strong biases against repetition.  In the analogy with playing cards, this is like saying that if I draw a club, the next card is substantially less likely to be a club, and moreover that the three remaining suits are not all equally likely either.  (The primes are weird!)  However, what appears to be true, and what some students might explore this summer, is that if you draw more cards from the deck of primes — say, 10 or 20 — there’s a certain regularity in the most common patterns of suits that begins to emerge.

Desired Background

To succeed at this project, students should be familiar with the basic idea of a proof, have some experience doing computations with a computer, and have some familiarity with concepts from discrete math or elementary number theory (including what it means to take one number modulo another).  No prior experience working with the primes will be assumed.