Divide and Conquer
Page 2 from the Prime Listening Guide

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There are infinitely many primes, and we could never discern infinitely many notes, so we will make the number of notes finite using a standard number theory tool--modular arithmetic.

Method Two: Rather than use the primes themselves, we divide each prime by a fixed number and then just play the remainders. The fixed number is then called the modulus or base. For example, if we choose seven as our modulus, then for the primes {2, 3, 5, 7, 11, 13, 17, 19, 23, ...} we would play {2, 3, 5, 0, 4, 6, 3, 5, 2, ...}. These would be rather low notes (using the midi standard), so to place them near the center of the keyboard I add a constant (such as 55).

Why not listen to the primes (modulo seven) as you answer the following questions? (You may also click on the eye to see the "musical" score.) EAREYE
You might be able to separate these notes better by ear if we use a different percussion sound for each note (each remainder modulo seven). EAR


Suppose we play all of the primes using this method.
  1. * Using the modulus seven (we often say "modulo seven"), will all seven notes be played?

    yes, no.

  2. ** There are infinitely many primes, so at least one of the notes must be played infinitely often. Modulo seven, how many of the seven notes are played infinitely often?

    1, 3, 6.

  3. ** If a key is played at least twice (for any fixed modulus), how many times is it played?

    at least twice, at least six times, infinitely many times!

  4. *** Playing modulo n we have n keys. Suppose that for a square-free n that less than half of the notes are played (that is, more than half are never played, not even once). What do we know about the number of prime divisors of n?

    nothing, there are at most two, there more than two.

The Prime Pages
Another prime page by Chris K. Caldwell <caldwell@utm.edu>