Page 3 from the Prime Listening Guide

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You have heard the primes modulo seven, and hopefully you noted that all seven notes were played but that the lowest note occurred just once. The lowest note was the fourth note played and came from the prime seven, which, of course, leaves a remainder of zero. (Any number that leaves a remainder of zero when divide by seven is a multiple of seven--so can only be prime if it is exactly seven.) By Dirichlet'ís theorem on primes in arithmetic progressions (a mouthful eh?), all of the other notes are played infinitely often as we play all of the primes.

Now listen to the primes modulo six (on a percussion organ) as we again pause for a question break. EAREYE
If you prefer, here are the same primes (the first 300) played with percussion only. EAR


If we play all of the primes modulo six...
  1. * How many of the notes repeat infinitely often?

    2, 3, 4, 5 or 6.

  2. ** How many different notes do you hear all together?

    2, 3, 4, 5 or 6.

  3. *** Can you list, modulo the positive integer n, which notes are played infinitely often? which just once? and which are never played?

    yes, no.

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