Marshall University Math Colloquium

October 22, 2004

“What is the opposite of a prime number?”

John Drost

Marshall University

Abstract

Prime numbers have only two divisors, themselves and 1. The opposite pole would be a number that has a lot of divisors. To quantify this further, we say a positive integer is highly composite if it has more divisors than any smaller positive integer. The sequence of highly composite numbers starts 1, 2, 4 (3 divisors), 6 (4 divisors), 12 (6 divisors), ... The concept is due to Ramanujan, but examples occur much further back, for example, Plato thought 5,040 a good number for the citizens of a city since it could be divided in so many ways.