# Glaisher's theorem

In number theory, Glaisher's theorem is an identity useful to the study of integer partitions. It is named for James Whitbread Lee Glaisher.

## Statement

It states that the number of partitions of an integer ${\displaystyle N}$ into parts not divisible by ${\displaystyle d}$ is equal to the number of partitions of the form

${\displaystyle N=N_{1}+\cdots +N_{k}}$

where

${\displaystyle N_{i}\geq N_{i+1}}$

and

${\displaystyle N_{i}\geq N_{i+d-1}+1,}$

that is, partitions in which no part is repeated d or more times.

When ${\displaystyle d=2}$ this becomes the special case, known as Euler's theorem, that the number of partitions of ${\displaystyle N}$ into distinct parts is the same as the number of partitions of ${\displaystyle N}$ into odd parts.

## Similar theorems

If instead of counting the number of partitions with distinct parts we count the number of partitions with parts differing by at least 2, a theorem similar to Euler's theorem known as Rogers' theorem (after Leonard James Rogers) is obtained:

The number of partitions whose parts differ by at least 2 is equal to the number of partitions involving only numbers congruent to 1 or 4 (mod 5).

For example, there are 6 partitions of 10 into parts differing by at least 2, namely 10, 9+1, 8+2, 7+3, 6+4, 6+3+1; and 6 partitions of 10 involving only 1, 4, 6, 9 ..., namely 9+1, 6+4, 6+1+1+1+1, 4+4+1+1, 4+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1. The theorem was discovered independently by Schur and Ramanujan.