Ore's conjecture

This conjecture states that there are no odd numbers satisfying a certain condition. Typically, there are known to be many even numbers (in some cases, there are known to be infinitely many even numbers) satisfying the condition.
View other such conjectures

Statement

The conjecture states that the harmonic mean of all the positive divisors of an odd natural number greater than $1$ cannot be an integer.

The harmonic mean of all the positive divisors is given by the expression:

$\frac{n{\sigma }_0(n)}{{\sigma }_1(n)}$,

where ${\sigma }_0$ is the divisor count function and ${\sigma }_1$ is the divisor sum function. Natural numbers $n$ for which this ratio is an integer are termed harmonic divisor numbers or Ore numbers, and Ore's conjecture can thus be stated more compactly as: there is no odd Ore number.

Related facts and conjectures

Weaker conjectures

• Odd perfect number conjecture: The harmonic mean of the divisors of a perfect number is an integer, hence Ore's conjecture implies that there does not exist any odd perfect number.

External links

• Ore's conjecture on Mathworld