info prev up next book cdrom email home

Euler's Rule

The numbers $2^n pq$ and $2^n r$ are Amicable Numbers if the three Integers

$\displaystyle p$ $\textstyle \equiv$ $\displaystyle 2^m (2^{n-m}+1)-1$  
$\displaystyle q$ $\textstyle \equiv$ $\displaystyle 2^m (2^{n-m}+1)-1$  
$\displaystyle r$ $\textstyle \equiv$ $\displaystyle 2^{n+m} (2^{n-m}+1)^2-1$  

are all Prime numbers for some Positive Integer $m$ satisfying $1\leq m \leq n-1$ (Dickson 1952, p. 42). However, there are exotic Amicable Numbers which do not satisfy Euler's rule, so it is a Sufficient but not Necessary condition for amicability.

See also Amicable Numbers


Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, 1952.

© 1996-9 Eric W. Weisstein