Euler's Rule

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

$$p \equiv 2^m (2^{n-m}+1)-1$$

$$q \equiv 2^m (2^{n-m}+1)-1$$

$$r \equiv 2^{n+m} (2^{n-m}+1)^2-1$$

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

References

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