A given number is perfect if it is equal to the sum of all its proper
divisors. This question was first posed by Euclid in ancient Greece.
This question is still open. Euler proved that if N is an odd
perfect number, then in the prime power decomposition of N, exactly
one exponent is congruent to 1 mod 4 and all the other exponents are
even. Furthermore, the prime occurring to an odd power must itself be
congruent to 1 mod 4. A sketch of the proof appears in Exercise 87,
page 203 of Underwood Dudley's Elementary Number Theory.
It has been shown that there are no odd perfect numbers 
.