Perfect Number - Minor Results

Minor Results

All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy's strong law of small numbers:

• An odd perfect number is not divisible by 105.
• Every odd perfect number is of the form N ≡ 1 (mod 12), N ≡ 117 (mod 468), or N ≡ 81 (mod 324).
• The only even perfect number of the form x3 + 1 is 28 (Makowski 1962).
• 28 is also the only even perfect number that is a sum of two positive integral cubes (Gallardo 2010).
• The reciprocals of the divisors of a perfect number N must add up to 2:
  • For 6, we have ;
  • For 28, we have, etc. (This is particularly easy to see, just by taking the definition of a perfect number, and dividing both sides by n.)
• The number of divisors of a perfect number (whether even or odd) must be even, because N cannot be a perfect square.
  • From these two results it follows that every perfect number is an Ore's harmonic number.
• The even perfect numbers are not trapezoidal numbers; that is, they cannot be represented as the difference of two positive non-consecutive triangular numbers. There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and a class of numbers formed from Fermat primes in a similar way to the construction of even perfect numbers from Mersenne primes.
• The number of perfect numbers less than n is less than, where c > 0 is a constant. In fact it is, using little-o notation.