# perfect number

A perfect number $P$ is equal to the sum of its divisors (where the divisors include $1$, but not $P$ itself).

• Euclid: If $2^n-1$ is prime then $2^{n-1}(2^n-1)$ is perfect
• Euler: all even perfect numbers are of the form $2^{p-1}(2^p-1)$, where $2^p-1$ is a Mersenne prime (and so $p$ is prime).
• 6 = 1 + 2 + 3 = 2 (22-1)
• 28 = 1 + 2 + 4 + 7 + 14 = 22 (23-1)
• 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 24 (25-1)
• 8128 = 1 + 2 + 4 + ... + 64 + 127 + ... + 4064 = 26 (27-1)
• 33,550,336 = 1 + ... + 4096 + 8191 + ... + 16775168 = 212 (213-1)
• 8,589,869,056 = 1 + ... + 65536 + 131071 + ... + 4294934528 = 216 (217-1)
• 137,438,691,328 = 1 + ... + 262144 + 524287 + ... + 68719345664 = 218 (219-1)
• ...
• Every even perfect number ends in a '6' or an '8'.
• All even perfect numbers are triangular numbers.
• 6 = 1 + 2 + 3 = 1 + 2 + (22-1)
• 28 = 1 + 2 + 3 + ... + 7 = 1 + 2 + 3 + ... + (23-1)
• 496 = 1 + 2 + 3 + ... + 31 = 1 + 2 + 3 + ... + (25-1)
• 8128 = 1 + 2 + 3 + ... + 127 = 1 + 2 + 3 + ... + (27-1)
• 33,550,336 = 1 + 2 + 3 + ... + 8191 = 1 + 2 + 3 + ... + (213-1)
• 8,589,869,056 = 1 + 2 + 3 + ... + 131071 = 1 + 2 + 3 + ... + (217-1)
• 137,438,691,328 = 1 + 2 + 3 + ... + 524287 = 1 + 2 + 3 + ... + (219-1)
• ...
• general result: $n: \mathbb{N} \vdash 2^{n-1} (2^n-1) = 1 + 2 + 3+ \ldots + (2^n-1)$
• Every even perfect number, other than 6, is the sum of consecutive odd cubes.
• (6, with p=2, does not fit the pattern)
• 28 = 13 + 33 = 13 + (2(3+1)/2-1)3
• 496 = 13 + 33 + 53 + 73 = 13 + 33 + 53 + (2(5+1)/2-1)3
• 8128 = 13 + 33 + ... + 153 = 13 + 33 + ... + (2(7+1)/2-1)3
• 33,550,336 = 13 + 33 + ... + 1273 = 13 + 33 + ... + (2(13+1)/2-1)3
• 8,589,869,056 = 13 + 33 + ... + 5113 = 13 + 33 + ... + (2(17+1)/2-1)3
• 137,438,691,328 = 13 + 33 + ... + 10233 = 13 + 33 + ... + (2(19+1)/2-1)3
• ...
• conjecture: $n: \mbox{Odd}\, ?{\vdash}\, 2^{n-1} (2^n-1) = 1^3 + 3^3 + \ldots + (2^{(n+1)/2}-1)^3$
• No odd perfect numbers are known, but if one does exist, a lot is known about it:
• it is a perfect square multiplied by an odd power of a single prime
• it has at least 8 distinct prime factors
• it has at least 75 prime factors (not necessarily distinct)
• its largest prime factor greater that 107
• its second largest prime factor is greater that 104
• its third largest prime factor is greater that 102
• it is divisible by a prime component greater that 1020
Exhaustive computer search has shown that there are no odd perfect numbers less than 10300.
[My thanks to Douglas Iannucci and Joshua Zelinsky for some of this information, some of which is from the work of Kevin Hare.]