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.]

• Conway and Guy. The Book of Numbers. chapter 5
• Devlin. All the Math that's Fit to Print. chapters 113, 138
• Hardy & Wright. Theory of Numbers. section 16.8