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Perfect Number Finder

Enter an upper limit to list every perfect number up to it.

How it is calculated

For every whole number n from 2 up to your limit, the tool adds up its proper divisors, meaning all divisors smaller than n including 1. It tests divisors only up to √n and counts each one together with its partner n ÷ d, which keeps the search fast. A number is perfect when sum of proper divisors = n. The output has three parts: the list of perfect numbers found, how many there are, and the largest one. If none is found, a dash is shown and the count is 0. The limit must be at least 1, and anything above 100,000 is silently reduced to 100,000, so the search never goes beyond that. Numbers in the list use thousands separators (8,128).

Worked example

Set the limit to 10,000. The tool checks every n from 2 to 10,000. Three matches are easy to verify by hand: 6 = 1 + 2 + 3, 28 = 1 + 2 + 4 + 7 + 14, and 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248. The fourth is 8,128. The result reads 6, 28, 496, 8,128, with count 4 and largest 8,128. The next perfect number, 33,550,336, is far beyond this range, which is why raising the limit to 100,000 adds nothing.

Try this example →

Common mistakes

  • Expecting more results from a bigger limit. Only four perfect numbers exist below 100,000 (6, 28, 496, 8,128), and the tool does not search above 100,000 anyway.
  • Entering a limit above 100,000 and assuming it was fully searched. The value is capped there, so a search for larger perfect numbers will not find them.
  • Forgetting that 1 is not perfect. Its only proper divisor sum is 0, so the list starts at 6.
  • Confusing perfect numbers with prime numbers or with perfect squares. A perfect number is defined by its divisor sum only.
  • Treating an empty result (a dash) as an error. A limit below 6, such as 5, correctly finds nothing.

About this calculator

A perfect number is a positive integer that equals the sum of its own proper divisors (all divisors except itself) — 6 is perfect because 1+2+3=6, and 28 is perfect because 1+2+4+7+14=28. These numbers are extremely rare: only four were known to the ancient Greeks (6, 28, 496, 8128), and even today only a few dozen are known in total.

Every known even perfect number follows the pattern discovered by Euclid and completed by Euler nearly two thousand years later — the Euclid–Euler theorem — which states that an even number is perfect if and only if it has the form 2^(p−1) × (2^p − 1), where 2^p − 1 is a Mersenne prime. Whether any odd perfect numbers exist at all is still an unsolved problem in number theory. This finder is a number-theory teaching tool: enter an upper limit and it lists every perfect number up to that bound by checking each candidate’s divisor sum.

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Comments (1)

  • Editorial team

    Run it with a limit of 10,000 and you get 6, 28, 496 and 8,128. Each of these is a power of two multiplied by a Mersenne prime, which is why such numbers are so rare: every even perfect number has this shape, and no odd perfect number has ever been found. The search stops at 100,000, so a larger limit adds nothing. Are you checking a homework claim or exploring divisor sums? Tell us which pattern you spotted.

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