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  "body": "\n## Statement\n\nLet $n$ be an even positive integer.\nThen $n$ is a [[perfect number|0SB7Q8SYSF]] if and only if $n = 2^{p-1} (2^p - 1)$, where $2^p - 1$ is a [[Mersenne prime|06GPJF9V7M]].\n\n## Remarks\n\nThe primes $2^p - 1$ for $p = 2, 3, 5, 7$ give the perfect numbers $6$, $28$, $496$ and $8128$.\nSo the even perfect numbers correspond one to one to the Mersenne primes.\nThe proof uses the multiplicativity of the [[sum of divisors|009QCP5T3Z]]: $\\sigma(2^{p-1} q) = (2^p - 1)(q + 1)$ for an odd prime $q$.\n\n## History\n\nEuclid proved in the Elements that $2^{p-1}(2^p - 1)$ is perfect when $2^p - 1$ is prime.\nIbn al-Haytham conjectured the converse around 1000 CE.\nEuler proved the converse in the 18th century, in a paper published after his death.\n\n## Explanations\n\n### L1 human\n\nEuclid found a recipe for perfect numbers: take a prime of the form $2^p - 1$ and multiply it by $2^{p-1}$.\nEuler showed that the recipe gives every even perfect number.\nWhether an odd perfect number exists is still unknown.\n",
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