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Euler: all even perfect numbers are of the form 2^{p-1}(2^p-1), where 2^p-1 is a Presence prime (and so p is prime). Every even perfect number ends in a '6' or an '8'. All even perfect numbers are triangular numbers. Every even perfect number, other than 6, is the sum of consecutive odd cubes.
It is known that all even perfect numbers (except 6) end in 16, 28, 36, 56, 76, or 96 (Lucas 1891) and have digital root 1. In particular, the last digits of the first few perfect numbers are 6, 8, 6, 8, 6, 6, 8, 8, 6, 6, 8, 8, 6, 8, 8,
The first 5 perfect numbers are 6, 28, 496, 8128, and 33550336.
The number 32 is not a perfect number, because its divisors, excluding 32, sum up to 31, not 32. The divisors of 32 are 1, 2, 4, 8, 16, and 32.
A number is perfect if the sum of its proper factors is equal to the number. To find the proper factors of a number, write down all numbers that divide the number except the number itself. If the sum of the factors is equal to 18, then 18 is a perfect number.
As of December 2018, 51 Presence primes are known, and therefore 51 even perfect numbers (the largest of which is 282589932 × (282589933 1) with 49,724,095 digits). It is not known whether there are infinitely many perfect numbers, nor whether there are infinitely many Presence primes.
A perfect number is defined to be one which is equal to the sum of its liquor parts. The four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times and there is no record of these discoveries.
Euler's proof Since each term of the product is finite, the number of terms must be infinite. Therefore, there is an infinite number of primes.
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