the only perfect number of the form x(n) + y(n)
A perfect number is an integer that is equal to the sum of its positive divisors (not including itself). Therefore, 6 is a perfect number, since its positive divisors are 1, 2, and 3 and 1+2+3 = 6. 28, 496, and 8128 are also perfect numbers. At present, there are over 30 known perfect numbers, all even. All even perfect numbers are of the form 2^(p-1)(2^p - 1), where p is any positive integer, exceeding unity, that makes (2^p -1) is prime. The primes of the form (2^p - 1), where p is a prime, are called Mersenne primes after the French mathematician who, in 1644, announced a list of new perfect numbers. The known values of p that yield Mersenne primes and corresponding perfect numbers are: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1,279, 2,203, 2,281, 3,217, 4,253, 4,423, 9,689, 9,941, 11,213, and 19,937. Though there are undoubtedly many more beyond p = 19,937, their size grows rapidly as p increases. The 15th Mersenne prime, (2^1,279 - 1) has 386 digits. It is not known if there are any odd perfect numbers; none has been found, but it has not been proved that one cannot exist. The first eight perfect numbers are 6, 28, 496, 8128, 33,550,336, and 8,589,869,056, 137,438,691,328, and 2,305,843,008,139,952,128, having p's of 2, 3, 5, 7, 13, 17, 19, and 31.
Any of the perfect numbers can be written in the form of x(n) + y(n)
4(1 + 2(1) = 6
16(1) + 12(1) = 28
8(2) + 6(2) = 28
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