Perfect Number Calculator

Sum of factors equal the perfect number

A perfect number is defined as any integer the sum of whose factors equal the number.

For some reason 1 is not considered a perfect number.

The calculator returns the list of perfect numbers up to the limit set by the user. To use the calculator simply enter the desired limit for the search.

From Wikipedia: Euclid proved that 2^P-1(2^P-1) is an even perfect number whenever 2^P-1 is a prime number. These primes are also known as Mersenne primes. Mersenne primes are very rare: only 45 of the first 2,270,720 primes yield a prime by this formula. Euler proved that all Mersenne primes yield all of the perfect numbers.

The following primes yield the first 45 perfect numbers: p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, and 37156667.

Yes, most perfect numbers known are massively large numbers. This calculator takes about a 10 seconds to sieve through the first 100,000 numbers.

for p = 2: 2¹(2² - 1) = 6

for p = 3: 2²(2³ - 1) = 28

for p = 5: 2⁴(2⁵ - 1) = 496

for p = 7: 2⁶(2⁷ - 1) = 8128

for p = 13: 2¹²(2¹³ - 1) = 33,550,336

for p = 17: 2¹⁶(2¹⁷ - 1) = 8,589,869,056

for p = 19: 2¹⁸(2¹⁹ - 1) = 137,438,691,328

Also from Wikipedia: each even perfect number is the (2^P - 1)th triangular number (and hence equal to the sum of the integers from 1 to 2^P - 1). So, for example, 6 is the 3rd triangular and is the sum of 1,2, & 3. Similarly, 8128 is the 7th triangular number and is the sum of the numbers from 1 to 127.

Enter a number and hit the button: Limit to Search

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