# What is a Perfect Number?

### ``` A number is called perfect if it is equal to the sum of its divisors. Six is perfect: 6 = 1 + 2 + 3. 28 is perfect: 28 = 1 + 2 + 4 + 7 + 14. N is an even perfect number if and only if N = 2^(q-1) * (2^q - 1) and 2^q - 1 is prime It should also be noted that for 2^q - 1 to be prime q must be prime. So when we search for even perfect numbers, we search on q equal to the primes. The numbers M(q) = 2^q - 1 (with q prime) are called Mersenne numbers. If M(q) = is prime then it is called a Mersenne prime. If q makes a Mersenne prime then P(q) = 2^(q-1) * (2^q - 1) is a Perfect number. The first few consecutive perfect numbers are - P( 2) = 2 * 3 = 6 P( 3) = 4 * 7 = 28 P( 5) = 16 * 31 = 496 P( 7) = 64 * 127 = 8128 P(13) = 4096 * 8191 = 33,550,336 P(17) = 65,536 * 131,071 = 8,589,869,056 P(19) = 262,144 * 524,287 = 137,438,691,328 P(31) = 1,073,741,824 * 2,147,483,647 = 2,305,843,008,139,952,128 P(61) = 1,152,921,504,606,846,976 * 2,305,843,009,213,693,951 = 2,658,455,991,569,831,744,654,692,615,953,842,176 P(61) can be computed with the program XICalc by: Command: p=PEq(61) p = 26,58455,99156,98317,44654,69261,59538,42176 (37 Digits) See q for all known perfect numbers It was pointed out to me by V. V. Raman that "Every Perfect Number is 1 (mod 9). V. V. Raman (1985)." This is true for all even perfect numbers except 6, but actually 2^(q-1) * (2^q - 1) = 1 (mod 9) iff q is an odd positive integer. This is easy to prove. For q = 1, 3, 5, ... 2^(q-1) mod 9 = 1, 4, 7, 1, 4, 7, ... and repeats the 1, 4, 7, 2^q - 1 mod 9 = 1, 7, 4, 1, 7, 4, ... and repeats the 1, 7, 4, so their product mod 9 = 1, 1, 1, 1, 1, 1 and repeats the 1. ```

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