A set X is said to be countable if the function f : N --> X is onto, where N is the set of natural numbers. An odd number M is said to be perfect if M is equal to the sum of its positive proper divisors (that is, including 1 but excluding M itself). An odd perfect number, according to an age-old result by Euler, has to have the form (p^k)(m^2), where gcd(p, m) = gcd(p^k, m^2) = 1 and p is congruent to k and also congruent to 1 modulo 4. If one exists, then an odd perfect number must be at least 10^500 (www.oddperfect.org), must have at least 9 distinct prime factors (Nielsen 2006) and must be a sum of two squares (Stuyvaert 1896).
2006-06-08
22:50:50
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JoseABDris
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Mathematics