Perfect NumbersDate: 10/17/96 at 20:18:58 From: Anonymous Subject: Perfect numbers Is there any way, other than by trial and error, to figure out what the perfect numbers are? Date: 11/05/96 From: Doctor Yiu Subject: Re: Perfect numbers Dear Tahquitz, A perfect number is a positive integer that is equal to the sum of all its divisors, including 1, but not the number itself. The two smallest perfect numbers are 6 = 1 + 2 + 3, 28 = 1 + 2 + 4 + 7 + 14. The search for perfect numbers began in ancient times. Euclid (3rd Century BC) gave a construction of perfect number based on the notion of prime numbers: If (2^k-1) is a prime number, then 2^(k-1)(2^k-1) is a perfect number. For example: If k = 2, 2^k-1 = 3 is prime, which gives 2^(2-1)(2^2-1) = 2 times 3 = 6, the first (even) perfect number. Similarly, for k = 3, 2^k-1 = 7 gives 28, for k = 5, 2^k-1 = 31 gives 2^4 x 31 = 496, etc. etc. (Note: When k = 4, (2^k-1) = 15 is NOT a prime number, and we CANNOT generate a perfect number by this method). Since the 17th Century, the number (2^k-1) has been known as a Mersenne number, denoted by M_k. In the 18th Century, the great Swiss mathematician Euler proved that all EVEN perfect numbers must be of the form 2^(k-1)(2^k-1), where (2^k-1) is PRIME. (It is not known whether or not there is an ODD perfect number.) A new record for the largest KNOWN perfect number was obtained in September 1996, along with a record for largest KNOWN Mersenne primes. Before that there were 33 known Mersenne primes. D. Slowinski and P. Gage also announced their discovery of a new Mersenne prime: 2^1257787 - 1, which has 378,632 digits. This gives 2^1257786 x 2^1257787 - 1, for a record largest KNOWN even perfect number. You can find more about all of these on the Internet at Neal Calkin's page: http://www.utm.edu/research/primes/mersenne.shtml -Doctor Yiu, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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