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Perfect numbers are related to Mersenne Primes (prime numbers that are one less than a power of 2) because a perfect number can be derived from a Mersenne prime. If M is a Mersenne prime, then M*(M+1)/2 is a perfect number. (This was proved by Euclid in the 4th century B.C.) All even perfect numbers are of this form (proved by Leonhard Euler in the 18th century). It is unknown whether there are any odd perfect numbers. |
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Perfect numbers are related to Mersenne Primes (prime numbers that are one less than a power of 2) because a perfect number can be derived from a Mersenne prime. If M is a Mersenne prime, then M*(M+1)/2 is a perfect number. (This was proved by Euclid in the 4th century B.C.) All even perfect numbers are of this form (as proved by Euler in the 18th century). So there is a one-to-one association between even perfect numbers and Mersenne primes. |
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Only a finite number (38) of Mersenne primes (hence perfect numbers) are presently known. It is unknown whether there are an infinite number. |
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Only a finite number of Mersenne primes (hence even perfect numbers) are presently known. It is unknown whether there are an infinite number of them. See the entry on Mersenne prime for additional information concerning the search for these numbers. |
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The first four perfect numbers were known in Antiquity: 6, 28, 496 and 8128. The fifth (33550336) was reported in 1461. The next two were found by Cataldi in 1588. The eigth was found by Euler in 1750. In the following two centuries four more were found. |
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It is unknown whether there are any odd perfect numbers. Various results have been obtained, but none that have helped to locate one or otherwise resolve the question of their existence. It is known that if an odd perfect number does exist, it must be greater than 10160. Also, it must have at least 8 distinct prime factors (and at least 11 if it is not divisible by 3), and it must have at least two prime factors greater than 138. |
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The first successful identification of a Mersenne prime by means of an electronic digital computer was achieved at 10:00 P.M. on January 30, 1952 using the U.S. National Bureau of Standard's Western Automatic Computer (SWAC) at the Institute for Numerical Analysis on the Los Angeles campus of the University of California, under the direction of D.H. Lehmer, using a computer program written by Prof. R.M. Robinson. It was the first Mersenne prime to be identified in seventy-five years. The next one was found by the computer a little less than two hours later. Three more were found by the same program in the next several months. Some related information can be found at http://xraysgi.ims.uconn.edu:8080/amicable.html |
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Some other related information can be found at http://xraysgi.ims.uconn.edu:8080/amicable.html |
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