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Perfect Numbers Today Are Defined in Terms of the Restricted Divisor Function, S(N), Which Is Inherently Defined by the Divisor Function, Σ(N)

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Perfect Numbers Today Are Defined in Terms of the Restricted Divisor Function, S(N), Which Is Inherently Defined by the Divisor Function, Σ(N) Is there an odd one, and are there are infinitely many? M. Alex W.-Higgins & Jin Cho Fullerton College Mentor and editor: Dr. Dana Clahane Definition of a Perfect Number A positive integer such that the sum of its proper divisors (divisors not including the integer itself) equals the integer. For example: Let us consider whether 6 is a perfect number Its proper divisors are 1, 2, and 3 (excluding 6), and 1+2+3=6 The sum of the divisors equal the integer, thus 6 is a perfect number (the smallest, in fact)! Perfect numbers today are defined in terms of the restricted divisor function, s(n), which is inherently defined by the divisor function, σ(n). For a perfect number n, note that by definition, n=s(n) In terms of the divisor function we find: n=s(n)=σ(n)-n or, equivocally σ(n)=2n For example: Let n=6. Note that : σ(6)=1+2+3+6=12=2n Thus (we have proved once again that) 6 is a perfect number! This form is useful for defining perfect numbers, and is used for demonstrating perfect numbers’ relationship to Mersenne primes Perfect numbers were thought to have significance in ancient societies, particularly Greece Pythagoras and his followers thought they had mystical properties Later, perfect numbers were investigated thoroughly by Euclid Specifically examined in detail around 300 BCE in Euclid’s Elements Euclid discovered that multiplying, for each prime p, the quantities 2p and 2p – 1, the second of which he notes is always prime when p is prime, yields a perfect number, at least for the first few primes. Euclid discovered the first four perfect numbers this way and generated the following function: 2p−1(2p − 1) is a perfect number for some p > 1 and if (2p − 1) is prime 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 Note: the form 2p − 1 relates to the Mersenne primes Around 100 CE Nicomachus of Gerasa claimed that the following statements regarding perfect numbers in his work Introductio Arethmetica, hold: I. The nth perfect number always has n digits** II. All perfect numbers are even III. All perfect numbers alternate 6 and 8 as their final digit** IV. 2p−1(2p − 1) is perfect for p is prime and 2p − 1 is prime (Euclid) V. There are infinitely many perfect numbers ** Indicates that the given statement has since then been disproved Hudalrichus Regius determined the fifth perfect number in 1536 (33550336), thus disproving Nicomachus’ first point, since this number has 8 digits, not 5. In 1603 Pietro Cataldi determined the sixth perfect number (8589869056), invalidating Nicomachus’ third claim of alternating final digits Later, in the seventeenth century, Descartes speculated the possibility of odd perfect numbers and believed that Euclid’s formula yields all possible even perfect numbers The is a close relationship between perfect numbers and Mersenne primes (it has neither been proven nor disproven that there infinitely many of these). Mersenne primes take the following form: M = 2n − 1 where M and n are both prime (otherwise M is simply called a Mersenne number) It was discovered that each Mersenne prime corresponds to exactly one perfect number, but not necessarily vice-versa There are thought to be infinitely many Mersenne primes, the first five being: 3, 7, 31, 127, 8191 The first 39 even perfect numbers, given by the form 2p−1(2p − 1), where p can be any of the following primes: 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 By 1911 mathematicians had determined the first 10 perfect numbers by hand Today there are 46 known perfect numbers, the latest and largest being discovered in 2008 There remain no known odd perfect numbers, nor is it know if one such number exists! Are there infinitely many even perfect numbers? Can we determine a more effective method of determining primes, and in turn, perfect numbers? Are there any odd perfect numbers? (oldest unanswered mathematics question in existence today) What is the relationship between perfect numbers and Mersenne primes, and how can this relationship be used to locate larger perfect numbers? 1. Greathouse, Charles and Weisstein, Eric W. “Odd Perfect Number.” From MathWolrd—A Wolfram Web Resource http://mathworld.wolfram.com/OddPerfectNumber.html 2. O’Conner, J.J. and Robertson, E.F. “Perfect Numbers.” University of St. Andrews. www.history.mcs.st- andrews.ac.uk/HistTopics/Perfect_numbers.html 3. Weisstein, Eric W. “Mersenne Prime.” From MathWolrd—A Wolfram Web Resource http://mathworld.wolfram.com/MersennePrime.html 4. Weisstein, Eric W. “Perfect Number.” From MathWolrd—A Wolfram Web Resource http://mathworld.wolfram.com/PerfectNumber.html Picture Credits: http://www.squarecirclez.com/blog/wp-content/uploads/2010/12/euclid.jpg http://www.whoguides.com/wp-content/uploads/2009/10/Pythagoras.jpg http://bhlspectrum.wikispaces.com/file/view/descartes_003.jpg/33543801/descartes_003.jpg .
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