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The Quest for Perfection The quest for perfection Paul Pollack History lessons The quest for perfection Even perfect numbers Odd perfect numbers The statistical Paul Pollack perspective Sociable numbers University of Georgia Oct. 12, 2012 1 / 68 Paul Pollack The quest for perfection For example, n = 28 is perfect, since 28 = 1 + 2 + 4 + 7 + 14: But who decided adding divisors was a reasonable thing to do in the first place? WHAT IS... a perfect number? The quest for perfection P Let σ(n) := djn d be the usual sum-of-divisors function, and Paul Pollack P let s(n) := djn;d<n d be the sum-of-proper-divisors function, History lessons so that s(n) = σ(n) − n. Even perfect numbers Definition Odd perfect A natural number n is called perfect if σ(n) = 2n, or numbers The statistical equivalently, if s(n) = n. perspective Sociable numbers 2 / 68 Paul Pollack The quest for perfection But who decided adding divisors was a reasonable thing to do in the first place? WHAT IS... a perfect number? The quest for perfection P Let σ(n) := djn d be the usual sum-of-divisors function, and Paul Pollack P let s(n) := djn;d<n d be the sum-of-proper-divisors function, History lessons so that s(n) = σ(n) − n. Even perfect numbers Definition Odd perfect A natural number n is called perfect if σ(n) = 2n, or numbers The statistical equivalently, if s(n) = n. perspective Sociable numbers For example, n = 28 is perfect, since 28 = 1 + 2 + 4 + 7 + 14: 3 / 68 Paul Pollack The quest for perfection WHAT IS... a perfect number? The quest for perfection P Let σ(n) := djn d be the usual sum-of-divisors function, and Paul Pollack P let s(n) := djn;d<n d be the sum-of-proper-divisors function, History lessons so that s(n) = σ(n) − n. Even perfect numbers Definition Odd perfect A natural number n is called perfect if σ(n) = 2n, or numbers The statistical equivalently, if s(n) = n. perspective Sociable numbers For example, n = 28 is perfect, since 28 = 1 + 2 + 4 + 7 + 14: But who decided adding divisors was a reasonable thing to do in the first place? 4 / 68 Paul Pollack The quest for perfection Carl Pomerance has called this the \Goldilox classification”. All Greek to us The quest for perfection Paul Pollack Among simple even numbers, some are superabundant, others are deficient: these two classes History lessons are as two extremes opposed one to the other; as for Even perfect numbers those that occupy the middle point between the two, Odd perfect they are said to be perfect. numbers The statistical perspective { Nicomachus (ca. 100 AD), Introductio Arithmetica Sociable numbers Abundant: s(n) > n, e.g., n = 12. Deficient: s(n) < n, e.g., n = 5. Perfect: s(n) = n, e.g., n = 6. 5 / 68 Paul Pollack The quest for perfection All Greek to us The quest for perfection Paul Pollack Among simple even numbers, some are superabundant, others are deficient: these two classes History lessons are as two extremes opposed one to the other; as for Even perfect numbers those that occupy the middle point between the two, Odd perfect they are said to be perfect. numbers The statistical perspective { Nicomachus (ca. 100 AD), Introductio Arithmetica Sociable numbers Abundant: s(n) > n, e.g., n = 12. Deficient: s(n) < n, e.g., n = 5. Perfect: s(n) = n, e.g., n = 6. Carl Pomerance has called this the \Goldilox classification”. 6 / 68 Paul Pollack The quest for perfection Goldilox explained The quest for perfection The superabundant number is . as if an adult animal Paul Pollack was formed from too many parts or members, having History \ten tongues", as the poet says, and ten mouths, or lessons nine lips, and provided with three lines of teeth; or Even perfect numbers with a hundred arms, or having too many fingers on Odd perfect one of its hands. The deficient number is . as if numbers an animal lacked members or natural parts . if he The statistical perspective does not have a tongue or something like that. Sociable numbers . In the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort | of which the most exemplary form is that type of number which is called perfect. 7 / 68 Paul Pollack The quest for perfection Just as . ugly and vile things abound, so superabundant and deficient numbers are plentiful and can be found without a rule. { Nicomachus You can see a lot just by looking The quest for perfection Let's list the first several terms of each of these sequences. Paul Pollack History Abundants: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, lessons 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, . Even perfect numbers Deficients: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, Odd perfect numbers 21, 22, 23, 25, 26, 27, . The statistical perspective Perfects: 6, 28, 496, 8128, 33550336, 8589869056, Sociable numbers 137438691328, 2305843008139952128, . 8 / 68 Paul Pollack The quest for perfection You can see a lot just by looking The quest for perfection Let's list the first several terms of each of these sequences. Paul Pollack History Abundants: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, lessons 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, . Even perfect numbers Deficients: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, Odd perfect numbers 21, 22, 23, 25, 26, 27, . The statistical perspective Perfects: 6, 28, 496, 8128, 33550336, 8589869056, Sociable numbers 137438691328, 2305843008139952128, . Just as . ugly and vile things abound, so superabundant and deficient numbers are plentiful and can be found without a rule. { Nicomachus 9 / 68 Paul Pollack The quest for perfection For example: N = 6 (n = 2), N = 28 (n = 3), and N = 8116868 ··· 22457856 corresponding to n = 3021377: (About 1.8 million middle digits omitted.) Here's looking at Euclid The quest for perfection A rule for generating perfect numbers was given by Euclid in Paul Pollack his Elements, written around 300 BCE. History lessons Theorem (Euclid) Even perfect numbers If as many numbers as we please beginning from Odd perfect a unit be set out continuously in double numbers proportion, until the sum of all becomes a prime, The statistical perspective and if the sum multiplied into the last make Sociable some number, the product will be perfect. numbers 10 / 68 Paul Pollack The quest for perfection Here's looking at Euclid The quest for perfection A rule for generating perfect numbers was given by Euclid in Paul Pollack his Elements, written around 300 BCE. History lessons Theorem (Euclid) Even perfect numbers If 2n − 1 is a prime number, then Odd perfect numbers N := 2n−1(2n − 1) The statistical perspective Sociable is a perfect number. numbers For example: N = 6 (n = 2), N = 28 (n = 3), and N = 8116868 ··· 22457856 corresponding to n = 3021377: (About 1.8 million middle digits omitted.) 11 / 68 Paul Pollack The quest for perfection A sequel two thousand years in the making The quest for perfection Paul Pollack Theorem (Euler) History lessons If N is an even perfect number, then N can be Even perfect written in the form numbers Odd perfect n−1 n numbers N = 2 (2 − 1); The statistical perspective where 2n − 1 is a prime number. Sociable numbers There are several proofs of Euler's theorem known. We choose to present an elegant argument of Graeme Cohen from the Math. Gazette. 12 / 68 Paul Pollack The quest for perfection The mouth speaketh The quest for perfection Let's define the abundancy of the number n as the ratio Paul Pollack h(n) := σ(n)=n: History lessons Even perfect We use the following simple lemma: numbers Odd perfect Lemma numbers If n j m, then h(n) ≤ h(m). Equality holds only if n = m. The statistical perspective Sociable numbers Proof. This is immediate upon observing that for every n, 1 X X d X 1 X 1 h(n) = d = = = : n n n=d e djn djn djn ejn 13 / 68 Paul Pollack The quest for perfection Using that σ is multiplicative, we find that 2k+1q = 2N = σ(N) = σ(2k)σ(q) = (2k+1 − 1)σ(q): It follows that 2k+1 − 1 divides q. Hence, σ(N) 2k+1 − 1 σ(q) 2 = = · N 2k q 2k+1 − 1 σ(2k+1 − 1) 2k+1 − 1 2k+1 ≥ · ≥ · = 2: 2k 2k+1 − 1 2k 2k+1 − 1 Making Euler slicker The quest for perfection We now prove Euler's characterization of even perfect Paul Pollack numbers, following Cohen. Say N is even perfect. Write N = 2kq, where q is odd and k ≥ 1. History lessons Even perfect numbers Odd perfect numbers The statistical perspective Sociable numbers 14 / 68 Paul Pollack The quest for perfection Hence, σ(N) 2k+1 − 1 σ(q) 2 = = · N 2k q 2k+1 − 1 σ(2k+1 − 1) 2k+1 − 1 2k+1 ≥ · ≥ · = 2: 2k 2k+1 − 1 2k 2k+1 − 1 Making Euler slicker The quest for perfection We now prove Euler's characterization of even perfect Paul Pollack numbers, following Cohen. Say N is even perfect. Write N = 2kq, where q is odd and k ≥ 1. History lessons Using that σ is multiplicative, we find that Even perfect numbers k+1 Odd perfect 2 q = 2N = σ(N) numbers k k+1 The statistical = σ(2 )σ(q) = (2 − 1)σ(q): perspective Sociable It follows that 2k+1 − 1 divides q.
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