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) := d|n d be the usual sum-of-divisors function, and Paul Pollack P let s(n) := d|n,d 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) := d|n d be the usual sum-of-divisors function, and Paul Pollack P let s(n) := d|n,d 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) := d|n d be the usual sum-of-divisors function, and Paul Pollack P let s(n) := d|n,d 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 | 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 d|n d|n d|n e|n 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. numbers 15 / 68 Paul Pollack The quest for perfection 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. Hence, numbers 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 16 / 68 Paul Pollack The quest for perfection Making Euler slicker The quest for perfection We’ve just seen that Paul Pollack σ(N) 2k+1 − 1 σ(q) History 2 = = · lessons N 2k q Even perfect 2k+1 − 1 σ(2k+1 − 1) 2k+1 − 1 2k+1 numbers ≥ · ≥ · = 2. Odd perfect 2k 2k+1 − 1 2k 2k+1 − 1 numbers The statistical So equality holds throughout. This means that perspective Sociable k+1 numbers 2 − 1 = q and that σ(2k+1 − 1) = 2k+1, which forces 2k+1 − 1 to be prime. So N = 2k(2k+1 − 1), 17 / 68 where the second factor is prime. Paul Pollack The quest for perfection But we don’t know if there are infinitely many Mersenne primes, though there is some good evidence for this. We don’t even have a proof that there are infinitely composite numbers of the form 2p − 1. The media’s anti-Platonist bias The quest for perfection So even perfect numbers are in one-to-one correspondence with Paul Pollack Mersenne primes. Great! History lessons Even perfect numbers Odd perfect numbers The statistical perspective Sociable numbers 18 / 68 Paul Pollack The quest for perfection We don’t even have a proof that there are infinitely composite numbers of the form 2p − 1. The media’s anti-Platonist bias The quest for perfection So even perfect numbers are in one-to-one correspondence with Paul Pollack Mersenne primes. Great! History lessons But we don’t know if there are infinitely many Mersenne Even perfect primes, though there is some good evidence for this. numbers Odd perfect numbers The statistical perspective Sociable numbers 19 / 68 Paul Pollack The quest for perfection The media’s anti-Platonist bias The quest for perfection So even perfect numbers are in one-to-one correspondence with Paul Pollack Mersenne primes. Great! History lessons But we don’t know if there are infinitely many Mersenne Even perfect primes, though there is some good evidence for this. numbers Odd perfect numbers The statistical perspective Sociable numbers We don’t even have a proof that there are infinitely composite p 20 / 68 numbers of the form 2 − 1. Paul Pollack The quest for perfection How many is finitely many? Well, 2 2k N < (4k)(4k) (Pomerance, 1977). Odd perfect numbers: the (probably non-existent) elephant in the room The quest for Maybe the oldest open problem in number theory is the perfection question of whether there exist odd perfect numbers. Paul Pollack History Is there a formula for odd perfect numbers, like the lessons Euclid–Euler formula? Probably not. Even perfect numbers Odd perfect numbers Theorem (Dickson, 1913) The statistical perspective For each positive integer k, there are only Sociable finitely many odd perfect numbers N with numbers precisely k distinct prime factors. 21 / 68 Paul Pollack The quest for perfection Odd perfect numbers: the (probably non-existent) elephant in the room The quest for Maybe the oldest open problem in number theory is the perfection question of whether there exist odd perfect numbers. Paul Pollack History Is there a formula for odd perfect numbers, like the lessons Euclid–Euler formula? Probably not. Even perfect numbers Odd perfect numbers Theorem (Dickson, 1913) The statistical perspective For each positive integer k, there are only Sociable finitely many odd perfect numbers N with numbers precisely k distinct prime factors. How many is finitely many? Well, 2 2k N < (4k)(4k) (Pomerance, 1977). 22 / 68 Paul Pollack The quest for perfection Theorem (P.) The number of odd perfect N with k distinct prime factors is at most 2 4k . Size matters The quest for perfection Paul Pollack Pomerance’s result was later refined by Heath-Brown, Cook, History and finally Nielsen: lessons Even perfect Theorem numbers If N is an odd perfect number with k distinct prime factors, Odd perfect numbers then 4k The statistical N < 2 . perspective Sociable numbers 23 / 68 Paul Pollack The quest for perfection Size matters The quest for perfection Paul Pollack Pomerance’s result was later refined by Heath-Brown, Cook, History and finally Nielsen: lessons Even perfect Theorem numbers If N is an odd perfect number with k distinct prime factors, Odd perfect numbers then 4k The statistical N < 2 . perspective Sociable numbers Theorem (P.) The number of odd perfect N with k distinct prime factors is at most 2 4k . 24 / 68 Paul Pollack The quest for perfection We know quite a few conditions an odd perfect number must satisfy. For instance: e 2 1 N has the form p M , where p ≡ e ≡ 1 (mod 4) (Euler), 2 N has at least 9 distinct prime factors (Nielsen) 3 N has at least 101 prime factors counted with multiplicity (Ochem and Rao), 4 N has more than 1500 decimal digits (Ochem and Rao) Sylvester’s web The quest for . . . a prolonged meditation has satisfied me perfection that the existence of [an odd perfect Paul Pollack number] - its escape, so to say, from the History complex web of conditions which hem it in lessons on all sides - would be little short of a Even perfect numbers miracle. – J. J. Sylvester Odd perfect numbers The statistical perspective Sociable numbers 25 / 68 Paul Pollack The quest for perfection Sylvester’s web The quest for . . . a prolonged meditation has satisfied me perfection that the existence of [an odd perfect Paul Pollack number] - its escape, so to say, from the History complex web of conditions which hem it in lessons on all sides - would be little short of a Even perfect numbers miracle. – J. J. Sylvester Odd perfect numbers The statistical We know quite a few conditions an odd perfect number must perspective satisfy. For instance: Sociable numbers e 2 1 N has the form p M , where p ≡ e ≡ 1 (mod 4) (Euler), 2 N has at least 9 distinct prime factors (Nielsen) 3 N has at least 101 prime factors counted with multiplicity (Ochem and Rao), 4 26 / 68 N has more than 1500 decimal digits (Ochem and Rao) Paul Pollack The quest for perfection Sylvester’s web The quest for perfection Paul Pollack We know quite a few conditions an odd perfect number must History satisfy. For instance: lessons e 2 Even perfect 1 N has the form p M , where p ≡ e ≡ 1 (mod 4) (Euler), numbers Odd perfect 2 N has at least 9 distinct prime factors (Nielsen) numbers 3 The statistical N has at least 101 prime factors counted with multiplicity perspective (Ochem and Rao), Sociable numbers 4 N has more than 1500 decimal digits (Ochem and Rao) Does this justify Sylvester’s conjecture that odd perfect numbers don’t exist? It’s amusing to note that almost all natural numbers satisfy these four conditions! 27 / 68 Paul Pollack The quest for perfection If A is a subset of N = {1, 2, 3,... }, define the density of A as #A ∩ [1, x] lim . x→∞ x For example, the even numbers have density 1/2, and the prime numbers have density 0. But the set of natural numbers with first digit 1 does not have a density. Question Does the set of abundant numbers have a density? What about the deficient numbers? The perfect numbers? A new hope The quest for perfection Rather than try to study the individual members of a set, one Paul Pollack can take a statistical perspective, examining how many elements of the set there are below a given bound. History lessons Even perfect numbers Odd perfect numbers The statistical perspective Sociable numbers 28 / 68 Paul Pollack The quest for perfection Question Does the set of abundant numbers have a density? What about the deficient numbers? The perfect numbers? A new hope The quest for perfection Rather than try to study the individual members of a set, one Paul Pollack can take a statistical perspective, examining how many elements of the set there are below a given bound. History lessons Even perfect If A is a subset of N = {1, 2, 3,... }, define the density of A as numbers Odd perfect #A ∩ [1, x] numbers lim . x→∞ x The statistical perspective Sociable For example, the even numbers have density 1/2, and the numbers prime numbers have density 0. But the set of natural numbers with first digit 1 does not have a density. 29 / 68 Paul Pollack The quest for perfection A new hope The quest for perfection Rather than try to study the individual members of a set, one Paul Pollack can take a statistical perspective, examining how many elements of the set there are below a given bound. History lessons Even perfect If A is a subset of N = {1, 2, 3,... }, define the density of A as numbers Odd perfect #A ∩ [1, x] numbers lim . x→∞ x The statistical perspective Sociable For example, the even numbers have density 1/2, and the numbers prime numbers have density 0. But the set of natural numbers with first digit 1 does not have a density. Question Does the set of abundant numbers have a density? What about the deficient numbers? The perfect numbers? 30 / 68 Paul Pollack The quest for perfection It’s OK to be dense The quest for perfection Theorem (Davenport, 1933) Paul Pollack For each real u ≥ 0, consider the set History lessons Ds(u) = {n : s(n)/n ≤ u}. Even perfect numbers Odd perfect This set always possesses an asymptotic numbers density Ds(u). Considered as a function of u, The statistical perspective the function Ds is continuous and strictly Sociable increasing, with Ds(0) = 0 and Ds(∞) = 1. numbers Corollary The perfect numbers have density 0, the deficient numbers have density Ds(1), and the abundant numbers have density 1 − Ds(1). 31 / 68 Paul Pollack The quest for perfection Numerics The quest for perfection Paul Pollack History The following theorem improves on earlier work of Behrend, lessons Sali´e,Wall, and Del´eglise: Even perfect numbers Theorem (Kobayashi, 2010) Odd perfect numbers For the density of abundant numbers, we have The statistical perspective Sociable 0.24761 < 1 − Ds(1) < 0.24765. numbers So just under 1 in every 4 natural numbers is abundant, and just over 3 in 4 are deficient. 32 / 68 Paul Pollack The quest for perfection Counting perfect numbers The quest for perfection Let V (x) denote the number of perfect numbers n ≤ x. Paul Pollack Davenport’s theorem says that V (x)/x → 0 as x → ∞. History Can we say anything more precise? lessons Even perfect numbers Even perfect numbers correspond to primes of the form 2n − 1. Odd perfect We know 47 such values of n, the largest being numbers The statistical perspective n = 42643801. Sociable numbers Conjecture The number of n ≤ x for which 2n − 1 is prime is asymptotic to eγ log x. log 2 33 / 68 Paul Pollack The quest for perfection An exercise in heuristic reasoning The quest for perfection n Paul Pollack Where does this conjecture come from? First, if 2 − 1 is prime, then n = p is also prime. A random number of size ≈ 2p History 1 1 p lessons is prime with probability about log(2p) = p log 2 . But 2 − 1 has Even perfect a ‘leg up’ on being prime: One can easily prove that it’s never numbers divisible by any prime < 2p. This suggests multiply our naive Odd perfect numbers probability by The statistical perspective Y (1 − 1/q)−1 ≈ eγ log p. Sociable numbers q<2p This gives a “corrected” probability of about eγ/ log 2 · log p/p. Summing this probability over p ≤ x gives us an expected eγ count of about log 2 log x. 34 / 68 Paul Pollack The quest for perfection An exercise in heuristic reasoning The quest for perfection Paul Pollack History This conjecture on the distribution of Mersenne primes lessons suggests a corresponding conjecture for even perfect numbers: Even perfect numbers Odd perfect Conjecture numbers The number of even perfect numbers up to x is asymptotic to The statistical eγ perspective log 2 log log x, as x → ∞. Sociable numbers Since we don’t think there are any odd perfect numbers, we can also erase the word “even”. 35 / 68 Paul Pollack The quest for perfection Volkmann, 1955 V (x) = O(x5/6) Hornfeck, 1955 V (x) = O(x1/2) Kanold, 1956 V (x) = o(x1/2) Erd˝os,1956 V (x) = O(x1/2−δ) log x Kanold, 1957 V (x) = O(x1/4 ) log log x Hornfeck & Wirsing, 1957 V (x) = O(x) A good half-decade for perfect numbers The quest for perfection Paul Pollack Theorem History We have the following estimates for V (x): lessons Even perfect numbers Odd perfect numbers The statistical perspective Sociable numbers 36 / 68 Paul Pollack The quest for perfection Hornfeck, 1955 V (x) = O(x1/2) Kanold, 1956 V (x) = o(x1/2) Erd˝os,1956 V (x) = O(x1/2−δ) log x Kanold, 1957 V (x) = O(x1/4 ) log log x Hornfeck & Wirsing, 1957 V (x) = O(x) A good half-decade for perfect numbers The quest for perfection Paul Pollack Theorem History We have the following estimates for V (x): lessons Even perfect 5/6 numbers Volkmann, 1955 V (x) = O(x ) Odd perfect numbers The statistical perspective Sociable numbers 37 / 68 Paul Pollack The quest for perfection Kanold, 1956 V (x) = o(x1/2) Erd˝os,1956 V (x) = O(x1/2−δ) log x Kanold, 1957 V (x) = O(x1/4 ) log log x Hornfeck & Wirsing, 1957 V (x) = O(x) A good half-decade for perfect numbers The quest for perfection Paul Pollack Theorem History We have the following estimates for V (x): lessons Even perfect 5/6 numbers Volkmann, 1955 V (x) = O(x ) Odd perfect 1/2 numbers Hornfeck, 1955 V (x) = O(x ) The statistical perspective Sociable numbers 38 / 68 Paul Pollack The quest for perfection Erd˝os,1956 V (x) = O(x1/2−δ) log x Kanold, 1957 V (x) = O(x1/4 ) log log x Hornfeck & Wirsing, 1957 V (x) = O(x) A good half-decade for perfect numbers The quest for perfection Paul Pollack Theorem History We have the following estimates for V (x): lessons Even perfect 5/6 numbers Volkmann, 1955 V (x) = O(x ) Odd perfect 1/2 numbers Hornfeck, 1955 V (x) = O(x ) The statistical Kanold, 1956 V (x) = o(x1/2) perspective Sociable numbers 39 / 68 Paul Pollack The quest for perfection log x Kanold, 1957 V (x) = O(x1/4 ) log log x Hornfeck & Wirsing, 1957 V (x) = O(x) A good half-decade for perfect numbers The quest for perfection Paul Pollack Theorem History We have the following estimates for V (x): lessons Even perfect 5/6 numbers Volkmann, 1955 V (x) = O(x ) Odd perfect 1/2 numbers Hornfeck, 1955 V (x) = O(x ) The statistical Kanold, 1956 V (x) = o(x1/2) perspective 1/2−δ Sociable Erd˝os,1956 V (x) = O(x ) numbers 40 / 68 Paul Pollack The quest for perfection Hornfeck & Wirsing, 1957 V (x) = O(x) A good half-decade for perfect numbers The quest for perfection Paul Pollack Theorem History We have the following estimates for V (x): lessons Even perfect 5/6 numbers Volkmann, 1955 V (x) = O(x ) Odd perfect 1/2 numbers Hornfeck, 1955 V (x) = O(x ) The statistical Kanold, 1956 V (x) = o(x1/2) perspective 1/2−δ Sociable Erd˝os,1956 V (x) = O(x ) numbers log x Kanold, 1957 V (x) = O(x1/4 ) log log x 41 / 68 Paul Pollack The quest for perfection A good half-decade for perfect numbers The quest for perfection Paul Pollack Theorem History We have the following estimates for V (x): lessons Even perfect 5/6 numbers Volkmann, 1955 V (x) = O(x ) Odd perfect 1/2 numbers Hornfeck, 1955 V (x) = O(x ) The statistical Kanold, 1956 V (x) = o(x1/2) perspective 1/2−δ Sociable Erd˝os,1956 V (x) = O(x ) numbers log x Kanold, 1957 V (x) = O(x1/4 ) log log x Hornfeck & Wirsing, 1957 V (x) = O(x) 42 / 68 Paul Pollack The quest for perfection A good half-decade for perfect numbers The quest for perfection The sharpest result to date is due to Wirsing Paul Pollack (1959): For all x > 3, History W/ log log x lessons V (x) ≤ x Even perfect numbers for a certain absolute constant W . This is no Odd perfect numbers doubt still very far from the truth. The statistical perspective Sociable numbers In the opposite direction, the following conjecture is wide open: Conjecture There are infinitely many perfect numbers, i.e., V (x) → ∞ as x → ∞. 43 / 68 Paul Pollack The quest for perfection Higher order generalizations of the perfect numbers The quest for perfection Paul Pollack Problem (Catalan, 1888) History Start with a natural number n. What is the lessons eventual behavior of the sequence of iterates Even perfect numbers n, s(n), s(s(n)), s(s(s(n))),... (the aliquot Odd perfect sequence at n)? numbers The statistical perspective Example Sociable numbers n = 20 leads to the sequence 20, 22, 14, 10, 8, 7, 1, 0. Example n = 25 leads to the sequence 25, 6, 6, 6, 6, 6,... . 44 / 68 Paul Pollack The quest for perfection As noticed almost immediately by Perrott, this is false: Example n = 220 leads to the sequence 220, 284, 220, 284, 220, 284,... . Conjecture (Catalan–Dickson, 1913) Every starting n leads to a bounded sequence, i.e., either a sequence terminating in 0 or reaching a cycle. The first n for which this conjecture is in doubt is n = 276. Higher order generalizations The quest for perfection Conjecture (Catalan) Paul Pollack Every starting n leads to a sequence terminating at 0, or hits a History perfect number. lessons Even perfect numbers Odd perfect numbers The statistical perspective Sociable numbers 45 / 68 Paul Pollack The quest for perfection Conjecture (Catalan–Dickson, 1913) Every starting n leads to a bounded sequence, i.e., either a sequence terminating in 0 or reaching a cycle. The first n for which this conjecture is in doubt is n = 276. Higher order generalizations The quest for perfection Conjecture (Catalan) Paul Pollack Every starting n leads to a sequence terminating at 0, or hits a History perfect number. lessons Even perfect numbers As noticed almost immediately by Perrott, this is false: Odd perfect numbers Example The statistical perspective n = 220 leads to the sequence 220, 284, 220, 284, 220, 284,... . Sociable numbers 46 / 68 Paul Pollack The quest for perfection Higher order generalizations The quest for perfection Conjecture (Catalan) Paul Pollack Every starting n leads to a sequence terminating at 0, or hits a History perfect number. lessons Even perfect numbers As noticed almost immediately by Perrott, this is false: Odd perfect numbers Example The statistical perspective n = 220 leads to the sequence 220, 284, 220, 284, 220, 284,... . Sociable numbers Conjecture (Catalan–Dickson, 1913) Every starting n leads to a bounded sequence, i.e., either a sequence terminating in 0 or reaching a cycle. 47 / 68 The first n for which this conjecture is in doubt is n = 276. Paul Pollack The quest for perfection Higher order generalizations The quest for perfection Paul Pollack History Definition lessons Even perfect We call n a sociable number if the aliquot sequence at n is numbers purely periodic; in this case, the length of the period is called Odd perfect numbers the order of sociability. The statistical perspective Definition Sociable numbers An amicable number is a sociable number of order 2. In this case, the pair {n, s(n)} is an amicable pair. 48 / 68 Paul Pollack The quest for perfection We tend to scoff at the beliefs of the ancients. But we can’t scoff at them personally, to their faces, and this is what annoys me. (Deep Thoughts, Jack Handey) The quest for perfection Paul Pollack Persons who have concerned History themselves with talismans affirm that lessons the amicable numbers 220 and 284 Even perfect numbers have an influence to establish a union Odd perfect or close friendship between two numbers individuals. – Ibn Khaldun The statistical perspective Sociable numbers Al-Majriti (ca. 1050 years ago) claims to have tested the erotic effect of giving any one the smaller number 220 to eat, and himself eating the larger number 284. 49 / 68 Paul Pollack The quest for perfection Conjecture (Erd˝os) 1− For each > 0, we have V2(x) > x once x > x0(). Though we know > 12 million amicable pairs, we still have no proof that there are infinitely many, or even any plausible strategy. The distribution of amicable numbers The quest for perfection Let V (x) denote the number of sociable numbers of order k Paul Pollack k not exceeding x. (So V (x) = V1(x).) History lessons Conjecture (Bratley, Lunnon, and McKay) Even perfect numbers 1/2 V2(x)/x → 0 as x → ∞. Odd perfect numbers The statistical perspective Sociable numbers 50 / 68 Paul Pollack The quest for perfection Though we know > 12 million amicable pairs, we still have no proof that there are infinitely many, or even any plausible strategy. The distribution of amicable numbers The quest for perfection Let V (x) denote the number of sociable numbers of order k Paul Pollack k not exceeding x. (So V (x) = V1(x).) History lessons Conjecture (Bratley, Lunnon, and McKay) Even perfect numbers 1/2 V2(x)/x → 0 as x → ∞. Odd perfect numbers The statistical perspective Conjecture (Erd˝os) Sociable 1− numbers For each > 0, we have V2(x) > x once x > x0(). 51 / 68 Paul Pollack The quest for perfection The distribution of amicable numbers The quest for perfection Let V (x) denote the number of sociable numbers of order k Paul Pollack k not exceeding x. (So V (x) = V1(x).) History lessons Conjecture (Bratley, Lunnon, and McKay) Even perfect numbers 1/2 V2(x)/x → 0 as x → ∞. Odd perfect numbers The statistical perspective Conjecture (Erd˝os) Sociable 1− numbers For each > 0, we have V2(x) > x once x > x0(). Though we know > 12 million amicable pairs, we still have no proof that there are infinitely many, or even any plausible strategy. 52 / 68 Paul Pollack The quest for perfection The distribution of amicable numbers The quest for perfection The first theorem in this area is due to Erd˝os. Paul Pollack Theorem (Erd˝os1955) History lessons The set of natural numbers which belong to an amicable pair Even perfect numbers has asymptotic density zero. Odd perfect numbers The statistical A more explicit result was obtained in joint work with Rieger perspective twenty years later. Sociable numbers Theorem (Erd˝osand Rieger 1975) The set of amicable numbers has asymptotic density zero. In fact, x V (x) . 2 log log log x 53 / 68 Paul Pollack The quest for perfection A wholly inadequate sketch of the proof that V2(x)/x → 0 The quest for perfection It’s enough to prove this for the set consisting of the smaller Paul Pollack member of each amicable pair. History lessons Applying s to any of the numbers n in this set, we jump to the Even perfect numbers larger member s(n) of the pair. So each n in this set is Odd perfect abundant. But if we apply s one more time, we jump back numbers down to n. So s(n) is deficient. The statistical perspective What Erd˝osreally proved was the following theorem: Sociable numbers Theorem If n is abundant, then almost all of the time, s(n) is also abundant. More precisely, the set of abundant n for which s(n) is non-abundant has density zero. 54 / 68 Paul Pollack The quest for perfection A wholly inadequate sketch of the proof that V2(x)/x → 0 The quest for perfection How do you prove the theorem? Notice that whether or not n Paul Pollack is abundant depends on the size of the ratio History lessons σ(n) Y 1 1 Even perfect = 1 + + ··· + , numbers n p pe pekn Odd perfect numbers The statistical and the size of this ratio is usually governed by the small prime perspective powers dividing n. Sociable numbers Erd˝os’sidea was to show that almost always, s(n) and n have the same set of small prime power divisors. He argues that if n is abundant, it’s probably the small prime powers that put it over the top, and since s(n) shares these small prime powers, s(n) should be abundant too! 55 / 68 Paul Pollack The quest for perfection Why should σ(n) be so highly divisible? Let’s start by looking at a special case: Why is σ(n) almost always divisible by 3? Sketchiness, continued The quest for perfection Paul Pollack Why should n and s(n) share the same small prime power History lessons factors? By definition, Even perfect numbers s(n) = σ(n) − n. Odd perfect numbers Erd˝osnoticed that σ(n) is usually divisible by high powers of The statistical perspective all the small primes (powers higher than those that appear in a Sociable typical number n). So the ultrametric inequality forces s(n) numbers and n to share the same small prime power divisors. 56 / 68 Paul Pollack The quest for perfection Sketchiness, continued The quest for perfection Paul Pollack Why should n and s(n) share the same small prime power History lessons factors? By definition, Even perfect numbers s(n) = σ(n) − n. Odd perfect numbers Erd˝osnoticed that σ(n) is usually divisible by high powers of The statistical perspective all the small primes (powers higher than those that appear in a Sociable typical number n). So the ultrametric inequality forces s(n) numbers and n to share the same small prime power divisors. Why should σ(n) be so highly divisible? Let’s start by looking at a special case: Why is σ(n) almost always divisible by 3? 57 / 68 Paul Pollack The quest for perfection If q is a prime that appears to the first power in the factorization of n, then q + 1 = σ(q) | σ(n). So if 3 | q + 1, then 3 | σ(n). We said above that there are log log n of these primes q (typically). Half of the primes are −1 (mod 3), so it’s highly unlikely that none of our q satisfy q ≡ −1 (mod 3). Sketchiness, continued The quest for perfection A classical result of Hardy and Ramanujan states that almost Paul Pollack all numbers n have roughly log log n distinct prime factors; History lessons moreover, this holds even if we count only primes that appear Even perfect only to the first power. numbers Odd perfect numbers The statistical perspective Sociable numbers 58 / 68 Paul Pollack The quest for perfection We said above that there are log log n of these primes q (typically). Half of the primes are −1 (mod 3), so it’s highly unlikely that none of our q satisfy q ≡ −1 (mod 3). Sketchiness, continued The quest for perfection A classical result of Hardy and Ramanujan states that almost Paul Pollack all numbers n have roughly log log n distinct prime factors; History lessons moreover, this holds even if we count only primes that appear Even perfect only to the first power. numbers Odd perfect If q is a prime that appears to the first power in the numbers factorization of n, then The statistical perspective Sociable q + 1 = σ(q) | σ(n). numbers So if 3 | q + 1, then 3 | σ(n). 59 / 68 Paul Pollack The quest for perfection Sketchiness, continued The quest for perfection A classical result of Hardy and Ramanujan states that almost Paul Pollack all numbers n have roughly log log n distinct prime factors; History lessons moreover, this holds even if we count only primes that appear Even perfect only to the first power. numbers Odd perfect If q is a prime that appears to the first power in the numbers factorization of n, then The statistical perspective Sociable q + 1 = σ(q) | σ(n). numbers So if 3 | q + 1, then 3 | σ(n). We said above that there are log log n of these primes q (typically). Half of the primes are −1 (mod 3), so it’s highly unlikely that none of our q satisfy q ≡ −1 (mod 3). 60 / 68 Paul Pollack The quest for perfection The quest for perfection Paul Pollack History lessons Even perfect numbers This (non-rigorous) argument suggests that σ(n) is, almost all Odd perfect of the time, divisible by every natural number up to about numbers log log n. One can prove a (slightly weaker) version of this The statistical perspective claim using sieve methods. Sociable numbers 61 / 68 Paul Pollack The quest for perfection The distribution of amicable numbers The quest for perfection Paul Pollack Pomerance proved a much sharper upper bound on V2(x) than History lessons Erd˝osand Rieger. Even perfect numbers Theorem (Pomerance, 1981) Odd perfect numbers For all large x, The statistical perspective x V (x) ≤ . Sociable 2 1/3 numbers exp((log x) ) As a consequence, the sum of the reciprocals of the amicable numbers converges. 62 / 68 Paul Pollack The quest for perfection Sociable numbers of higher order, or “three is a crowd” The quest for perfection Example Paul Pollack Here is a sociable cycle of order 5 (found by Poulet in 1918): History lessons 12496 → 14288 → 15472 → 14536 → 14264 → 12496 → ... Even perfect numbers Odd perfect order of the cycle number of known examples numbers 1 47 The statistical perspective 2 > 12 million Sociable 4 206 numbers 5 1 6 5 8 3 9 1 28 1 63 / 68 Paul Pollack The quest for perfection x V (x) , k log log log log ··· log x where the denominator is a (3k)-fold iterated log. With Kobayashi and Pomerance, we improved this. For example, we can replace the denominator with any fixed power of log3 x. The distribution of sociable numbers The quest for perfection Theorem (Erd˝os,1976) Paul Pollack For each fixed k, the set of sociable numbers of order k has History density zero. lessons Even perfect numbers Erd˝os’sbounds for V (x) were very weak. How weak? His Odd perfect k numbers method proves: The statistical perspective Sociable numbers 64 / 68 Paul Pollack The quest for perfection The distribution of sociable numbers The quest for perfection Theorem (Erd˝os,1976) Paul Pollack For each fixed k, the set of sociable numbers of order k has History density zero. lessons Even perfect numbers Erd˝os’sbounds for V (x) were very weak. How weak? His Odd perfect k numbers method proves: The statistical perspective x Sociable Vk(x) , numbers log log log log ··· log x where the denominator is a (3k)-fold iterated log. With Kobayashi and Pomerance, we improved this. For example, we can replace the denominator with any fixed power 65 / 68 of log3 x. Paul Pollack The quest for perfection Theorem (Kobayashi, Pomerance, P., 2009) The set of deficient sociable numbers has density zero. The set of even abundant sociable numbers has density zero. Finally, the set of odd abundant numbers has density ≈ 1/500. All together now The quest for perfection Put Paul Pollack ∗ V (x) = V1(x) + V2(x) + V3(x) + ..., History ∗ lessons so that V (x) is the counting function of all the sociable Even perfect numbers. numbers Odd perfect Conjecture numbers ∗ The statistical As x → ∞, we have V (x)/x → 0. In other words, the set of perspective sociable numbers has density zero. Sociable numbers 66 / 68 Paul Pollack The quest for perfection All together now The quest for perfection Put Paul Pollack ∗ V (x) = V1(x) + V2(x) + V3(x) + ..., History ∗ lessons so that V (x) is the counting function of all the sociable Even perfect numbers. numbers Odd perfect Conjecture numbers ∗ The statistical As x → ∞, we have V (x)/x → 0. In other words, the set of perspective sociable numbers has density zero. Sociable numbers Theorem (Kobayashi, Pomerance, P., 2009) The set of deficient sociable numbers has density zero. The set of even abundant sociable numbers has density zero. Finally, the set of odd abundant numbers has density ≈ 1/500. 67 / 68 Paul Pollack The quest for perfection The quest for perfection Paul Pollack History lessons Even perfect numbers Odd perfect numbers Thank you for your attention! The statistical perspective Sociable numbers 68 / 68 Paul Pollack The quest for perfection