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Bounding the Factors of Odd Perfect Numbers Charles Greathouse Miami University

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Perfect Numbers σ(n) is the sum of divisors function: σ(n) = Σd σ(n) is the sum of divisors function: σ(n) = Σd A number N is called perfect iff σ(N)=2N: A number N is called perfect iff σ(N)=2N: σ(6) = 1 + 2 + 3 + 6 = 2 · 6 σ(496) = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 + 496 = 2 · 496 The 42 nd known perfect number was discovered in February of 2005. All 42 are even. The 42 nd known perfect number was discovered in February of 2005. All 42 are even. d|n

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Odd Perfect Numbers Odd Perfect Conjecture: There exist no odd perfect numbers. Odd Perfect Conjecture: There exist no odd perfect numbers. Euler: All perfect numbers are of the form p a x², with the p≡a≡1 (mod 4) and gcd(p, x) = 1. Euler: All perfect numbers are of the form p a x², with the p≡a≡1 (mod 4) and gcd(p, x) = 1. This means that for an odd perfect n = p 1 a 1 p 2 a 2 …p t a t (with the p i distinct primes) all but the “special prime” will have even exponents. This means that for an odd perfect n = p 1 a 1 p 2 a 2 …p t a t (with the p i distinct primes) all but the “special prime” will have even exponents.

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Abundancy σ -1 (n) := Σ d -1 = σ(n) / n. σ -1 (n) is clearly multiplicative since σ(n) is multiplicative. σ -1 (n) := Σ d -1 = σ(n) / n. σ -1 (n) is clearly multiplicative since σ(n) is multiplicative. Note that 1 < (p + 1) / p ≤ σ -1 (p a ) < p / (p – 1) and σ -1 (p a ) < σ -1 (p a+1 ). We can bound σ -1 (p a ) without knowing a. Note that 1 < (p + 1) / p ≤ σ -1 (p a ) < p / (p – 1) and σ -1 (p a ) < σ -1 (p a+1 ). We can bound σ -1 (p a ) without knowing a. Since N is perfect iff σ -1 (n) = 2, if we have N=p 1 p 2 …p k …p m and σ -1 (p 1 p 2 …p k ) > 2 then n is not perfect since σ -1 is strictly increasing on prime powers. Since N is perfect iff σ -1 (n) = 2, if we have N=p 1 p 2 …p k …p m and σ -1 (p 1 p 2 …p k ) > 2 then n is not perfect since σ -1 is strictly increasing on prime powers. d|n

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Bounding with Abundancy Suppose the odd perfect N = 3 a 5 b 7 c x with integral variables and {3, 5, 7, x} pairwise relatively prime. Suppose the odd perfect N = 3 a 5 b 7 c x with integral variables and {3, 5, 7, x} pairwise relatively prime. 3 and 7 can’t be the special prime, so a,c ≥ 2. 3 and 7 can’t be the special prime, so a,c ≥ 2. σ -1 (N) = σ -1 (3 a 5 b 7 c x) ≥ σ -1 (3 2 5 1 7 2 x) = σ -1 (3 2 ) σ -1 (5) σ -1 (7 2 ) σ -1 (x) = 13/9 · 6/5 · 57/49 · σ -1 (x) ≥ 2.016. σ -1 (N) = σ -1 (3 a 5 b 7 c x) ≥ σ -1 (3 2 5 1 7 2 x) = σ -1 (3 2 ) σ -1 (5) σ -1 (7 2 ) σ -1 (x) = 13/9 · 6/5 · 57/49 · σ -1 (x) ≥ 2.016. No such N can ever be perfect! No such N can ever be perfect!

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Bounding with Abundancy So if 3, 5, and 7 cannot all divide odd perfect n, we know that the third-smallest divisor of an odd perfect number is at least 11. So if 3, 5, and 7 cannot all divide odd perfect n, we know that the third-smallest divisor of an odd perfect number is at least 11. I will write N = p 1 a 1 p 2 a 2 …p t a t with p 1 < p 2 < … < p t for all odd perfect N, with the p i distinct primes. Thus the above result is p 3 ≥ 11. I will write N = p 1 a 1 p 2 a 2 …p t a t with p 1 < p 2 < … < p t for all odd perfect N, with the p i distinct primes. Thus the above result is p 3 ≥ 11.

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Bounds for t = 9 31 ≤ p 9 31 ≤ p 9 29 ≤ p 8 29 ≤ p 8 23 ≤ p 7 23 ≤ p 7 19 ≤ p 6 19 ≤ p 6 17 ≤ p 5 17 ≤ p 5 13 ≤ p 4 13 ≤ p 4 11 ≤ p 3 11 ≤ p 3 5 ≤ p 2 5 ≤ p 2 3 ≤ p 1 3 ≤ p 1

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Literature Using previous notation: Using previous notation: p t > 10 7 (Jenkins 2003) p t-1 > 10 4 (Iannucci 2000) p t-2 > 10 2 (Iannucci 1999) p 1 = 3 when t < 11 (Hagis 1983 and Kishore 1983)

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Bounds for t = 9 10 7 ≤ p 9 10 7 ≤ p 9 10 4 ≤ p 8 10 4 ≤ p 8 101 ≤ p 7 101 ≤ p 7 19 ≤ p 6 19 ≤ p 6 17 ≤ p 5 17 ≤ p 5 13 ≤ p 4 13 ≤ p 4 11 ≤ p 3 11 ≤ p 3 5 ≤ p 2 5 ≤ p 2 3 ≤ p 1 3 ≤ p 1

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Literature Using previous notation: Using previous notation: p t > 10 7 (Jenkins 2003) p t-1 > 10 4 (Iannucci 2000) p t-2 > 10 2 (Iannucci 1999) p 1 = 3 when t < 11 (Hagis 1983 and Kishore 1983) p i < 2 2 i-1 (t-i+1) for 2 ≤ i ≤ 6 (Kishore 1981) N < 2 4 t (Nielsen 2003)

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Bounds for t = 9 10 7 < p 9 < 2 4 9 ≈ 10 78914 10 7 < p 9 < 2 4 9 ≈ 10 78914 10 4 < p 8 < 2 4 9 ≈ 10 78914 10 4 < p 8 < 2 4 9 ≈ 10 78914 101 < p 7 < 2 4 9 ≈ 10 78914 101 < p 7 < 2 4 9 ≈ 10 78914 19 ≤ p 6 < 2 2 5 · 4 = 17 179 869 184 19 ≤ p 6 < 2 2 5 · 4 = 17 179 869 184 17 ≤ p 5 < 2 2 4 · 5 = 327 680 17 ≤ p 5 < 2 2 4 · 5 = 327 680 13 ≤ p 4 < 2 2 3 · 6 = 1536 13 ≤ p 4 < 2 2 3 · 6 = 1536 11 ≤ p 3 < 2 2 2 · 7 = 112 11 ≤ p 3 < 2 2 2 · 7 = 112 5 ≤ p 2 < 2 2 1 · 8 = 32 5 ≤ p 2 < 2 2 1 · 8 = 32 p 1 = 3 p 1 = 3

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Bounds for t = 9 10 7 < p 9 < 2 4 9 ≈ 10 78914 10 7 < p 9 < 2 4 9 ≈ 10 78914 10 4 < p 8 < 2 4 9 ≈ 10 78914 10 4 < p 8 < 2 4 9 ≈ 10 78914 101 < p 7 < 2 4 9 ≈ 10 78914 101 < p 7 < 2 4 9 ≈ 10 78914 19 ≤ p 6 ≤ 17 179 869 143 19 ≤ p 6 ≤ 17 179 869 143 17 ≤ p 5 ≤ 327 673 17 ≤ p 5 ≤ 327 673 13 ≤ p 4 ≤ 1531 13 ≤ p 4 ≤ 1531 11 ≤ p 3 ≤ 109 11 ≤ p 3 ≤ 109 5 ≤ p 2 ≤ 31 5 ≤ p 2 ≤ 31 p 1 = 3 p 1 = 3

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More with Abundncy So p 2 ≤ 31, but can we improve this? So p 2 ≤ 31, but can we improve this? Suppose p 2 = 31. Then σ -1 (n) ≤ σ -1 (3 a 31 b 37 c 41 d 43 e 47 f 101 g 10007 h 10000019 i ) < 1.7254 Even if p 2 = 13 we have σ -1 (n) ≤ σ -1 (3 a 13 b 17 c 19 d 23 e 29 f 101 g 10007 h 10000019 i ) < 1.9934 We can use this technique to bound p 3 as well, but no further. Without other restrictions we have 1.905 < σ -1 (3 a 5 b 11 c ) < 2.084.

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Bounds for t = 9 10 7 < p 9 < 2 4 9 ≈ 10 78914 10 7 < p 9 < 2 4 9 ≈ 10 78914 10 4 < p 8 < 2 4 9 ≈ 10 78914 10 4 < p 8 < 2 4 9 ≈ 10 78914 101 < p 7 < 2 4 9 ≈ 10 78914 101 < p 7 < 2 4 9 ≈ 10 78914 19 ≤ p 6 ≤ 17 179 869 143 19 ≤ p 6 ≤ 17 179 869 143 17 ≤ p 5 ≤ 327 673 17 ≤ p 5 ≤ 327 673 13 ≤ p 4 ≤ 1531 13 ≤ p 4 ≤ 1531 11 ≤ p 3 ≤ 31 11 ≤ p 3 ≤ 31 5 ≤ p 2 ≤ 11 5 ≤ p 2 ≤ 11 p 1 = 3 p 1 = 3

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Additional Results There are a number of other, less general results that can be used to narrow the bounds on odd perfect numbers: There are a number of other, less general results that can be used to narrow the bounds on odd perfect numbers: With care, we can tighten the bounds from Nielsen’s result: it is a bound on the whole number and not its individual factors. One known, these can be used as inputs for further abundancy restrictions.

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Bounds for t = 9 10 7 < p 9 < 2 4 9 ≈ 10 78914 10 7 < p 9 < 2 4 9 ≈ 10 78914 10 4 < p 8 < 2 4 9 ≈ 10 26305 10 4 < p 8 < 2 4 9 ≈ 10 26305 101 < p 7 < 2 4 9 ≈ 10 15783 101 < p 7 < 2 4 9 ≈ 10 15783 29 ≤ p 6 ≤ 17 179 869 143 29 ≤ p 6 ≤ 17 179 869 143 19 ≤ p 5 ≤ 327 673 19 ≤ p 5 ≤ 327 673 13 ≤ p 4 ≤ 1531 13 ≤ p 4 ≤ 1531 11 ≤ p 3 ≤ 31 11 ≤ p 3 ≤ 31 5 ≤ p 2 ≤ 11 5 ≤ p 2 ≤ 11 p 1 = 3 p 1 = 3

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Combined Sigma-Abundancy Consider the statement “13 ≤ p 4 ”. For equality to hold, we must have p 1 = 3, p 2 = 7, p 3 = 11, and p 4 = 13. Otherwise, σ -1 (N) > 2.063. Consider the statement “13 ≤ p 4 ”. For equality to hold, we must have p 1 = 3, p 2 = 7, p 3 = 11, and p 4 = 13. Otherwise, σ -1 (N) > 2.063. If a 3 = 2 then since σ(11²) = 7∙19 we see 19 | N, and this makes σ -1 (N) > 2.099. Thus a 3 ≥ 4. If a 3 = 2 then since σ(11²) = 7∙19 we see 19 | N, and this makes σ -1 (N) > 2.099. Thus a 3 ≥ 4. Continuing in this fashion, we find that a 3 ≥ 10, a 2 ≥ 4, and a 1 = 2. Continuing in this fashion, we find that a 3 ≥ 10, a 2 ≥ 4, and a 1 = 2. If a 4 ≥ 2 then σ -1 (N) > 2.0079. Thus 13 is the “special prime” with exponent 1. If a 4 ≥ 2 then σ -1 (N) > 2.0079. Thus 13 is the “special prime” with exponent 1.

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Combined Sigma-Abundancy So we now (under this scenario) have p 1 = 3, p 2 = 7, p 3 = 11, p 4 = 13, a 1 = 2, a 2 ≥ 4, a 3 ≥ 10, and a 3 = 1. So we now (under this scenario) have p 1 = 3, p 2 = 7, p 3 = 11, p 4 = 13, a 1 = 2, a 2 ≥ 4, a 3 ≥ 10, and a 3 = 1. Suppose further that a 2 = 4, a 3 = 10. p 5 is then restricted to the impossible 673 ≤ p 5 ≤ 661, so either a 2 > 4 or a 3 > 10. Suppose further that a 2 = 4, a 3 = 10. p 5 is then restricted to the impossible 673 ≤ p 5 ≤ 661, so either a 2 > 4 or a 3 > 10.

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t > 9 Most of the previous results can be extended in some fashion to OPNs with more than 9 distinct primes. Most of the previous results can be extended in some fashion to OPNs with more than 9 distinct primes. Unfortunately, few methods can be generalized sufficiently with current methods. These computational results seem useful only for extending various bounds, not for proving the Odd Perfect Conjecture. Unfortunately, few methods can be generalized sufficiently with current methods. These computational results seem useful only for extending various bounds, not for proving the Odd Perfect Conjecture.

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Skepticism Sylvester's Web of Conditions Sylvester's Web of Conditions “…a prolonged meditation on the subject has satisfied me that the existence of any one such—its escape, so to say, from the complex web of conditions which hem it in on all sides—would be little short of a miracle.”

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Skepticism Pomerance Heuristic Pomerance Heuristic If n=pm² is odd perfect then p | σ(m²). Thus there are at most log m possibilities for p. The ‘probability’ that σ(n) is divisible by n is p/n = 1/m². The sum over m converges, so there ‘should’ exist at most a finite number of odd perfects. Since there are no OPNs up to 10 300, “it may be more appropriate to sum (log m)/m² for m > 10 75.” This is 10 -70, so it is reasonable to conjecture that no odd perfect numbers exist.

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Where from here? Kevin Hare has published a number of papers recently restricting the number of total prime factors (to a minimum of 75 in his latest preprint). Kevin Hare has published a number of papers recently restricting the number of total prime factors (to a minimum of 75 in his latest preprint). Suryanarayana and Hagis have a paper discussing the sum of reciprocals of the distinct prime factors of OPNs. Suryanarayana and Hagis have a paper discussing the sum of reciprocals of the distinct prime factors of OPNs. If the methods of Kishore’s 1981 paper could be extended to include more factors, stricter bounds could be place on these numbers. If the methods of Kishore’s 1981 paper could be extended to include more factors, stricter bounds could be place on these numbers.

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Bibliography Peter Hagis, Jr., “Sketch of a proof that an odd perfect number relatively prime to 3 has at least eleven prime factors,” Mathematics of Computation, Vol. 40, No. 161 (Jan., 1983), pp. 399-404. Peter Hagis, Jr., “Sketch of a proof that an odd perfect number relatively prime to 3 has at least eleven prime factors,” Mathematics of Computation, Vol. 40, No. 161 (Jan., 1983), pp. 399-404. Douglas E. Iannucci, “The third largest prime divisor of an odd perfect number exceeds one hundred,” Mathematics of Computation, Vol. 69, No. 230 (2000), pp. 867-879. Douglas E. Iannucci, “The third largest prime divisor of an odd perfect number exceeds one hundred,” Mathematics of Computation, Vol. 69, No. 230 (2000), pp. 867-879. Douglas E. Iannucci, “The second largest prime divisor of an odd perfect number exceeds ten thousand,” Mathematics of Computation, Vol. 68, No. 228 (1999), pp. 1749-1760. Douglas E. Iannucci, “The second largest prime divisor of an odd perfect number exceeds ten thousand,” Mathematics of Computation, Vol. 68, No. 228 (1999), pp. 1749-1760. Paul M. Jenkins, “Odd perfect numbers have a prime factor exceeding 10 7,” Mathematics of Computation, Vol. 72, No. 243 (2003), pp. 1549-1554. Paul M. Jenkins, “Odd perfect numbers have a prime factor exceeding 10 7,” Mathematics of Computation, Vol. 72, No. 243 (2003), pp. 1549-1554. Masao Kishore, “Odd perfect numbers not divisible by 3. II,” Mathematics of Computation, Vol. 40, No. 161 (Jan., 1983), pp. 405-411. Masao Kishore, “Odd perfect numbers not divisible by 3. II,” Mathematics of Computation, Vol. 40, No. 161 (Jan., 1983), pp. 405-411. Masao Kishore, “On odd perfect, quasiperfect, and odd almost perfect numbers,” Mathematics of Computation, Vol. 36, No. 154 (Apr., 1981), pp. 583-586. Masao Kishore, “On odd perfect, quasiperfect, and odd almost perfect numbers,” Mathematics of Computation, Vol. 36, No. 154 (Apr., 1981), pp. 583-586. William Lipp, “Odd Perfect Number Search,” http://www.oddperfect.org/. William Lipp, “Odd Perfect Number Search,” http://www.oddperfect.org/. Pace P. Nielsen, “An upper bound for odd perfect numbers,” INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 3 (2003), #A14. Pace P. Nielsen, “An upper bound for odd perfect numbers,” INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 3 (2003), #A14.

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1 In this lecture Number Theory ● Rational numbers ● Divisibility Proofs ● Direct proofs (cont.) ● Common mistakes in proofs ● Disproof by counterexample.

1 In this lecture Number Theory ● Rational numbers ● Divisibility Proofs ● Direct proofs (cont.) ● Common mistakes in proofs ● Disproof by counterexample.

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