On the total number of prime factors of an odd perfect number. (English) Zbl 1487.11007
For any natural number \(n\), as usual let \(\omega(n)\) denote the number of distinct prime factors of \(n\), and let \(\Omega(n)\) denote the total number of prime factors (counting multiplicity) of \(n\). The first half of the work under review is devoted to proving that if \(N\) is an odd perfect number, then \(\frac{66}{25}\omega(N)-5\leq \Omega(N)\). A slightly better inequality is obtained when \(\gcd(3,N)=1\). These bounds improve on a string of results due to Ochem, Rao, and Zelinsky.
The main component of the proof is the following: Let \(\Phi_n(x)\) denote the \(n\)th cyclotomic polynomial. The author considers the situation when \[ \Phi_3(x)=\Phi_3(a)\Phi_3(b)\Phi_3(c) \] and where each of the seven quantities \(x,a,b,c,\Phi_3(a),\Phi_3(b),\Phi_3(c)\) is prime. The author calls these “triple threats” and he conjectures that this situation is impossible. Using a mixture of case analysis and integer inequalities some special cases of the conjecture are established. Recently, C. S. Hansen and the reviewer have shown that triple threats are in fact impossible, which extends and simplifies the main result of this paper; see [“Prime factors of \(\Phi_3(x)\) of the same form”, Preprint, arXiv:2204.08971].
The second half of the paper under review focuses on improving some results of Grün and Norton related to the smallest prime factor of an odd perfect number. Letting \(P_n\) denote the \(n\)th prime number, we can define a quantity \(a(n)\), for \(n>1\), by means of the inequalities \[ \prod_{r=n}^{n+a(n)-2}\frac{P_r}{P_r-1}<2< \prod_{r=n}^{n+a(n)-1}\frac{P_r}{P_r-1}. \] Norton showed that \[ a(n)=\frac{1}{2}n^2\log n+\frac{1}{2}n^2\log\log n-\frac{3}{4}n^2+\frac{n^2\log\log n}{2\log n} +O\left(\frac{n^2}{\log n}\right), \] where the bound is ineffective. He also gave the effective bound \[ a(n)>n^2-2n-\frac{n+1}{\log n}-\frac{5}{4}-\frac{1}{2n}-\frac{1}{4n\log n}. \] In the paper under review, the effective bound is improved to \[ a(n)>\frac{1}{2}n^2\log n-\frac{3}{4}n^2\log\log n+\frac{1}{40}n^2+\frac{n^2\log\log n}{2\log n}-n+1, \] which now has the correct leading term.
Putting these two halves together, other sorts of bounds are obtained. For instance, the author finds a lower bound on an odd perfect number in terms of its smallest prime divisor.
The main component of the proof is the following: Let \(\Phi_n(x)\) denote the \(n\)th cyclotomic polynomial. The author considers the situation when \[ \Phi_3(x)=\Phi_3(a)\Phi_3(b)\Phi_3(c) \] and where each of the seven quantities \(x,a,b,c,\Phi_3(a),\Phi_3(b),\Phi_3(c)\) is prime. The author calls these “triple threats” and he conjectures that this situation is impossible. Using a mixture of case analysis and integer inequalities some special cases of the conjecture are established. Recently, C. S. Hansen and the reviewer have shown that triple threats are in fact impossible, which extends and simplifies the main result of this paper; see [“Prime factors of \(\Phi_3(x)\) of the same form”, Preprint, arXiv:2204.08971].
The second half of the paper under review focuses on improving some results of Grün and Norton related to the smallest prime factor of an odd perfect number. Letting \(P_n\) denote the \(n\)th prime number, we can define a quantity \(a(n)\), for \(n>1\), by means of the inequalities \[ \prod_{r=n}^{n+a(n)-2}\frac{P_r}{P_r-1}<2< \prod_{r=n}^{n+a(n)-1}\frac{P_r}{P_r-1}. \] Norton showed that \[ a(n)=\frac{1}{2}n^2\log n+\frac{1}{2}n^2\log\log n-\frac{3}{4}n^2+\frac{n^2\log\log n}{2\log n} +O\left(\frac{n^2}{\log n}\right), \] where the bound is ineffective. He also gave the effective bound \[ a(n)>n^2-2n-\frac{n+1}{\log n}-\frac{5}{4}-\frac{1}{2n}-\frac{1}{4n\log n}. \] In the paper under review, the effective bound is improved to \[ a(n)>\frac{1}{2}n^2\log n-\frac{3}{4}n^2\log\log n+\frac{1}{40}n^2+\frac{n^2\log\log n}{2\log n}-n+1, \] which now has the correct leading term.
Putting these two halves together, other sorts of bounds are obtained. For instance, the author finds a lower bound on an odd perfect number in terms of its smallest prime divisor.
Reviewer: Pace Nielsen (Provo)
MSC:
| 11A25 | Arithmetic functions; related numbers; inversion formulas |
| 11A51 | Factorization; primality |
| 11N32 | Primes represented by polynomials; other multiplicative structures of polynomial values |
References:
| [1] | P. Batemen, P. Erdős, C. Pomerance, and E. Straus, The arithmetic mean of the divisors of an integer, Analtyic Number Theory, Lecture Notes in Mathematics, 899, Springer-Verlag, Berlin-New York, 1981. · Zbl 0478.10027 |
| [2] | S. Colton, Refactorable numbers -a machine invention, J. Integer Seq. 2 (1992), Article 99.1.2. · Zbl 1036.11540 |
| [3] | P. Dusart, The kth prime is greater than k(ln k + ln ln k − 1) for k ≥ 2, Math. Comp. 68 (1999), 411-415. · Zbl 0913.11039 |
| [4] | S. Fletcher, P. Nielsen, P. Ochem, Sieve Methods for Odd Perfect Numbers, Math. Comp. 81 (2012), 1753-1776. · Zbl 1268.11004 |
| [5] | S. Gimbel, J. Jaroma, Sylvester: Ushering in the Modern Era of Research on Odd Perfect Numbers, Integers 3 (2003). #A16 · Zbl 1128.01302 |
| [6] | T. Goto and Y. Ohno, Odd perfect numbers have a prime factor exceeding 10 8 , Math. Comp. 77 (2008), 1859-1868. · Zbl 1206.11009 |
| [7] | O. Grün,Über ungerade vollkommene Zahlen, Math. Z. 55 (3) (1952), 353-354. · Zbl 0046.27107 |
| [8] | G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1985. |
| [9] | K. Hare New techniques for bounds on the total number of prime factors of an odd perfect number, Math. Comp. 76 (2007), 2241-2248. · Zbl 1139.11006 |
| [10] | R. E. Kennedy and C. N. Cooper, Tau numbers, natural density, and Hardy and Wright’s Theorem 437, Internat. J. Math. Math. Sci. 13 (1990), 383-386. · Zbl 0713.11002 |
| [11] | P. Nielsen, Odd perfect numbers have at least nine distinct prime factors, Math. Comp. 76 (2007), 2109-2126. · Zbl 1142.11086 |
| [12] | P. Nielsen, An upper bound for odd perfect numbers, Integers 3 (2003), #A14. · Zbl 1085.11003 |
| [13] | P. Nielsen. Odd perfect numbers, Diophantine equations, and upper bounds, Math. Comp. 84 (2015), 2549-2567. · Zbl 1325.11009 |
| [14] | K. Norton, Remarks on the number of factors of an odd perfect number, Acta Arith. 6 (1960/1961), 365-374. · Zbl 0102.03203 |
| [15] | P. Ochem, M. Rao Odd perfect numbers are greater than 10 1500 , Math. Comp. 81 (2012), 1869-1877. · Zbl 1263.11005 |
| [16] | P. Ochem, M. Rao, On the number of prime factors of an odd perfect number, Math. Comp. 83 (2014), 2435-2439. · Zbl 1370.11009 |
| [17] | V. Pambuccian, Problem E3081 Solution, Amer. Math. Monthly 94 (1987), 794-795. |
| [18] | L. Reis, On the factorization of iterated polynomials, https://arxiv.org/abs/1810.07715 · Zbl 1473.12001 |
| [19] | J. Rosser, L. Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and Ψ(x), Math. Comp. 29 (1975), 243-269. · Zbl 0295.10036 |
| [20] | H. Salié,Über abundante Zahlen, Math. Nachr. 9 (1953), 217-220. · Zbl 0050.04201 |
| [21] | C. Servais, Sur les nombres parfaits, Mathesis 8 (1888), 92-93. · JFM 20.0174.01 |
| [22] | M. Sha, Counting decomposable polynomials with integer coefficients, https://arxiv.org/abs/1803.08755 . |
| [23] | J. Zelinsky, An Improvement of an Inequality of Ochem and Rao Concerning Odd Perfect Numbers, Integers 18 (2018), #A48. · Zbl 1448.11013 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
