- Email: [email protected]
On the divisors of xn−1
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 43 (2013) 141–149 www.elsevier.com/locate/endm
On the divisors of xn − 1 Lola Thompson
1,2
Department of Mathematics University of Georgia Athens, GA, United States Abstract We examine two natural questions concerning the polynomial divisors of xn − 1: “For a given integer n, how large can the coefficients of divisors of xn − 1 be?” and “How often does xn − 1 have a divisor of every degree between 1 and n?” We consider the latter question when xn − 1 is factored in both Z[x] and Fp [x]. The primary tools used in our investigation arise the study of the anatomy of integers. We also make use of several results on the size of the multiplicative order function (which stem from Hooley’s conditional proof of Artin’s Primitive Root Conjecture) in our work over Fp [x]. Keywords: Cyclotomic polynomials, Practical numbers, Euler totient function, Multiplicative orders, Sieve methods
1
Coefficients of divisors of xn − 1
Cyclotomic polynomials are intrinsic to our study of the divisors of xn − 1. The identity Y Φd (x) xn − 1 = d|n
1 This article summarizes a portion of the work contained in my Ph.D. thesis. I would like to thank my thesis adviser, Carl Pomerance, for his guidance throughout the process of completing the original research. I would also like to thank my postdoctoral mentor, Paul Pollack, for helpful discussions related to these topics. 2 Email: [email protected]
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.07.024
142
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
shows that the monic irreducible divisors of xn − 1 in Z[x] are precisely the dth cyclotomic polynomials, for values of d dividing n. As a result, every monic divisor of xn − 1 in Z[x] is a product of distinct cyclotomic polynomials. The coefficients of cyclotomic polynomials have been studied for some time. In 1883, Migotti [10] showed that Φ105 (x) is the first cyclotomic polynomial to have coefficients that lie outside the set {±1, 0}: Φ105 (x) = x48 + x47 + x46 − x43 − x42 − 2x41 − x40 − x39 + x36 + x35 + x34 + x32 + x31 − x28 − x26 − x24 − x22 − x20 + x17 + x16 + x15 + x14 + x13 + x12 − x9 − x8 − 2x7 − x6 − x5 + x2 + x + 1. The fact that the new coefficient appearing in Φ105 (x) has absolute value 2 leads to the natural question, “How do the coefficients grow (in absolute value) as n increases?” One approach has been to bound the magnitude of the maximal coefficient of Φn (x) for integers n with an arbitrary but fixed number of prime factors. We will denote this magnitude by A(n). Bateman [1] was the first to obtain a bound for A(n) when n has k distinct odd prime factors, where k ranges over all positive integers. He gave a simple argument in 1949 which showed that k−1 the height of Φn (x) is at most n2 . There were a number of improvements on Bateman’s result in papers of Erd˝os [3], Vaughan [19] and Bateman, Pomer2k−1
ance, and Vaughan [2], the last of which gives an upper bound of n k −1 . We remark that this bound is nearly best possible; in the same paper, Bateman, 2k−1 k−1 Pomerance and Vaughan showed that A(n) ≥ n k −1 /(5 log n)2 holds for infinitely many n with exactly k distinct odd prime factors. Moreover, under the assumption of the prime k-tuples conjecture, they proved that for each 2k−1 k there exists a constant ck such that A(n) ≥ ck n k −1 holds for infinitely many n with exactly k distinct odd prime factors. We can re-state these results without the dependence on k by using the fact that the maximal order of ω(n) is logloglogn n . This yields an upper bound of A(n) ≤ en
(log 2+o(1))/ log log n
(1)
that holds for all positive integers n. Furthermore, the same argument can be used to show that the inequality in (1) can be reversed for infinitely many values of n. Maier showed in [8] and [9] that stronger results can be obtained for “typical” n; that is, for all n except for a set with asymptotic density 0. In particular, he proved that if ψ(n) is any function defined for all positive in-
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
143
tegers such that ψ(n) tends to infinity as n tends to infinity, then the height of Φn (x) is at most nψ(n) for almost all n. Moreover, he was able to obtain a complementary lower bound, showing that if ε(n) is any function that tends to 0 as n tends to infinity, then the height of Φn (x) is at least nε(n) for almost all n. We consider the maximal height over all (not necessarily irreducible) divisors of xn − 1, which we denote by B(n). In general, much less is known about B(n) than A(n). In 2005, Pomerance and Ryan [11] proved that as n → ∞, we have log B(n) ≤ n(log 3+o(1))/ log log n . They also showed that this inequality can be reversed for infinitely many n. In the spirit of Maier, we obtain [15, Theorem 1.2] an upper bound for B(n) that holds for “typical” n. Theorem 1.1 Let ψ(n) be a function defined for all positive integers such that ψ(n) → ∞ as n → ∞. Then B(n) ≤ nτ (n)ψ(n) for almost all n, i.e., for all n except for a set with asymptotic density 0.
2
Degrees of divisors of xn − 1
In addition to describing the behavior of the coefficients, we could also examine the degrees of divisors of xn − 1 and see how they behave. When xn − 1 has a divisor of every degree between 1 and n in Z[x], we say that n is ϕ-practical. An alternative characterization of ϕ-practical numbers is given by the following condition: n is ϕ-practical if and only if every integer m with 1 ≤ m ≤ n can be written in the form X m= ϕ(d), d∈D
where D is a subset of divisors of n and ϕ is the Euler totient function. The nomenclature stems from the similarity between this second characterization of ϕ-practical numbers and the definition of a practical number. An integer n is practical if every m with 1 ≤ m ≤ n can be written as a sum of distinct divisors of n. The practical numbers have been well-studied by Erd˝os [4], Stewart [13], Tenenbaum [14], Saias [12] and others. Recent work has focused on estimating the size of the set of practical numbers up to X, which we denote by P R(X). The sharpest bounds were given by Saias in [12], who showed that there exist two constants C1 and C2 such that C1
X X ≤ P R(X) ≤ C2 . log X log X
The practical numbers have an intimate connection with the integers with
144
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
(n) 2-dense divisors, integers n for which max1≤i≤τ (n)−1 di+1 ≤ 2, where d1 (n) < di (n) d2 (n) < · · · < dτ (n) is the increasing sequence of divisors of n. Examining the integers with 2-dense divisors is part of a larger program to the study of the “anatomy of integers.” The underlying principle is to dissect the integers in different ways and see what additional information can be gleaned through this process. Instead of focusing solely on the size and quantity of prime factors in a given family of integers, one might instead try to understand the sizes of the gaps between their divisors, as this may provide a new approach to some classical integer counting problems. In spite of their importance in answering statistical questions about the practical numbers, the 2-dense numbers are not, in general, well-understood. There have been a handful of papers written by Tenenbaum, Saias and others on this topic, but there is still much that is not known. In order to find bounds for the size of the set of ϕ-practical numbers, we have developed a theory of strictly 2-dense numbers, integers whose divisors satisfy the inequality
dτ (n) (n) d2 (n) di+1 (n) < 2, with = = 2. 1
We can use new sieving techniques introduced by Tenenbaum [14] and Saias [12], along with some classical estimation methods from multiplicative number theory, in order to obtain bounds for the size of the set of strictly 2-dense numbers up to X. Since it can be shown that all strictly 2-dense numbers are ϕ-practical, this process allows us to prove [16, Theorem 1.2]: Theorem 2.1 Let F (X) = #{n ≤ X : n is ϕ-practical}. There exist two positive constants c1 and c2 such that for X ≥ 2, we have c1
X X ≤ F (X) ≤ c2 . log X log X
In particular, our work on strictly 2-dense numbers implies that there are ≫ logXX integers n ≤ X that are both practical and ϕ-practical. Using an analogous argument to that which we used to prove the lower bound in Theorem 2.1, we can also obtain the following corollary: Corollary 2.2 For X sufficiently large, we have #{n ≤ X : n is practical but not ϕ-practical} ≫
X log X
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
145
and #{n ≤ X : n is ϕ-practical but not practical} ≫
X . log X
Instead of factoring xn − 1 in Z[x], we could examine its factorization in other rings. We define an integer n to be p-practical if xn − 1 has a divisor in Fp [x] of every degree less than or equal to n. In order to better understand the relationship between ϕ-practical numbers and p-practical numbers, we define an intermediate set of integers that we call the λ-practical numbers. We say that an integer n is λ-practical if and only if it is p-practical for every rational prime p. From the definitions, it is clear that every ϕ-practical number is λ-practical. It is not difficult to come up with families of examples to demonstrate that there are infinitely many λ-practicals that are not ϕpractical and, for each prime p, there are infinitely many p-practicals that are not λ-practical. In fact, we can use the same sieving argument from the proof of Theorem 2.1 in order to prove a much stronger result for the λ-practicals (cf. [18, Theorems 1.1 and 1.2]): Theorem 2.3 For X sufficiently large, the order of magnitude of λ-practicals in [1, X] that are not ϕ-practical is logXX . Moreover, for each prime p, the order of magnitude of p-practicals in [1, X] that are not λ-practical is at least logXX . One of our main objectives has been to obtain sharp bounds for the size of the set of p-practical numbers. Our work on the ϕ-practicals implies that #{n ≤ X : n is p-practical} ≫ logXX , but the task of finding an upper bound is much more challenging. By assuming the Generalized Riemann Hypothesis, we can prove the following [17, Theorem 1.2]: Theorem 2.4 Let X ≥ 2 and Fp (X) = #{n ≤ X : n is p-practical}. Then, assuming that the Generalized Riemann Hypothesis holds, we have Fp (X) = O X
s
log log X log X
!
.
In order to understand the basic ingredients for this argument, one observes that the multiplicative orders of p modulo d (as d ranges over the divisors of n) represent the degrees of the irreducible divisors of xn − 1 when factored in Fp [x]. The key to our proof is then to show that the multiplicative order of p modulo d is usually not too small, which relies on some technical results ([5], [6], [7]) related to Artin’s primitive root conjecture.
146
3
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
Open conjectures
The work discussed in the preceding sections can be extended in a number of ways. First, it would be natural to try to extend Maier’s method for obtaining a lower bound for A(n) in order to prove a complementary result for our B(n) upper bound. Certainly, Maier’s lower bound will also be a lower bound for B(n). However, we believe that the lower bound can be improved: Conjecture 3.1 Let ε(n) be any function defined for all positive integers n that tends to 0 as n tends to infinity. Then, for almost all n, we have B(n) ≥ nτ (n)ε(n) . Second, there is an obvious parallel between the bounds that we obtain for the size of the set of ϕ-practical numbers and the bounds given by Chebyshev’s inequality: Theorem 3.2 (Chebyshev, 1852) Let π(X) denote the number of primes in [1, X]. There exist positive constants c1 and c2 such that c1
X X ≤ π(X) ≤ c2 . log X log X
Of course, we know that the story does not end with Chebyshev’s inequality. Nearly half of a century later, the celebrated Prime Number Theorem was proven. Theorem 3.3 (Hadamard & de la Vall´ ee Poussin, 1896) Let π(X) denote the number of primes in [1, X]. Then, we have π(X) = 1. X→∞ X/ log X lim
It would be interesting to know whether F (X) X→∞ X/ log X lim
exists and, if so, what it approaches. Using Sage, we have been able to compute the following table of ratios:
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
X
F (X)
102
28
1.289448
103
174
1.201949
104
1198
1.103399
105
9301
1.070817
106
74461
1.028717
107
635528
1.024350
108
5525973
1.017922
109
48386047
1.002717
147
F (X)/(X/ log X)
Table 1 Ratios for ϕ-practicals
The table seems to suggest the following conjecture:
Conjecture 3.4 Let F (X) = #{n ≤ X : n is ϕ-practical}. Then,
F (X) = 1. X→∞ X/ log X lim
Proving this may be exceedingly difficult; for one thing, there does not appear to be an L-function whose zeros correspond to the distribution of ϕ-practical numbers, so it does not seem likely that any of the analytic approaches for proving the Prime Number Theorem will be useful in this scenario. Moreover, it appears that there is no hope of adapting the Erd˝os-Selberg elementary proof of the Prime Number Theorem in order to say anything about the size of the set of practical numbers. Another natural goal would be to find a sharper estimate for the order of magnitude of Fp (X). For example, when p = 2, we can use Sage to compute a table of ratios of F2 (X)/ logXX .
148
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
X
F2 (X)
F2 (X)/(X/ log X)
102
34
1.565758
103
243
1.678585
104
1790
1.648651
105
14703
1.692745
106
120276
1.661674
107
1030279
1.660614
Table 2 Ratios for 2-practicals
The table looks similar for other small values of p. The fact that the sequence of ratios appears to be bounded suggests the following conjecture: Conjecture 3.5 For each rational prime p, limX→∞ Fp (X)/ logXX exists and is finite. At the very least, it would be nice to establish the true order of magnitude of Fp (X). The table seems to suggest that Fp (X) is on the order of X/ log X, which given Theorem 2.1 requires establishing: Conjecture 3.6 For each rational prime p, we have Fp (X) ≪
X . log X
References [1] P. T. Bateman, Note on the coefficients of the cyclotomic polynomials, Bull. Amer. Math. Soc. 55 (1949), 1180 – 1181. [2] P.T. Bateman, C. Pomerance, and R.C. Vaughan, On the size of the coefficients of the cyclotomic polynomial, Topics in classical number theory, Vol. I, II (1981), Colloq. Math. Soc. Janos Bolyai 34 (1984), 171 – 202. [3] P. Erd˝ os, On the coefficients of the cyclotomic polynomial, Bull. Amer. Math. Soc. 52 (1946), 179 – 184. [4] P. Erd˝ os, On a Diophantine equation, Mat. Lapok 1 (1950), 192 – 210.
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
149
[5] J. Friedlander, C. Pomerance and I. E. Shparlinski, Period of the power generator and small values of the Carmichael function, Math. Comp. 70 (2001), 1591 – 1605. [6] P. Kurlberg, C. Pomerance, On a problem of Arnold: the average multiplicative order of a given integer, Algebra and Number Theory, to appear. [7] S. Li and C. Pomerance, On generalizing Artin’s conjecture on primitive roots to composite moduli, J. Reine Angew. Math. 556 (2003), 205 – 224. [8] H. Maier, Cyclotomic polynomials with large coefficients, Acta Arith. 64 (1993), 227 – 235. [9] H. Maier, The coefficients of cyclotomic polynomials, Proc. Conf. in Honor of Paul T. Bateman, Progr. Math. 85 (1990), 349 – 366. [10] A. Migotti, Zur theorie der kreisteilungsgleichung, Z. B. der Math.-Naturwiss, Classe der Kaiserlichen Akademie der Wissenschaften, Wien, 87 (1883), 7 – 14. [11] C. Pomerance and N. Ryan, Maximal height of divisors of xn − 1. Illinois J. Math. 51 no. 2 (2007), 597 – 604 (electronic). [12] E. Saias, Entiers ` a diviseurs denses I., J. Number Theory 62 (1997), 163 – 191. [13] B. M. Stewart, Sums of distinct divisors, Amer. J. Math. 76 no. 4 (1954), 779 – 785. ´ [14] G. Tenenbaum, Sur un probl`eme de crible et ses applications, Ann. Sci. Ecole Norm. Sup. 19 no. 4 (1986), 1 – 30. [15] L. Thompson, Heights of divisors of xn − 1, Integers 11A. Proceedings of the Integers Conference 2009 (2011), Article 20, 1 – 9. [16] L. Thompson, Polynomials with divisors of every degree, J. Number Theory 132 (2012), 1038 – 1053. [17] L. Thompson, On the divisors of xn − 1 in Fp [x], Int. J. Number Theory 9 no. 2 (2013), 421 – 430. [18] L. Thompson, Variations on a question concerning the degrees of divisors of xn − 1, arXiv:1206.4355. [19] R. C. Vaughan, Bounds for the coefficients of cyclotomic polynomials, Michigan Math. J. 21 (1974), 289 – 295.
Electronic Notes in Discrete Mathematics 43 (2013) 141–149 www.elsevier.com/locate/endm
On the divisors of xn − 1 Lola Thompson
1,2
Department of Mathematics University of Georgia Athens, GA, United States Abstract We examine two natural questions concerning the polynomial divisors of xn − 1: “For a given integer n, how large can the coefficients of divisors of xn − 1 be?” and “How often does xn − 1 have a divisor of every degree between 1 and n?” We consider the latter question when xn − 1 is factored in both Z[x] and Fp [x]. The primary tools used in our investigation arise the study of the anatomy of integers. We also make use of several results on the size of the multiplicative order function (which stem from Hooley’s conditional proof of Artin’s Primitive Root Conjecture) in our work over Fp [x]. Keywords: Cyclotomic polynomials, Practical numbers, Euler totient function, Multiplicative orders, Sieve methods
1
Coefficients of divisors of xn − 1
Cyclotomic polynomials are intrinsic to our study of the divisors of xn − 1. The identity Y Φd (x) xn − 1 = d|n
1 This article summarizes a portion of the work contained in my Ph.D. thesis. I would like to thank my thesis adviser, Carl Pomerance, for his guidance throughout the process of completing the original research. I would also like to thank my postdoctoral mentor, Paul Pollack, for helpful discussions related to these topics. 2 Email: [email protected]
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.07.024
142
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
shows that the monic irreducible divisors of xn − 1 in Z[x] are precisely the dth cyclotomic polynomials, for values of d dividing n. As a result, every monic divisor of xn − 1 in Z[x] is a product of distinct cyclotomic polynomials. The coefficients of cyclotomic polynomials have been studied for some time. In 1883, Migotti [10] showed that Φ105 (x) is the first cyclotomic polynomial to have coefficients that lie outside the set {±1, 0}: Φ105 (x) = x48 + x47 + x46 − x43 − x42 − 2x41 − x40 − x39 + x36 + x35 + x34 + x32 + x31 − x28 − x26 − x24 − x22 − x20 + x17 + x16 + x15 + x14 + x13 + x12 − x9 − x8 − 2x7 − x6 − x5 + x2 + x + 1. The fact that the new coefficient appearing in Φ105 (x) has absolute value 2 leads to the natural question, “How do the coefficients grow (in absolute value) as n increases?” One approach has been to bound the magnitude of the maximal coefficient of Φn (x) for integers n with an arbitrary but fixed number of prime factors. We will denote this magnitude by A(n). Bateman [1] was the first to obtain a bound for A(n) when n has k distinct odd prime factors, where k ranges over all positive integers. He gave a simple argument in 1949 which showed that k−1 the height of Φn (x) is at most n2 . There were a number of improvements on Bateman’s result in papers of Erd˝os [3], Vaughan [19] and Bateman, Pomer2k−1
ance, and Vaughan [2], the last of which gives an upper bound of n k −1 . We remark that this bound is nearly best possible; in the same paper, Bateman, 2k−1 k−1 Pomerance and Vaughan showed that A(n) ≥ n k −1 /(5 log n)2 holds for infinitely many n with exactly k distinct odd prime factors. Moreover, under the assumption of the prime k-tuples conjecture, they proved that for each 2k−1 k there exists a constant ck such that A(n) ≥ ck n k −1 holds for infinitely many n with exactly k distinct odd prime factors. We can re-state these results without the dependence on k by using the fact that the maximal order of ω(n) is logloglogn n . This yields an upper bound of A(n) ≤ en
(log 2+o(1))/ log log n
(1)
that holds for all positive integers n. Furthermore, the same argument can be used to show that the inequality in (1) can be reversed for infinitely many values of n. Maier showed in [8] and [9] that stronger results can be obtained for “typical” n; that is, for all n except for a set with asymptotic density 0. In particular, he proved that if ψ(n) is any function defined for all positive in-
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
143
tegers such that ψ(n) tends to infinity as n tends to infinity, then the height of Φn (x) is at most nψ(n) for almost all n. Moreover, he was able to obtain a complementary lower bound, showing that if ε(n) is any function that tends to 0 as n tends to infinity, then the height of Φn (x) is at least nε(n) for almost all n. We consider the maximal height over all (not necessarily irreducible) divisors of xn − 1, which we denote by B(n). In general, much less is known about B(n) than A(n). In 2005, Pomerance and Ryan [11] proved that as n → ∞, we have log B(n) ≤ n(log 3+o(1))/ log log n . They also showed that this inequality can be reversed for infinitely many n. In the spirit of Maier, we obtain [15, Theorem 1.2] an upper bound for B(n) that holds for “typical” n. Theorem 1.1 Let ψ(n) be a function defined for all positive integers such that ψ(n) → ∞ as n → ∞. Then B(n) ≤ nτ (n)ψ(n) for almost all n, i.e., for all n except for a set with asymptotic density 0.
2
Degrees of divisors of xn − 1
In addition to describing the behavior of the coefficients, we could also examine the degrees of divisors of xn − 1 and see how they behave. When xn − 1 has a divisor of every degree between 1 and n in Z[x], we say that n is ϕ-practical. An alternative characterization of ϕ-practical numbers is given by the following condition: n is ϕ-practical if and only if every integer m with 1 ≤ m ≤ n can be written in the form X m= ϕ(d), d∈D
where D is a subset of divisors of n and ϕ is the Euler totient function. The nomenclature stems from the similarity between this second characterization of ϕ-practical numbers and the definition of a practical number. An integer n is practical if every m with 1 ≤ m ≤ n can be written as a sum of distinct divisors of n. The practical numbers have been well-studied by Erd˝os [4], Stewart [13], Tenenbaum [14], Saias [12] and others. Recent work has focused on estimating the size of the set of practical numbers up to X, which we denote by P R(X). The sharpest bounds were given by Saias in [12], who showed that there exist two constants C1 and C2 such that C1
X X ≤ P R(X) ≤ C2 . log X log X
The practical numbers have an intimate connection with the integers with
144
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
(n) 2-dense divisors, integers n for which max1≤i≤τ (n)−1 di+1 ≤ 2, where d1 (n) < di (n) d2 (n) < · · · < dτ (n) is the increasing sequence of divisors of n. Examining the integers with 2-dense divisors is part of a larger program to the study of the “anatomy of integers.” The underlying principle is to dissect the integers in different ways and see what additional information can be gleaned through this process. Instead of focusing solely on the size and quantity of prime factors in a given family of integers, one might instead try to understand the sizes of the gaps between their divisors, as this may provide a new approach to some classical integer counting problems. In spite of their importance in answering statistical questions about the practical numbers, the 2-dense numbers are not, in general, well-understood. There have been a handful of papers written by Tenenbaum, Saias and others on this topic, but there is still much that is not known. In order to find bounds for the size of the set of ϕ-practical numbers, we have developed a theory of strictly 2-dense numbers, integers whose divisors satisfy the inequality
dτ (n) (n) d2 (n) di+1 (n) < 2, with = = 2. 1
We can use new sieving techniques introduced by Tenenbaum [14] and Saias [12], along with some classical estimation methods from multiplicative number theory, in order to obtain bounds for the size of the set of strictly 2-dense numbers up to X. Since it can be shown that all strictly 2-dense numbers are ϕ-practical, this process allows us to prove [16, Theorem 1.2]: Theorem 2.1 Let F (X) = #{n ≤ X : n is ϕ-practical}. There exist two positive constants c1 and c2 such that for X ≥ 2, we have c1
X X ≤ F (X) ≤ c2 . log X log X
In particular, our work on strictly 2-dense numbers implies that there are ≫ logXX integers n ≤ X that are both practical and ϕ-practical. Using an analogous argument to that which we used to prove the lower bound in Theorem 2.1, we can also obtain the following corollary: Corollary 2.2 For X sufficiently large, we have #{n ≤ X : n is practical but not ϕ-practical} ≫
X log X
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
145
and #{n ≤ X : n is ϕ-practical but not practical} ≫
X . log X
Instead of factoring xn − 1 in Z[x], we could examine its factorization in other rings. We define an integer n to be p-practical if xn − 1 has a divisor in Fp [x] of every degree less than or equal to n. In order to better understand the relationship between ϕ-practical numbers and p-practical numbers, we define an intermediate set of integers that we call the λ-practical numbers. We say that an integer n is λ-practical if and only if it is p-practical for every rational prime p. From the definitions, it is clear that every ϕ-practical number is λ-practical. It is not difficult to come up with families of examples to demonstrate that there are infinitely many λ-practicals that are not ϕpractical and, for each prime p, there are infinitely many p-practicals that are not λ-practical. In fact, we can use the same sieving argument from the proof of Theorem 2.1 in order to prove a much stronger result for the λ-practicals (cf. [18, Theorems 1.1 and 1.2]): Theorem 2.3 For X sufficiently large, the order of magnitude of λ-practicals in [1, X] that are not ϕ-practical is logXX . Moreover, for each prime p, the order of magnitude of p-practicals in [1, X] that are not λ-practical is at least logXX . One of our main objectives has been to obtain sharp bounds for the size of the set of p-practical numbers. Our work on the ϕ-practicals implies that #{n ≤ X : n is p-practical} ≫ logXX , but the task of finding an upper bound is much more challenging. By assuming the Generalized Riemann Hypothesis, we can prove the following [17, Theorem 1.2]: Theorem 2.4 Let X ≥ 2 and Fp (X) = #{n ≤ X : n is p-practical}. Then, assuming that the Generalized Riemann Hypothesis holds, we have Fp (X) = O X
s
log log X log X
!
.
In order to understand the basic ingredients for this argument, one observes that the multiplicative orders of p modulo d (as d ranges over the divisors of n) represent the degrees of the irreducible divisors of xn − 1 when factored in Fp [x]. The key to our proof is then to show that the multiplicative order of p modulo d is usually not too small, which relies on some technical results ([5], [6], [7]) related to Artin’s primitive root conjecture.
146
3
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
Open conjectures
The work discussed in the preceding sections can be extended in a number of ways. First, it would be natural to try to extend Maier’s method for obtaining a lower bound for A(n) in order to prove a complementary result for our B(n) upper bound. Certainly, Maier’s lower bound will also be a lower bound for B(n). However, we believe that the lower bound can be improved: Conjecture 3.1 Let ε(n) be any function defined for all positive integers n that tends to 0 as n tends to infinity. Then, for almost all n, we have B(n) ≥ nτ (n)ε(n) . Second, there is an obvious parallel between the bounds that we obtain for the size of the set of ϕ-practical numbers and the bounds given by Chebyshev’s inequality: Theorem 3.2 (Chebyshev, 1852) Let π(X) denote the number of primes in [1, X]. There exist positive constants c1 and c2 such that c1
X X ≤ π(X) ≤ c2 . log X log X
Of course, we know that the story does not end with Chebyshev’s inequality. Nearly half of a century later, the celebrated Prime Number Theorem was proven. Theorem 3.3 (Hadamard & de la Vall´ ee Poussin, 1896) Let π(X) denote the number of primes in [1, X]. Then, we have π(X) = 1. X→∞ X/ log X lim
It would be interesting to know whether F (X) X→∞ X/ log X lim
exists and, if so, what it approaches. Using Sage, we have been able to compute the following table of ratios:
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
X
F (X)
102
28
1.289448
103
174
1.201949
104
1198
1.103399
105
9301
1.070817
106
74461
1.028717
107
635528
1.024350
108
5525973
1.017922
109
48386047
1.002717
147
F (X)/(X/ log X)
Table 1 Ratios for ϕ-practicals
The table seems to suggest the following conjecture:
Conjecture 3.4 Let F (X) = #{n ≤ X : n is ϕ-practical}. Then,
F (X) = 1. X→∞ X/ log X lim
Proving this may be exceedingly difficult; for one thing, there does not appear to be an L-function whose zeros correspond to the distribution of ϕ-practical numbers, so it does not seem likely that any of the analytic approaches for proving the Prime Number Theorem will be useful in this scenario. Moreover, it appears that there is no hope of adapting the Erd˝os-Selberg elementary proof of the Prime Number Theorem in order to say anything about the size of the set of practical numbers. Another natural goal would be to find a sharper estimate for the order of magnitude of Fp (X). For example, when p = 2, we can use Sage to compute a table of ratios of F2 (X)/ logXX .
148
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
X
F2 (X)
F2 (X)/(X/ log X)
102
34
1.565758
103
243
1.678585
104
1790
1.648651
105
14703
1.692745
106
120276
1.661674
107
1030279
1.660614
Table 2 Ratios for 2-practicals
The table looks similar for other small values of p. The fact that the sequence of ratios appears to be bounded suggests the following conjecture: Conjecture 3.5 For each rational prime p, limX→∞ Fp (X)/ logXX exists and is finite. At the very least, it would be nice to establish the true order of magnitude of Fp (X). The table seems to suggest that Fp (X) is on the order of X/ log X, which given Theorem 2.1 requires establishing: Conjecture 3.6 For each rational prime p, we have Fp (X) ≪
X . log X
References [1] P. T. Bateman, Note on the coefficients of the cyclotomic polynomials, Bull. Amer. Math. Soc. 55 (1949), 1180 – 1181. [2] P.T. Bateman, C. Pomerance, and R.C. Vaughan, On the size of the coefficients of the cyclotomic polynomial, Topics in classical number theory, Vol. I, II (1981), Colloq. Math. Soc. Janos Bolyai 34 (1984), 171 – 202. [3] P. Erd˝ os, On the coefficients of the cyclotomic polynomial, Bull. Amer. Math. Soc. 52 (1946), 179 – 184. [4] P. Erd˝ os, On a Diophantine equation, Mat. Lapok 1 (1950), 192 – 210.
L. Thompson / Electronic Notes in Discrete Mathematics 43 (2013) 141–149
149
[5] J. Friedlander, C. Pomerance and I. E. Shparlinski, Period of the power generator and small values of the Carmichael function, Math. Comp. 70 (2001), 1591 – 1605. [6] P. Kurlberg, C. Pomerance, On a problem of Arnold: the average multiplicative order of a given integer, Algebra and Number Theory, to appear. [7] S. Li and C. Pomerance, On generalizing Artin’s conjecture on primitive roots to composite moduli, J. Reine Angew. Math. 556 (2003), 205 – 224. [8] H. Maier, Cyclotomic polynomials with large coefficients, Acta Arith. 64 (1993), 227 – 235. [9] H. Maier, The coefficients of cyclotomic polynomials, Proc. Conf. in Honor of Paul T. Bateman, Progr. Math. 85 (1990), 349 – 366. [10] A. Migotti, Zur theorie der kreisteilungsgleichung, Z. B. der Math.-Naturwiss, Classe der Kaiserlichen Akademie der Wissenschaften, Wien, 87 (1883), 7 – 14. [11] C. Pomerance and N. Ryan, Maximal height of divisors of xn − 1. Illinois J. Math. 51 no. 2 (2007), 597 – 604 (electronic). [12] E. Saias, Entiers ` a diviseurs denses I., J. Number Theory 62 (1997), 163 – 191. [13] B. M. Stewart, Sums of distinct divisors, Amer. J. Math. 76 no. 4 (1954), 779 – 785. ´ [14] G. Tenenbaum, Sur un probl`eme de crible et ses applications, Ann. Sci. Ecole Norm. Sup. 19 no. 4 (1986), 1 – 30. [15] L. Thompson, Heights of divisors of xn − 1, Integers 11A. Proceedings of the Integers Conference 2009 (2011), Article 20, 1 – 9. [16] L. Thompson, Polynomials with divisors of every degree, J. Number Theory 132 (2012), 1038 – 1053. [17] L. Thompson, On the divisors of xn − 1 in Fp [x], Int. J. Number Theory 9 no. 2 (2013), 421 – 430. [18] L. Thompson, Variations on a question concerning the degrees of divisors of xn − 1, arXiv:1206.4355. [19] R. C. Vaughan, Bounds for the coefficients of cyclotomic polynomials, Michigan Math. J. 21 (1974), 289 – 295.