On the Distribution of Carmichael Numbers

Transcription

1 On the Distribution of Carmichael Numbers Aran Nayebi arxiv: v10 [math.nt] 12 Aug 2009 Abstract {1+o(1)} log log log x 1 Pomerance conjectured that there are x log log x Carmichael numbers up to x. At the time, his data tables up to appeared to support his conjecture. However, Pinch extended this data and showed that up to 10 21, Pomerance's conjecture did not appear well-supported. Thus, we build upon the work of Carl Pomerance and others to formulate an alternative conjecture regarding the distribution of Carmichael numbers that ts proven bounds and is roughly supported by Richard Pinch's data. All tables are included in the Appendix, and they are fully explained in the text. 1 Introduction Fermat's little theorem, also known as Fermat's criterion, states that for any integer b, n is prime if b n b (mod n). (1.0.1) When gcd(b, n) = 1, we can divide by b, b n 1 1 (mod n). (1.0.2) A composite natural number n for which b n 1 1 (mod n) for any xed integer b 2 is a pseudoprime. A positive composite integer n is a Carmichael number if b n 1 1 (mod n) for all integers b 2 with gcd(b, n) = 1. The importance of Carmichael numbers is that they test the limits of the Fermat primality test, and hence their distribution will be the interest of this paper. Let P b (x) denote the number of base b pseudoprimes x; and let C(x) denote the number of Carmichael numbers x. Pomerance [8] conjectures that for all large x, {1+o(1)} log log log x 1 C(x) = x log log x. (1.0.3) Unfortunately, according to Richard Pinch [19], Pomerance's conjecture does not appear to be well-supported by the data, an assertion which will be explained later in the discourse. As a result, we present an alternate conjecture based on a strongly supported conjecture that, C(x) C 3(x)P b (x), (1.0.4) P b,2 (x) where P b,2 (x) is the number of two-prime base b pseudoprimes x and C 3 (x) is the number of three-prime Carmichael numbers x. Instead, we propose that the number of Carmichael 1

2 numbers up to x suciently large is, C(x) = ψ x 5 6 log x L(x) = 1 x 3 ψ 1x 1 2 log 2 x 2 L(x) dt log 3 t, (1.0.5) log x log log log x where L(x) := exp{ }, ψ = τ 3 log log x C, ψ 1 = τ 3. If we let p, q, and d be odd primes, 27C and we dene ω a,b,c (p) as the number of distinct residues modulo p represented by a, b, c, then the constants C and τ 3 are explicitly given as such, κ 3 = n 1 C = 4T s 1 gcd(n, 6) n 4/3 r>s gcd(r,s)=1 δ(rs)ρ(rs(r s)) (rs) 3 2 T = 2 1 2/d (1 1/d), 2 d ρ(m) = d 1 d 2, d m 2, if 4 m; δ(m) = 1, if otherwise. (1.0.6) τ 3 = κ 3 λ, (1.0.7) λ := ( ) 1 3/p, (1 1/p) 3 p>3 p n p>3 p p 3 a<b<c, n=abc a,b,c pairwise coprime δ (a, b, c) p n p>3 2, if a b c 0 (mod 3); δ (a, b, c) =. 1, if otherwise. p ω a,b,c (p), p 3 We believe that ψ will approach and ψ 1 will approach We also demonstrate that this conjecture is roughly supported by Pinch's data and that it ts the proven upper and lower bounds for C(x). 2 Previous Results Before delving into the main results of this paper, we shall rst present previous results that we will explicitly use later on in our derivations. 2

3 2.1 Pseudoprimes Pseudoprime Distribution Erd s [1] proved that for the pseudoprimes dened in (1.0.2), Theorem 1 (Erd s 1950). lim x P π b (x) = o(π(x)), where P π b (x) is the number of base-b pseudoprimes x for b 2. Thus, the pseudoprimes are rarer than the primes. Although this claim is rather elementary, Erd s [11] went on to formulate an upper-bound for the pseudoprime counting function. Theorem 2 (Lehmer 1936, Erd s 1956). For some constants ɛ 11 and ɛ 2, ɛ 1 log x 2 P π 2 (x) x exp{ɛ 2 log x log log x}. In 1989, Pomerance [6] [11] proved tighter bounds for pseudoprime distribution. Theorem 3 (R.A. Mollin ed. 1989, Pomerance 1981). For the base-2 pseudoprime counting function, exp{(log x) } P 2 (x) x L(x) 1 log x log log log x 2, where L(x) = exp{ }. These log log x bounds are applicable to P b (x) for x x 0 (b). Theorem 4 (Pomerance 1981). Pomerance [11] showed that if we allow l 2 (n) to denote the exponent to which 2 belongs modulo n, then n is a pseudoprime (base 2) if and only if l 2 (n) n 1. Conjecture 5 (Pomerance 1981). Pomerance conjectured that the number of solutions w for all n is, #{w x : l 2 (w) = n} x L(x) 1+θ(x), lim θ(x) = 0. (2.1.1) x As a result, the number of base b pseudoprimes for suciently large x x 0 (b) is conjectured to be, P b (x) x L(x) 1. (2.1.2) Two-Prime Pseudoprime Distribution In 2004, William Galway conjectured a formula for the distribution of pseudoprimes with two distinct prime factors, p and q [3]. He noticed that a majority of these pseudoprimes satisfy the relation p 1 = r, where r and s are small coprime integers. Thus, q 1 s Conjecture 6 (Galway 2004). Allow p, q, and d be odd primes, allow P b,2 (x) to represent the counting function for odd pseudoprimes with two distinct prime factors, and P b,2 (x) := #{n x : n = pq, p < q, P b (n)}. Hence, 1 5 ɛ 1 can be taken as 8 log 2 [10]. 2 The lower bound is accredited to Lehmer [9]. lim P b,2(x) = Cx 1 2 x log 2 x, (2.1.3) 3

4 where C = 4T s 1 r>s gcd(r,s)=1 T = 2 d δ(rs)ρ(rs(r s)) (rs) , (2.1.4) 1 2/d 1.32, (2.1.5) (1 1/d) 2 ρ(m) = d 1 d 2, (2.1.6) d m 2, if 4 m; δ(m) = (2.1.7) 1, if otherwise. Galway's conjecture is somewhat supported by the data in Table 3 for it appears that C is slowly approaching its predicted constant value of (2.1.4). 2.2 Carmichael Numbers Carmichael Number Distribution Erd s proved an upper-bound for C(x) in 1956 [17]. Theorem 7 (Erd s 1956). where k is a constant. { k log x log log log x } C(x) < x exp, (2.2.1) log log x In the other direction, Alford, Granville, and Pomerance a lower-bound for C(x) for x suciently large [15]. Theorem 8 (Alford-Granville-Pomerance 1994). thus there are innitely many Carmichael numbers. Recently, Harman improved this lower-bound [18]. Theorem 9 (Harman 2005). just below x 1 3. It is not even known if C(x) > x 1 2. C(x) > x 2 7, (2.2.2) C(x) > x 0.332, (2.2.3) 4

5 Conjecture 10 (Pomerance 1981). The distribution of Carmichael numbers [8] is given by, for x suciently large. {1+o(1)} log log log x 1 C(x) = x log log x, (2.2.4) Remark 11. It is obvious that Conjecture 10 holds if and only if lim k = 1 in Theorem 7. However, Pinch [19] explains that the decrease in k is reversed between and 10 14, which is presented in Table 6. In fact, in Table 6, there is no clear evidence that suggests lim k = Three-Prime Carmichael Number Distribution In 2001, Granville and Pomerance [8] conjectured a formula for the distribution of Carmichael numbers with three distinct prime factors. Conjecture 12 (Granville-Pomerance 2001). If we let C 3 (x) be the counting function for Carmichael numbers with 3 distinct prime factors, then where κ 3 = n 1 gcd(n, 6) n 4/3 x 1 3 C 3 (x) τ 3 log 3 x τ 1 x dt log 3 t, (2.2.5) τ 3 = κ 3 λ 2100, (2.2.6) λ := ( ) 1 3/p , (2.2.7) (1 1/p) 3 p>3 p n p>3 p p 3 a<b<c, n=abc a,b,c pairwise coprime δ (a, b, c) p n p>3 p ω a,b,c (p), (2.2.8) p 3 2, if a b c 0 (mod 3); δ (a, b, c) =, (2.2.9) 1, if otherwise. and ω a,b,c (p) is the number of distinct residues modulo p represented by a, b, c. Recent provisional estimates by Chick and Davies [14] of the slowly converging innite series κ 3 suggest that κ 3 = which gives τ 3 = In Table 4, it is evident that τ 3 is in fact approaching We should note that we will use this value for τ 3 rather than Preliminaries We now present the main results achieved in the paper by rst introducing three important lemmas. 5

6 3.1 Three Lemmas Regarding Base b Pseudoprimes with k Distinct Prime Factors Let ω(n) represent the number of dierent prime factors of n. Also, given an integer sequence {m i } i=1, note that a prime p is said to be a primitive prime factor of m i if p divides m i but does not divide any m j for j < i. Lemma 13 (Erd s 1949). Let n be a base-2 pseudoprime. For every k, there exist innitely many squarefree base-2 pseudoprimes with ω(n) = k [2]. Lemma 14. We closely follow Erd s' proof [2] of Lemma 13 to show that there exist in- nitely many squarefree base b pseudoprimes n for any b 2 with ω(n) = k distinct prime factors. Proof. Let {n j } j=1 be an integer sequence of base b pseudoprimes such that each term is greater than its preceding term, and ω(n i ) = k 1, for any n i in {n j } j=1. Let p i be one of the primitive prime factors of b n i 1 1. Since b n i 1 1 (mod p i n i ) and b p i 1 1 (mod p i ), p i n i is a pseudoprime to base b. We observe that b p i 1 1 (mod n i ) because p i 1 0 (mod (n i 1)). As a result, it follows that b n i 1 1 (mod n i ). Also, b n ip i 1 1 (mod p i n i ) since b n ip i 1 = b (n i 1)(p i 1) b n i 1 b p i 1. Hence, p i n i is squarefree and ω(p i n i ) = k. Moreover, every integer satisfying p i n i is dierent because n i is composite, p i > n i, and p i 1 (mod (n i 1)). Lemma 15. We extend Szymiczek's proof [12] to demonstrate that for any base b pseudoprime, b 2, having k 2 distinct prime factors and for x suciently large, P b,k+1 (x) P b,k (log b x). (3.1.1) Proof. Let n be a pseudoprime with k > 1 distinct prime factors. Since n 1 is the smallest exponent ɛ such that p b ɛ 1 and ɛ divides an exponent h such that p b h 1, it follows from Fermat's little theorem that p b p 1 1. Thus, from Zsigmondy's theorem, there exists a prime p > n for which p b n 1 1 and n 1 p 1 for b 2. As a result, np b n 1 1. (3.1.2) On the other hand, since np 1 = n(p 1) + n 1 and n 1 p 1, n 1 np 1 and np b np 1 1. If we let n, m N, the set of positive natural numbers, such that n m and p > n, q > m, then np mq for primes p and q. However, suppose we let np = mq and p > n, then m p. Hence, m p and m > n. Unfortunately, the latter statement is contradictory, and as a result np mq. If n and m are two dierent base b pseudoprimes with k 2 distinct prime factors, then np and mq are distinct pseudoprimes as well. From (3.1.2), p (b n 1 2 1)(b n ), (3.1.3) and p b n < b n 2. (3.1.4) 6

7 If n log b x, then pn < x 1 2 log b x < x. It then follows that for every base b pseudoprime n with k distinct prime factors, n = p 1 p 2 p k log b x, there is at least one base b pseudoprime such that p 1 p 2 p k p < x. Our proof of Lemma 15 closely models Szymiczek's proof in [12] until (3.1.3), where the relation p 2 n < e n 2 given in [12] does not hold for bases b > 2. This is easily overcome by using the relation p b n < b n 2 instead. 3.2 Conjecture 16 and Conjecture 18 We conjecture the following relations: Conjecture 16. For any xed k 2, let P b,k (x) denote the counting function for base b pseudoprimes with k distinct prime factors, and let P b (x) denote the counting function for base b pseudoprimes. Asymptotically, P b,k (x) P b (x) = o(1). (3.2.1) as, log x log log x We are only able to partially support Conjecture 16. First, we express the ratio P b,k(x) P b (x) P b,k (x) P b (x) = P b,k(x), (3.2.2) g(x) P b,i (x) where the maximum number of distinct prime factors, g(x), of any integer x is g(x). Let log(j) b x denote the the j-fold iteration of the base b logarithm. Thus, i=2 g(x) k 1 P b,i (x) = P b,i (x) + P b,k + i=2 i=2 g(x) i=k+1 Due to Lemma 15, for any h k in (3.2.3), P b,k (x) P b,h (log (k h) b in (3.2.3), P b,w (x) P b,k (log (w k) b x). Hence, i=2 P b,i (x). (3.2.3) x), and for any w k k 1 P b,i (x) P b,2 (log (k 2) b x) + P b,3 (log (k 3) b x) + + P b,k 1 (log b x) (3.2.4) We cut o the terms from proceeding until could be suciently large to satisfy (3.2.5). g(x) i=k+1 log x log log x because if such were the case, then no x P b,i (x) P b,k+1 (log b x) + + P b,r(x) (log (r(x) k) b x), (3.2.5) 7

8 where r(x) is any function that grows slower than log x, the iterated logarithm. We explicitly dene log x as { log 0 if x 1; x := 1 + log (log x) if x > 1. (3.2.6) Remark 17. We should note that the support for Conjecture 16 is rather weak. This is largely due to the weakness of Szymiczek's construction, P b,k+1 (x) P b,k (log b x), in Lemma 15. We believe that the latter relation can be strengthened if a polynomial decrease can be proven. In other words, if P b,k+1 (x) P b,k (x c ) for some c (0, 1). However, we neither know how to prove this nor do we know what value c can acquire. Similarly, in our proof of Conjecture 16, we dened the function r(x) as any function that grows slower than log x, the iterated logarithm. Although it is not hard to see that any function growing faster than log x will fail, it is not obvious whether any function growing at the same rate as log x will succeed. However, we have several reasons to strongly believe that r(x) = log x. First, for practical values of x the iterated logarithm grows much more slowly than the logarithm. Second, log x = slog e (x), where slog e (x) is the super-logarithm, the inverse of tetration. Thus, the iterated logarithm's relation to the super-logarithm also supports its slow growth. Third, higher bases give smaller iterated logarithms, and log x is well dened for { any base greater than exp 1 e will grow even more slowly for higher pseudoprime bases. }. This implies that for any base b 2, the iterated logarithm Conjecture 18. For any xed k 3, let C k (x) denote the number of k-prime Carmichael numbers up to x, and let C(x) denote the Carmichael number counting function. Asymptotically, C k (x) = o(1). (3.2.7) C(x) 3.3 Support for Conjecture 16 and Conjecture 18 So far, the claim established by Conjecture 16 is not yet borne out by the data in Table 1. We believe that the ratio P b,2(x) P 2 will approach 0, but may do so slowly. On the other hand, it (x) appears that the ratio C 3(x) in Table 2 rapidly approaches 0, thereby supporting Conjecture C(x) 18. Furthermore, Pomerance, Selfridge, and Wagsta's results [10] support both conjectures as well. In Conjecture 1 of their paper, they believe that for each ɛ > 0, there is an x 0 (ɛ) such that for all x x 0 (ɛ), { {2 + ɛ} log x log log log x } C(x) > x exp. (3.3.1) log log x Pomerance, Selfridge, and Wagsta [10] show that P b,k (x) O k (x 2k/(2k+1) ). Hence, if (3.3.1) is true, then the pseudoprimes with exactly k prime factors form a set of relative density 0 in the set of all [pseudoprimes] [10]. Similarly, they show that C k (x) x (k 1)/k, and if 8

9 (3.3.1) holds, then for each k, C k (x) = o(c(x)) [10]. Interestingly, we can also support the statements in Conjecture 16 and Conjecture 18 by relating them to their composite superset. Let the number of composites x with k distinct prime factors be denoted by π k (x) and let the number of composites x with k prime factors (not necessarily distinct) be represented by τ k (x). Hence, we can prove upper and lower bounds for π k (x). In of Hardy and Wright [7] for k 1, where Π k (x) = ϑ k(x) log x +O( x kx(log log x) k 1 for k 2 and follows that k!π k (x) Π k (x) k!τ k (x), (3.3.2) x ) in In , since ϑ log x k(x) = Π k (x) log x x 2 Π k (x) dt = O(x), Π k (x) t 2 Π k (x) dt t kx(log log x)k 1. As a result, it log x x(log log x)k 1 π k (x) (1 + o(1)) (k 1)! log x. (3.3.3) In the same respect, a lower bound for π k can be formulated. In it is proven that, τ k (x) π k (x) 1 1 := Π k 1 (x). (3.3.4) p 1 p 2 p 2 k 1 x p 1 p 2 p k 1 x Since π k (x) τ k (x) Π k 1 (x) and π k (x) Π k(x) Π k! k 1 (x), ( ) x(log log x) k 1 (k 1)x(log log x)k 2 π k (x) O (k 1)! log x log x ( ) x + O. (3.3.5) log x Although the lower-bound in (3.3.5) is meaningless, we can improve the upper-bound given in (3.3.3) to an equality, x(log log x)k 1 π k (x) (k 1)! log x. (3.3.6) By the Erd s-kac Theorem, we can formulate the probability that a number near x has k distinct prime factors using the fact that these numbers are distributed with a mean and variance of log log x. Hence, setting log log x as the λ of the Poisson distribution P(k; λ) and taking its limit for any xed k, (log log x) k 1 exp{ log log x} lim P(k; λ) = lim x x (k 1)! = 0, (3.3.7) 1 where the asymptotic error bound is given by O( ) [13]. However, we caution the reader log log x to consider that just because the probability of a general composite near x having k distinct prime factors goes to 0, does not necessitate that this probability will hold for either P b,k (x) or C k (x). 9

10 4 On the Distribution of Carmichael Numbers From Conjecture 16 and Conjecture 18, it is obvious that the k-prime pseudoprimes and the k-prime Carmichael numbers are much more sparsely distributed than the set of all pseudoprimes and Carmichael numbers, respectively. We hypothesize that if k is minimized for both the k-prime pseudoprimes and the k-prime Carmichael numbers, then the ratios P b,2 (x) P b and C 3(x) (x) C(x) will roughly achieve the same values for large enough x. We also recommend using the minimum number of distinct prime factors for both the pseudoprimes and the Carmichael numbers because rst, there is no overlap between the three-prime Carmichael numbers and two-prime pseudoprimes and second, the distinct prime factors cannot be arbitrarily chosen. This idea leads us to believe that, Conjecture 19. C(x) C 3(x)P b (x). (4.0.8) P b,2 (x) As a result, assuming Conjecture 5, Conjecture 6, Conjecture 12, and Conjecture 19, Conjecture 20. The amount of Carmichael numbers x given by the counting function C(x) is conjectured to be for x suciently large, where and C(x) = ψ x 5 6 log x L(x) = 1 x 3 ψ 1x 1 2 log 2 x 2 L(x) ψ = τ 3 C We propose that ψ approaches and that ψ 1 dt log 3 t, (4.0.9) (4.0.10) ψ 1 = τ 3 27C. (4.0.11) approaches Conclusions Conjecture 20 ts the proven bounds for C(x) given in Theorems 7-9, and both ψ and ψ 1 appear to be approaching constant values. However, there are several reasons as to why Conjecture 20 may not be necessarily borne out by the data in Table 5. The estimate of C (2.1.4) was found by summing roughly 10 9 of the most signicant terms of the dening sum [3], and Galway proposes to make the computation of C rigorous, with explicit error bounds [3]. Similarly, the innite series κ 3 is slowly convergent, and it is not until in Table 4 that κ 3 appears to approach its estimated value of However, the primary source of inaccuracy is due to Conjecture 5. Since Pomerance's conjecture for the distribution of pseudoprimes is applicable for suciently large x and pseudoprime counts have only been 10

11 conducted to 10 15, we are not sure how suciently large x must be for Conjecture 5 to be an accurate model for pseudoprime distribution. Lastly, x must also be immensely large in order for P b,2(x) P b (x) = o(1); a claim which can be interpreted from Table 1. 6 Acknowledgements I would like to express my gratitude to Charles R. Greathouse IV who provided invaluable guidance in the direction of this paper; Johan B. Henkens who helped me count the base-2 pseudoprimes and 2-strong pseudoprimes up to using Galway's data; Professor Kazimierz Szymiczek who claried Lemma 15; and David H. Low who assisted me with formatting and typesetting errors. I would also like to thank Professor William F. Galway for sharing his viewpoints regarding the pseudoprimes with k distinct prime factors. 11

12 References [1] Crandall, Richard, and Carl Pomerance. Prime Numbers: A Computational Perspective. New York: Springer, [2] Erd s, Paul. On the Converse of Fermat's Theorem. Amer. Math. Monthly 56 (1949): [3] Galway, William F. Research Statement galway/research-statement.pdf. [4] Galway, William F. Tables of pseudoprimes and related data. 4 Nov Apr [5] Galway, William F. The Pseudoprimes below Nov Simon Fraser University. 8 Apr [6] Gordon, Daniel M., and Carl Pomerance. The Distribution of Lucas and Elliptic Pseudoprimes. Math. Comp. 57 (1991): [7] Hardy, Godfrey H., and Edward M. Wright. An Introduction to the Theory of Numbers. Oxford: Clarendon P, Oxford UP, [8] Pomerance, Carl, and Andrew Granville. Two Contradictory Conjectures Concerning Carmichael Numbers. Math. Comp. 71 (2001): [9] Pomerance, Carl. A New Lower Bound for the Pseudoprime Counting Function. Illinois J. Math. 26 (1982): 4-9. [10] Pomerance, Carl, J. L. Wagsta, and Samuel S. Wagsta, jr. Pseudoprimes to Math. Comp. 35 (1980): [11] Pomerance, Carl. On the Distribution of Pseudoprimes. Math. Comp. 37 (1981): [12] Szymiczek, Kazimierz. On Pseudoprimes which are Products of Distinct Primes. Amer. Math. Monthly 74 (1967): [13] Tenenbaum, Gérald. Introduction to Analytic and Probabilistic Number Theory. Cambridge: Cambridge UP, [14] Chick, J.M. and G.H. Davies. The Evaluation of κ 3. Math. Comp. 77 (2008): [15] Alford, W.R., Andrew Granville, and Carl Pomerance. There are Innitely Many Carmichael Numbers. Ann. of Math. 140 (1994):

13 [16] Pinch, Richard G.E. The Carmichael Numbers up to Proceedings Conference on Algorithmic Number Theory, Turku, May Turku Centre for Computer Science General Publications 46, edited by Anne-Maria Ernvall-Hytönen, Matti Jutila, Juhani Karhumäki and Arto Lepistö. [17] Erd s, Paul. On pseudoprimes and Carmichael numbers. Publ. Math. Debrecen 4 (1956), [18] Harman, Glyn. On the number of Carmichael numbers up to x. Bull. Lond. Math. Soc. 37 (2005), [19] Pinch, Richard G.E. The Carmichael Numbers up to 10 to the 21. Eighth Algorithmic Number Theory Symposium ANTS-VIII May 17-22, 2008 Ban Centre, Ban, Alberta (Canada). 13

14 APPENDIX P b,2 (x) P 2 (x) Bound P b,2 (x) P 2 (x) Table 1: Comparison between P b,2 (x) and P 2 (x) up to C 3 (x) C(x) Bound C 3 (x) C(x) Table 2: Comparison between C 3 (x) and C(x) up to

15 Bound P b,2 (x) C Table 3: Values of C up to Bound τ Table 4: Values of τ 3 up to

16 Bound C(x) 69.51x 5 6 log x L(x) 2.57x 1 2 log 2 x R x dt log 3 t L(x) ψ ψ Table 5: Values of ψ and ψ 1 up to

17 Bound k Table 6: Values of k up to Data Sources: The counts of P b,2 (x) in Table 1 and Table 3 as well as the values of C in Table 3 are from [5]. The counts of P 2 (x) in Table 1 were performed by myself using W.F. Galway's data in [4]. The counts of C 3 (x) in Table 2 and the counts of C(x) in Table 2 and Table 5 are from [16]. Table 4 is a replica of Table 3 in [14]. Table 6 is based o of values of k found in [19]. 17