On Values Taken by the Largest Prime Factor of Shifted Primes

Transcription

1 On Values Taken by the Largest Prime Factor of Shifted Primes William D. Banks Department of Mathematics, University of Missouri Columbia, MO 652 USA Igor E. Shparlinski Department of Computing, Macquarie University Sydney, NSW 209, Australia September 6, 20 Abstract Let P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n >. We show that, for every real number 32/7 < η < ( )/4, there exists a constant c(η) > such that for every integer a 0, the set { p P : p = P(q a) for some prime q with p η < q < c(η)p η} has relative asymptotic density one in the set of all prime numbers. Moreover, in the range 2 η < (4+3 2)/4, one can take c(η) = +ε for any fixed ε > 0. In particular, our results imply that for every real number ϑ 0.53, the relation P(q a) q ϑ holds for infinitely many primes q. We use this result to derive a lower bound on the number of distinct prime divisors of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q P(q a) for a > 0, and show that for infinitely many primes q, this map can be iterated at least (log log q) +o() times before it terminates.

2 Introduction. Background Let P be the set of prime numbers, and for every integer n >, let P(n) P be the largest prime factor of n. The function P : {2, 3,...} P arises naturally in many number theoretic situations and has been the subject of numerous investigations; see, for example, [5, 6, 9, 5, 7, 8, 9, 22, 25, 29] and the references contained therein. Recently, driven in part by applications to cryptography, there has been a surge of interest in studying the largest prime factors of the shifted primes {q ± : q P}. Improving on earlier results of Pomerance [26], Balog [4], Fouvry and Grupp [3], and Friedlander [4], Baker and Harman [3] proved the existence of infinitely many primes q for which P(q a) q 0.296, where a 0 is any fixed integer. In the same paper, they also showed the existence of infinitely many primes q for which P(q a) q 0.677, improving earlier results of Hooley [20], Deshouillers and Iwaniec [], Fouvry [2], and others. In this paper, we study the related problem of estimating the number of primes p that occur as the largest prime factor of a shifted prime q a when q P lies in a certain interval determined by p. Interestingly, questions of this sort also have applications in theoretical computer science and, in a different form, have been considered by Vishnoi [30]. We also study iterates of the map q P(q a) for a > 0, and show that for infinitely many primes q, this map can be iterated at least (log log q) +o() times before it terminates..2 Main Results For an integer a 0 and real numbers η > 0 and c >, let P a,η,c be the set of primes: P a,η,c = { p P : p = P(q a) for some prime q with p η < q < c p η}, and let π a,η,c (x) denote its counting function: π a,η,c (x) = #{p x : p P a,η,c }. Denoting by π(x), as usual, the number of primes p x, we show that if η lies in a suitable range, then there exists a constant c = c(η) such that π a,η,c (x) lim x π(x) 2 =

3 holds for every integer a 0. In other words, P a,η,c has relative asymptotic density one in the set of all prime numbers. More precisely, we prove: Theorem. For every real number 32/7 < η < ( )/4, there exists a constant c = c(η) > such that the estimate ( ) x π a,η,c (x) = π(x) + O, log K x holds for every integer a 0 and real number K, where the implied constant depends only on a, η, and K. Moreover, if 2 η < (4+3 2)/4, this estimate holds for any constant c >. Let I a denote the set of all limit points of the set of ratios: { } log P(q a) : q P. log q Certainly, there is no reason to doubt that I a = [0, ]; however, our present knowledge about the structure of I a is rather limited. Using the results of [3] mentioned above, it is easy to see that inf I a and sup I a for any a 0. In view of Theorem, we immediately deduce the following: Corollary. For every integer a 0, the set I a contains the closed interval [0.486, 0.53]. We remark that, under the Elliott-Halberstam conjecture, which asserts that the bound max π(y) gcd(a,m)= π(y; m, a) ϕ(m) x log C x max y x m x ε holds for any fixed real numbers ε, C > 0, our approach yields an extension of Theorem to the range < η < ( )/4. The same conjecture also implies that I a = [0, ]. Indeed, if π a (x, y) denotes the number of primes q x for which P(q a) y, then it is natural to expect that the asymptotic relation π a (x, y) ρ(u)π(x) () 3

4 holds over a wide range in the xy-plane, where u = (log x)/(log y) and ρ(u) is the Dickman function (see [5, 8, 28]). The statement () is a well known consequence of the Elliott-Halberstam conjecture (see [, 5]), and using () it is easy to see that I a = [0, ]. Next, recall that the Carmichael function λ(n) is defined for n as the maximal order of any element in the multiplicative group (Z/nZ). More explicitly, for a prime power p ν, one has { p λ(p ν ) = ν (p ), if p 3 or ν 2; 2 ν 2, if p = 2 and ν 3; and for an arbitrary integer n 2, λ(n) = lcm ( λ(p ν ),...,λ(pν k k )), where n = p ν pν k k is the prime factorization of n. Clearly, λ() =. We also use ω(n), as usual, to denote the number of distinct prime divisors of n ; in particular, ω() = 0. Theorem 2. For a fixed integer a 0, let Q a (x) = q P a<q x (q a) and W a (x) = ω (λ(q a (x))). Then, for sufficiently large x, the following bound holds: W a (x) x Again, it is an easy matter to verify that, under the Elliott-Halberstam conjecture, the estimate W a (x) x +o() holds for any fixed a. Now, let a > 0 be fixed, and put Q a,0 = {q P : q a + }. We define sets of primes {Q a,k : k } recursively by Q a,k = {q P : q a + 2, P(q a) Q a,k }, k =, 2,..., and consider the corresponding counting functions: ρ a,k (x) = #{q x : q Q a,k }, k =, 2,.... 4

5 Theorem 3. For every integer a > 0, the bound ρ a,k (x) 2 a 3 k+ x exp( (log x) /k )(log x) ak holds for all x > x 0 (a), where x 0 (a) depends only on a, and k. For fixed a > 0 and an arbitrary prime q, consider the chain given by q 0 = q, and q j = P(q j a), j =, 2,..., and define k a (q) as the smallest nonnegative integer k for which q k a +. Corollary 2. Let a > 0 be fixed. Then for all but o(π(x)) primes q x, the following lower bound holds: log log x k a (q) ( + o()) log log log x We remark that the lower bound of Corollary 2 is closely related to (and complements) certain results from [23]. We also observe that k (q) gives a lower bound for the height of the tree representing the Pratt primality certificate [27] associated to q. This primality certificate is a recursively-defined construction which consists of a primitive root g modulo q and a list of the prime divisors p,...,p s of q together with their certificates of primality; accordingly, the whole certificate has the natural structure of a tree. Clearly, the height H(q) of this tree satisfies the trivial bound H(q) log q. On the other hand, our Corollary 2 implies that the lower bound log log x H(q) ( + o()) log log log x holds for all but o(π(x)) primes q x..3 Acknowledgements The authors would like to thank Carl Pomerance for many useful comments and, in particular, for the observation that the Elliott-Halberstam conjecture implies I a = [0, ]. We also thank Kevin Ford for bringing to our attention the link between our results and the Pratt primality certificate. This work was done during a visit by the second author to the University of Missouri-Columbia; the support and hospitality of this institution are gratefully acknowledged. During the preparation of this paper, I. S. was supported in part by ARC grant DP (2)

6 2 Preliminaries 2. Notation Throughout the paper, we adopt the following conventions. Any implied constants in the symbols O, and may depend (where obvious) on the parameters a, η, and K, but are absolute otherwise. We recall that the statements A B and B A are equivalent to A = O(B) for positive functions A and B. The letters p, q, r, l are always used to denote prime numbers, and m, n always denote positive integers. As usual, we write π(x; m, a) for the number of primes p x in the arithmetic progression a (mod m). For simplicity, we use log x to denote the maximum of and the natural logarithm of x > 0, and we write log 2 x = log(log x). Finally, we use ϕ(n) to denote the value of the Euler function at the positive integer n. 2.2 Necessary Tools Our principal tool is the following result, which follows immediately from the Bombieri-Vinogradov theorem (see [0]) in the range 0 < ϑ < /2, and from Theorem of [2] in the range /2 ϑ 3/25, and from the main theorem of [24] in the range 3/25 < ϑ < 7/32: Lemma. There exist functions C 2 (ϑ) > C (ϑ) > 0, defined for real numbers ϑ in the open interval (0, 7/32), such that for every integer a 0 and real number K, the inequalities C (ϑ) y ϕ(p) log y < π(y; p, a) < C 2(ϑ) y ϕ(p) log y hold for all primes p y ϑ, with at most O(y ϑ / log K y) exceptions, where the implied constant depends only on a, ϑ, and K. Moreover, for any fixed ε > 0, these functions can be chosen to satisfy the following properties: C (ϑ) is monotonic decreasing, and C 2 (ϑ) is monotonic increasing; C (/2) = ε, and C 2 (/2) = + ε. 6

7 We also need a result from sieve theory, which is an application of Brun s method. The following statement is Theorem 6.7 of [2] (see also Theorem 5.7 of [6]): Lemma 2. Let g be a natural number, and let a j, b j (j =,..., g) be integers such that E 0, where E = g j= a j i<k g (a i b k a k b i ). For a prime number r, let ρ(r) be the number of solutions n modulo r to the congruence g (a j n + b j ) 0 (mod r), j= and suppose that ρ(r) < r for every r. Then, for X Y >, # { Y X < n X : (a j n + b j ) is prime for j =,..., g } 2 g g! ( ρ(r) ) ( ) g { ( )} Y log2 Y + log r r log g + O 2 E, Y log Y r P where the implied constant depends only on g. Finally, we need the following technical result: Lemma 3. For every positive integer n, let ψ(n) = Then the following estimate holds: k z gcd(k,a)=, 2 ka ψ(k) k r P r n, r 2 = ( c 2 + o() ) ϕ(2a) 2a ( + r 2 log z ). r P r a, r 2 ( where c 2 is the twin primes constant given by: c 2 = ( ) = (r ) 2 r P r 2 7 ), (r ) 2

8 Proof. Since ψ(n) = d n, 2 d where µ(d) is the Möbius function, and F(n) = r P r n µ 2 (d) F(d), (r 2), it follows that k z gcd(k,a)=, 2 ka ψ(k) k = = k z gcd(k,a)=, 2 ka d z gcd(d,2a)= µ 2 (d) df(d) µ 2 (d) k F(d) d k 2 d h z/d gcd(h,a)=, 2 ha h. In the case that a is odd, we have h z/d gcd(h,a)=, 2 ha h = = 2 = 2 h z/2d gcd(h,a)= δ a δ a = ϕ(a) 2a µ(δ) δ µ(δ) δ 2h = 2 h z/2dδ h z/2d h h (log(z/d) + O()) log(z/d) + O(), δ gcd(h,a) µ(δ) 8

9 whereas for even a, h z/d gcd(h,a)=, 2 ha h = Hence, in either case, we have k z gcd(k,a)=, 2 ka ψ(k) k h z/d gcd(h,a)= = δ a = δ a = ϕ(a) a = ϕ(2a) 2a µ(δ) δ h = h z/dδ h z/d h h δ gcd(h,a) µ(δ) (log(z/d) + O()) δ log(z/d) + O(). d z gcd(d,2a)= µ(δ) µ 2 (d) log(z/d) + O(). df(d) Now we split the summation on the right according to whether d w or d > w, where w = exp ( log z ). Since F(d) d for all odd squarefree integers d, it follows that and d w gcd(d,2a)= The result follows. w<d z gcd(d,2a)= µ 2 (d) df(d) log z log(z/d) = o(), w/2 µ 2 (d) log(z/d) = ( + o()) log z df(d) = ( + o()) log z r P r 2a = ( + o()) c 2 log z 9 d= gcd(d,2a)= ( + r P r a, r 2 µ 2 (d) df(d) ) r(r 2) ( ). (r ) 2

10 3 Proofs 3. Proof of Theorem Let the numbers a, η, and K be fixed as in the statement of Theorem, and put ϑ = /η. In what follows, the real numbers λ > and, µ > 0 are constants that depend only on a, η, and K. Let x be a large positive real number, put y = x η. Then x = y ϑ and log y = η log x. (3) Applying Lemma, first with y and then with λy, we see that the inequalities and C (ϑ) y ϕ(p) log y < π(y; p, a) < C 2(ϑ) y ϕ(p) log y C (ϑ) λy ϕ(p) log y < π(λy; p, a) < C 2(ϑ) λy ϕ(p) log y hold for all primes p x, with at most O ( x/ log K x ) exceptions. Hence, if we define the set { A = p x : π(y; p, a) C 2(ϑ) y p log y and π(λy; p, a) C } (ϑ) λy, p log y it follows that Next, let ( ) x #A = π(x) + O. log K x B C = {p A : p ( ) x}, = {p A : ( ) x < p x}. Since A is the disjoint union of B and C, and ( ) x #B π(( ) x) = ( ) π(x) + O, log K x we see that ( ) x #C π(x) + O. (4) log K x 0

11 For a fixed prime p C, let D p E p = {y < q λy : q a (mod p) and P(q a) > p}, = {y < q λy : P(q a) = p}, and observe that #E p = π(λy; p, a) π(y; p, a) #D p y (C (ϑ)λ C 2 (ϑ)) p log y #D p. (5) If q D p, there exists a prime l > p and an integer k such that q = pkl + a and P(q a) = l. In fact, the condition P(q a) = l is redundant. Indeed, since l > p > ( ) x, we have: k = q a pl y x 2 = xη 2, and since η < 3, it follows that l > p > k once x is sufficiently large. Moreover, the preceding estimate implies that D p = for all p C if η < 2 and x is large enough. Next, we estimate #D p in the case that η 2. For each prime p C and integer k x η 2, let F p,k = {y < q λy : q = pkl + a for some prime l > p}. Clearly, D p k F p,k. Moreover, assuming that x is large enough, we have F p,k G p,k, where G p,k = {( µ)y/pk < l ( + µ)λy/pk : both l and (pkl + a) are prime}. To estimate #G p,k, we first observe that G p,k = if either gcd(k, a) or 2 ka. On the other hand, if gcd(k, a) = and 2 ka, then we apply Lemma 2 with the choices g = 2, a =, b = 0, a 2 = pk, and b 2 = a. Note that E = pka 0. For every prime r, the number ρ(r) of solutions modulo r to the congruence n(pkn + a) 0 (mod r) is one if r pka, and two otherwise; in particular, ρ(r) < r for every prime r. Finally, taking X = ( + µ)λy/pk and Y = (λµ + λ + µ )y/pk in the

12 statement of Lemma 2 (thus, X Y = ( µ)y/pk), we obtain the following bound: ( ρ(r) ) ( ) 2 γy r r pk log 2 (γy/pk) #G p,k 8 r P { + O ( log2 (γy/pk) + log 2 (pka) log(γy/pk) )}, where γ = λµ + λ + µ. Noting that γ > λ > 0, and using the simple estimates pka x η γy and pk x, we deduce that #G p,k (8γ + o()) r P ( ρ(r) ) ( ) 2 y r r pk log 2 x. Now, since p, k, and a are pairwise coprime, and 2 ka, it follows that r P ( ρ(r) ) ( = 2c 2 ψ(p)ψ(k)ψ(a), r r) where the constant c 2 and the function ψ(k) are defined as in Lemma 3. Therefore, #G p,k (6c 2 γ + o()) ψ(p)ψ(k)ψ(a) y pk log 2 x. Summing this estimate over k, applying Lemma 3, and using the fact that ψ(p) = ( + o()), we derive that #D p k x η 2 gcd(k,a)=, 2 ka #G p,k (6c 2 γ + o()) ψ(a) = (8γ(η 2) + o()) y p log 2 x k x η 2 gcd(k,a)=, 2 ka y ψ(a)ϕ(2a) p log x a 2 ψ(k) k r P r a, r 2 ( ). (r ) 2

13 It is easy to verify that, for every integer a 0, one has ψ(a)ϕ(2a) ( ) =, a (r ) 2 and therefore, using (3), we have #D p (8γ(η 2) + o()) r P r a, r 2 y p log x = (8γη(η 2) + o()) y p log y. (6) We now turn to the selection of the constants c, λ > and, µ > 0. Our first goal is to show, for x sufficiently large, that C P a,η,c. Suppose that p C and q E p ; then, p η x η = y < q λy = λx η < λ ( ) η pη. From these inequalities, it follows that C P a,η,c if E p for all p C, and the constants c, λ, and satisfy the relation: c = λ ( ) η. (7) In the case that η < 2, we have already seen that D p = for all p C once x is large enough. By (5), it follows that E p for all p C if λ > C 2(ϑ) C (ϑ). (8) Since ϑ > 0.5 in this case, Lemma implies that for any c C 2 (ϑ)/c 2 (ϑ), both relations (7) and (8) can be simultaneously satisfied for an appropriate choice of λ > and > 0, provided that η > 32/7. In the case that η 2, after substituting (6) into (5), we see that E p for all p C if x is sufficiently large, and C (ϑ)λ C 2 (ϑ) > 8γη(η 2). (9) Let ε > 0 be a fixed constant, and put c = + ε. Choosing λ = + ε/2 > and = 3 ( ) ϑ + ε/2 > 0, + ε

14 we see that relation (7) is satisfied. For any fixed constant δ > 0, we can assume C (ϑ) = δ and C 2 (ϑ) = + δ according to Lemma. Since the left-hand side of (9) and γ = λµ + λ + µ can each be made arbitrarily close to λ = ε/2 by choosing δ and µ sufficiently close to 0, it follows that relation (8) is also satisfied for an appropriate choice of δ and µ, provided that 8η(η 2) <, that is, η < ( )/4. Taking into account (4), we have therefore shown that if 32/7 < η < ( )/4, there exists a constant c = c(η) such that at least ( ) x π(x) + O log K x primes p in the interval ( )x < p x lie in the set P a,η,c, and the theorem follows. 3.2 Proof of Theorem 2 For fixed η in the range 32/7 < η < (4+3 2)/4, put ϑ = /η, and consider the counting function: a(y) = # {p y : p = P(q a) for some q P with q c p η }, where c = c(η) is the constant described in Theorem. According to that theorem, we have the following estimate: ( ) y a(y) = π(y) + O log 2. y Defining y = (c x) ϑ, it follows that there are ( + o())π(y) primes p y such that p Q a (x); let S denote this set of primes. By a result of [3], there are at least y +o() primes p S with P(p a) y ; let R denote this set of primes. Then ( ( )) ( ( )) W a (x) ω λ p ω λ p. p S Let L be the set of primes l for which l = P(p ) for some p R. 4 p R

15 Clearly, a prime l y cannot have the property that l = P(p ) for more than y primes p R; consequently, W a (x) #L y o() = x 0.677ϑ+o(). Taking η = ϑ sufficiently close to 32/7, we obtain the stated result. 3.3 Proof of Theorem 3 Let ψ(x, y) = #{n x : P(n) y}. We recall the well known bound on ψ(x, y) (see [8, 8, 28]): ψ(x, e w ) x exp( ( + o())u log u), (0) which holds uniformly as u = (log x)/(log y) with u y /2. If k =, then the set Q a, [, x] consists of all prime numbers q of the form m + a, where P(m) a + and m x a < x; therefore, the bound ρ a, (x) (2 log x) π(a+) (2 log x) a holds uniformly for x 2. We finish the proof by induction. Suppose that the result is true up to k, where k 2. We can assume that k log log x log log log x, () for otherwise the bound of the theorem holds trivially. For every integer w 0, we have: ρ a,k (x) ψ(x, e w ) + x/q q Q a,k e w <q x ψ(x, e w ) + x q Q a,k e w <q x log x ψ(x, e w ) + x ν=w q e ν q Q a,k e ν <q e ν+ log x ψ(x, e w ) + x e ν ρ a,k (e ν+ ). ν=w 5

16 We now choose and put u = (log x) /k, w = log x u = (log x) /k. Since u log log x by (), and u e w/2 if x is large enough (independent of k or a), we can use the bound (0); in a weaker form, this gives ψ(x, e w ) xe u = x exp( (log x) /k ) if x is sufficiently large. Using the induction hypothesis, we derive that log x ν=w Therefore e ν ρ a,k (e ν+ ) log x ν=w 2 a 3 k e exp( (ν + ) /(k ) )(log x) a(k ) 2 a 3 k e exp( w /(k ) )(log x) a(k )+. ρ a,k (x) x exp( (log x) /k ) + 2 a 3 k e x exp( w /(k ) )(log x) a(k )+. Since w /(k ) = (log x) /k, a, and + 3 k e 3 k+, we conclude the proof. 4 Concluding Remarks As we have already remarked, the Elliott-Halberstam conjecture leads to an extension of Theorem to the range < η < ( )/4. We also note that the factor 2 g g! in Lemma 2 is probably unnecessary. In the absence of this factor, the stronger estimate of Lemma 2 would lead to a corresponding extension of Theorem to the range 32/7 < η < + 2. Clearly, Theorem 2 implies that ( ( )) λ λ q exp(x ). q P, q x It would also be interesting to estimate k-fold iterates of the Carmichael function applied to the product of the primes q x. 6

17 We now recall the asymptotic formula x 2 n x log P(n) log n = o() for the average logarithmic size of the largest prime factor (see Exercise 3 in Chapter III.5 of [28]). Assuming that the shifts q j a, where q 0 = q, and q j = P(q j a), j =, 2,..., behave as typical integers, then it is reasonable to expect that the bound k a (q) log log q holds for almost all primes q. In particular, the lower bound of Corollary 2 is probably rather tight. On the other hand, it should be possible to improve the logarithmic factor (log x) k in the bound of Theorem and thus obtain a slightly better bound for k a (q), although the technical details are more involved. Similarly, although we expect that Theorem and Corollary 2 also hold for negative integers a (with appropriate modifications), the proof appears to be more complicated as the induction step must be handled in a different way to retain uniformity of the bound with respect to k. Finally, it would be interesting to know whether the lower bound (2) for the height of the Pratt tree is tight. References [] W. R. Alford, A. Granville and C. Pomerance, There are infinitely many Carmichael numbers, Annals Math., 40 (994), [2] R. C. Baker and G. Harman, The Brun-Titchmarsh theorem on average, Proc. Conf. in Honor of Heini Halberstam (Allerton Park, IL, 995), Progr. Math., vol. 38, Birkhäuser, Boston, 996, [3] R. C. Baker and G. Harman, Shifted primes without large prime factors, Acta Arith., 83 (998), [4] A. Balog, p+a without large prime factors, Sém. Théorie des Nombres Bordeaux (983 84), exposé 3. 7

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