AUTHOR’S COPY | AUTORENEXEMPLAR
Integers 11 (2011), 307–313 DOI 10.1515/INTEG.2011.024 © de Gruyter 2011
Power Totients with Almost Primes
William D. Banks and Florian Luca
Abstract. A natural number n is called a k-almost prime if n has precisely k prime factors, counted with multiplicity. For any fixed k > 2, let Fk.X/ be the number of k-th powers k k m 6 X such that !.n/ m for some squarefree k-almost prime n, where !. / is the D 1=k 2k ! Euler function. We show that the lower bound Fk.X/ X =.log X/ holds, where " the implied constant is absolute and the lower bound is uniform over a certain range of k relative to X. In particular, our results imply that there are infinitely many pairs .p; q/ of distinct primes such that .p 1/.q 1/ is a perfect square. # # Keywords. Squares, Euler Function. 2010 Mathematics Subject Classification. 11N60.
– Dedicated to Carl Pomerance on the occasion of his 65th birthday
1 Introduction
A longstanding conjecture in number theory asserts the existence of infinitely many primes of the form m2 1. Although the problem is intractable at present, C there have been a number of partial steps in the direction of this result. For in- 2 stance, thanks to Brun, one knows that the number of integers m 1 6 X that C are prime is at most O.X 1=2= log X/. In the opposite direction, Iwaniec [6] has shown that m2 1 is the product of at most two primes infinitely often. C For any prime p, we clearly have
p m2 1 !.p/ m2; D C ” D where !. / is the Euler function, and hence the m2 1 conjecture can be refor- ! C mulated as the assertion that the set
S2 n N !.n/ is a perfect square D¹ 2 W º contains infinitely many primes. Motivated by this observation, the set S2 was first studied by Banks, Friedlander, Pomerance and Shparlinski [3]; they showed that
0:7038 n 6 X n S2 > X ¹ W 2 º ˇ ˇ for all sufficiently large valuesˇ of X. ˇ
AUTHOR’S COPY | AUTORENEXEMPLAR AUTHOR’S COPY | AUTORENEXEMPLAR
308 W. D. Banks and F. Luca
Although we cannot prove that S2 contains infinitely many primes, it is in- teresting to ask whether other thin sets of integers enjoy an infinite intersection with S2. Recently, Banks [2] showed that S2 contains infinitely many Carmichael numbers, and he asked whether S2 contains infinitely many integers with at most two prime factors. In this note, we give an affirmative answer to this question by showing that there exist infinitely many pairs .p; q/ of distinct primes such that !.pq/ .p 1/.q 1/ is a perfect square. D # # Recall that a natural number n is called a k-almost prime if n has precisely k prime factors, counted with multiplicity. Our main result is the following:
k Theorem. For each k > 2, let Fk.X/ be the number of k-th powers m 6 X such k that !.n/ m for some squarefree k-almost prime n. There is a constant X0 D such that the bound 4X 1=k log X Fk.X/ > holds for 2 6 k 6 9e.log X/2k s12 log log X whenever X > X0.
2 Notation and Outline of Proof
In what follows, the letters p and q always denote prime numbers. As is customary, we use ".x/ to denote the number of primes p 6 x and ".x d; a/ the number of I such primes in the arithmetic progression a mod d. Below, any constants implied by the symbols O, , and are absolute. % " & In particular, the notation x 1 means that x exceeds some positive absolute " constant. Our approach to the proof of the theorem is as follows. Let x X 1=k. We D begin by constructing a certain set Q of primes close to x1=.3k/. Next, we take P k to be the set of primes p 6 x such that p 1 aq for some prime q Q and an # D 2 integer a that is not divisible by any prime in Q. The number ap a is uniquely 2=3 D determined by p, and ap x for all p P , whereas the cardinality of P sat- % 2 isfies the lower bound P x2=3 1=3k.log x/ 2, and hence it follows that P has j j" C ! a large subset of the form Pa p P ap a . For every k-element subset D¹ 2 W D º k p1; : : : ; p of Pa, the number n p1 p does not exceed x X, and n ¹ kº D !!! k D is a squarefree k-almost prime for which the totient !.n/ is a k-th power. Indeed, k writing pj aq 1 with qj Q for each j , we have D j C 2 k !.n/ !.p1 p / .p1 1/ .p 1/ .aq1 q / : D !!! k D # !!! k # D !!! k Thus, to obtain a lower bound for Fk.X/, it suffices to count the number of k-ele- ment subsets of P that are contained in one of the sets Pa.
AUTHOR’S COPY | AUTORENEXEMPLAR AUTHOR’S COPY | AUTORENEXEMPLAR
Power Totients with Almost Primes 309
3 Proof of the Theorem
Following Alford, Granville and Pomerance [1], let B denote the set of numbers B .0; 1/ for which there is a number x0.B/ > 0 and an integer DB > 1 such 2 B 1 B that whenever x > x0.B/, gcd.a; d/ 1 and 1 6 d 6 min x ; y=x ! , one D ¹ º has ".y/ ".y d; a/ > (1) I 2!.d/ provided that d is not divisible by some member of DB .x/, a set consisting of at most DB integers, each of which exceeds log x. In [1, Section 2], it is shown that the interval .0; 5=12/ is contained in B. Let B 1=3 B, let x > x0.1=3/, and let k > 2 be an integer such that D 2 log x k 6 : (2) 12 log log x Observe that if x > 3, then k 6 log x, and we have log x k log k : (3) 6 12 Let Q be the set of primes q in the range
1=.3k/ 1=.3k/ x
AUTHOR’S COPY | AUTORENEXEMPLAR AUTHOR’S COPY | AUTORENEXEMPLAR
310 W. D. Banks and F. Luca
k Let P be the set of primes p 6 x such that q p 1 for some q Q, and j # 2 a .p 1/=qk is not divisible by any prime in Q. Clearly, D # k k P > ".x q ; 1/ ".x q1 q2; 1/: (5) j j I # I q Q q1;q2 Q X2 X2 Taking y x, d qk, a 1 in (1), we have D D D k ".x/ c2x ".x q ; 1/ > > .q Q;x 1/; I 2!.qk/ qk log x 2 " where c2 0:46 (say). Using this bound together with (4), it follows that D k c2x 1 c2x Q ".x q ; 1/ > > j j I log x qk log x ! .x1=.3k//k.1 1=k/k q Q q Q X2 X2 C (6) 2=3 1=.3k/ c1c2x C > e.log x/2 if x is sufficiently large. On the other hand, using the Montgomery–Vaughan large sieve estimate (see [7]) one has 2x ".x qkq ; 1/ : 1 2 6 k k I q1 q2 log.x=.q1 q2//
For all primes q1;q2 Q, we have 2 k k 1 1=3 1=.3k/ 2=3 q1 q2 6 .1 1=k/ C x C 6 x C for all large x and uniformly in k > 2. Therefore, taking (4) into account we derive the bound 6x 1 1 ".x qkq ; 1/ 1 2 6 k I log x q q2 q1;q2 Q q1 Q 1 q2 Q X2 X2 X2 6x c2x2=.3k/ 1 (7) 6 1=.3k/ k 1 2 log x ! .x / C .log x/ 2=3 1=.3k/ 96x C 6 .log x/3 provided that x is large. Here, we have used the fact that c1 6 4. Inserting the bounds (6) and (7) into (5), and taking into account that c1c2=e > 1=3, we obtain the lower bound 2=3 1=.3k/ x C P > .x 1/: (8) j j 3.log x/2 "
AUTHOR’S COPY | AUTORENEXEMPLAR AUTHOR’S COPY | AUTORENEXEMPLAR
Power Totients with Almost Primes 311
By construction, every prime p P has a unique representation of the form k 2 p apq 1, where ap is a natural number and qp is a prime in Q. Let D p C A a N a ap for some p P : D¹ 2 W D 2 º 2=3 Since every ap is a positive integer that does not exceed x , we have the trivial bound 2=3 A 6 x : (9) j j We also note that the inequality k A 1 j j 6 (10) P k 1 j j C holds for all large x and uniformly for all k satisfying (2). Indeed, in view of (8) and (9) this inequality is implied by 2 1=.3k/ 3k.k 1/.log x/ 6 x : C 2 Since k satisfies (2), it follows that 3k.k 1/ 6 .log x/ for all large x, and hence C 4 1=.3k/ it suffices to observe that the inequality .log x/ 6 x is equivalent to (2). For every a A, let 2 Pa p P ap a : D¹ 2 W D º For an arbitrary subset S of P satisfying the properties (i) S k, j jD (ii) S Pa for some a A, ' 2 we put n p: S D p S Y2 Then nS is a squarefree k-almost prime, and the totient !.nS / is a k-th power since k k !.n / .p 1/ .aq / a qp : S D # D p D p S p S Â p S Ã Y2 Y2 Y2 Moreover, the k-th powers constructed in this way are pairwise distinct as S varies over the subsets of P satisfying (i) and (ii) since the set S is uniquely determined by the number m !.n /1=k. Indeed, the number a is the largest divisor of m D S that is coprime to every element of Q, and after factoring m aq1 qk, one k D !!! recovers the set S pj aqj 1 j 1; : : : ; k . k D¹ D C W D º Put X x . Since !.nS / 6 nS 6 X for every subset S P satisfying (i) D ' and (ii), we see that Fk.X/ is bounded below by the number of such subsets S;
AUTHOR’S COPY | AUTORENEXEMPLAR AUTHOR’S COPY | AUTORENEXEMPLAR
312 W. D. Banks and F. Luca therefore, Pa Pa Fk.X/ > j j j j ; (11) k D k a A ! a A0 ! X2 X2 where A0 denotes the set of a A such that Pa > k. Note that A0 is nonempty 2 j j j j for all large X, for if A0 ¿ it follows that P 6 k A , which is untenable in D j j j j view of (10). Now, for fixed k > 2 the function y y.y 1/ .y k 1/ fk.y/ # !!! # C D k! D kŠ is convex as a function of y > k, and hence using (11) together with Jensen’s in- equality, we have 1 1 1 Fk.X/ > fk. Pa / > fk Pa A0 A0 j j A0 j j a A0  a A0 à j j j j X2 j j X2 1 A A0 > fk Pa k j j#j j A0 j j# A0  a A  Ãà j j X2 j j P k A fk j j# j j k : D A0 C  j j à Since y.y 1/ .y k 1/ y k k fk.y/ # !!! # C > # .y > k > 2/; D kŠ k  à it follows that P k A k P k k A k F .X/ A 1 k > 0 j j# j j k j j k 1 j j j j k A0 D k A0 # P  j j à j j !  j j à (12) P k k A k > j j 1 j j : kk A k 1 # P j j !  j j à Taking into account (10) we see that k k k A 1 1 1 j j > 1 >e! : # P # k 1  j j à  C à Using this result in (12) along with (8) and (9), we derive that x X 1=k Fk.X/ > : e.3k/k.log x/2k D e.3=k/k.log X/2k
AUTHOR’S COPY | AUTORENEXEMPLAR AUTHOR’S COPY | AUTORENEXEMPLAR
Power Totients with Almost Primes 313
2 k Since .3=2/ 9=4 and .3=k/ 6 1 for all k > 3, this proves the desired inequal- D ity for those k > 2 that satisfy (2). To finish the proof, we observe that for any integer k such that log X 2 6 k 6 ; s12 log log X we clearly have 2 log X k log x k 6 6 : 12 log log X 12 log log x Hence, (2) holds for any such k.
Bibliography
[1] W. R. Alford, A. Granville and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), no. 3, 703–722. [2] W. D. Banks, Carmichael numbers with a square totient, Canad. Math. Bull. 52 (2009), no. 1, 3–8. [3] W. D. Banks, J. B. Friedlander, C. Pomerance and I. E. Shparlinski, Multiplicative structure of values of the Euler function, in: High Primes and Misdemeanours: Lec- tures in Honour of the 60th Birthday of Hugh Cowie Williams, pp. 29–47, Fields In- stitute Communications 41, American Mathematical Society, Providence, RI, 2004. [4] D. R. Heath-Brown, The number of primes in a short interval, J. Reine Angew. Math. 389 (1988), 22–63. [5] M. N. Huxley, On the difference between consecutive primes, Invent. Math. 15 (1972), 164–170. [6] H. Iwaniec, Almost-primes represented by quadratic polynomials, Invent. Math. 47 (1978), no. 2, 171–188. [7] H. L. Montogomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119–134.
Received November 11, 2009; accepted February 14, 2010.
Author information William D. Banks, Department of Mathematics, University of Missouri, Columbia, Missouri, USA. E-mail: [email protected] Florian Luca, Instituto de Matemáticas, Universidad Nacional Autónoma de México, Morelia, Michoacán, México. E-mail: [email protected]
AUTHOR’S COPY | AUTORENEXEMPLAR