Almost All Palindromes Are Composite

Transcription

1 Almost All Palindromes Are Comosite William D Banks Det of Mathematics, University of Missouri Columbia, MO 65211, USA bbanks@mathmissouriedu Derrick N Hart Det of Mathematics, University of Missouri Columbia, MO 65211, USA hart@mathmissouriedu Mayumi Sakata Det of Mathematics, University of Missouri Columbia, MO 65211, USA sakata@mathmissouriedu Never odd or even Abstract We study the distribution of alindromic numbers with resect to a fixed base g 2 over certain congruence classes, and we derive a nontrivial uer bound for the number of rime alindromes n x as x Our results show that almost all alindromes in a given base are comosite MSC Numbers: 11A63, 11L07, 11N69 Corresonding author 1

2 1 Introduction Fix once and for all an integer g 2, and consider the base g reresentation of an arbitrary natural number n N: L 1 n a k ng k k0 Here a k n {0, 1,, } for each k 0, 1,, L 1, and we assume that the leading digit a L 1 n is nonzero The integer n is said to be a alindrome if its digits satisfy the symmetry condition: a k n a L 1 k n, k 0, 1,, L 1 Let P N denote the set of alindromes in base g, and for every ositive real number x, let Px {n x n P} In this aer, we study the distribution of alindromes in congruence classes Using the Weil bound for mixed Kloosterman sums, we bound exonential sums over the set P L of alindromes with recisely L digits and use this result to show that the set Px becomes uniformly distributed as x over the congruence classes modulo, where > g is any rime number for which the multilicative order ord g of g in the grou Z/Z is at least 3 1/2 ; see Corollary 44 for a recise statement We remark that, thanks to the work of Paalardi [4], almost all rimes satisfy the stronger condition ord g 1/2 exlog c where c is any constant less than 1 log 2/2; see also [2, 3] Using a variation of these techniues, we also show that the set Px becomes uniformly distributed as x over the congruence classes modulo, where 2 is any integer corime to gg 2 1; see Corollary 45 This latter result, though weaker than that obtained for rimes satisfying the condition ord g 3 1/2, allows us to deduce the main result of this aer: almost all alindromes in a given base are comosite More recisely, in Theorem 51, we show that # { n Px n is rime } O # Px 2 log log log x, x, log log x

3 where the imlied constant deends only on the base g This result aears to be the first of its kind in the literature Acknowledgments The authors would like to thank Florian Luca and Igor Sharlinski, whose valuable observations on the original manuscrit led to significant imrovements in our estimates During the rearation of this aer, W B was suorted in art by NSF grant DMS Preliminary Estimates For any integer 2, let e x denote the exonential function ex2πix/, which is defined for all x R For any integer c that is relatively rime to, let c denote an arbitrary multilicative inverse for c modulo ; that is, c c 1 mod Finally, let d be the number of ositive integral divisors of, and let ord g be the smallest integer t 1 such that g t 1 mod Lemma 21 For any rime with gcd, g 1 and all a, b Z, we have ord g e ag k bg k 21/2 gcda, b, 1/2 k1 Proof Consider the mixed Kloosterman sum 1 K χ a, b; χc e ac bc, c1 where χ is a Dirichlet character modulo From the work of Weil [6], it follows that the bound K χ a, b; 2 1/2 gcda, b, 1/2 holds for all such sums Averaging over all Dirichlet characters χ modulo for which χg 1, it follows that ord g ϕ K χ a, b; χ The result follows 1 c 1 c g k mod, k 3 e ac bc ord g k1 e ag k bg k

4 Lemma 22 The following bound holds for all 2, k 2 and h Z rovided that h: k 1 e ha k ex 4 gcdh, 2 2 Proof Let us write a0 k 1 s, k, h e ha If d gcdh,, then s, k, h s/d, k, h/d, hence it suffices to rove the assertion for the secial case where gcdh, 1, which we now assume Without loss of generality, we may also suose that k Indeed, if k 1, then we can exress k m r with 0 r 1 and simly observe that s, k, h s, r, h r 1 1 ex 4/ 2 k ex 4/ 2 If gcdh, 1 and 2 k, we have therefore, s, k, h 2 s, k, h 2 k 1 a,b0 k 2 1 k a0 e ha b k k 1 a,b0 a b 2πha b cos k kk 1 cos2π/; 1 1 cos2π/ 1 1 cos2π/ k 2 Using the fact that 1 cosx 2 ex x 2 /4 for 0 x π, we obtain the desired result 3 Exonential Sums over Palindromes For every L 1, let P L denote the set of alindromes in base g with recisely L digits; that is, P L {n P g L 1 n < g L } 4

5 Lemma 31 Let > g be a rime number such that ord g > 2 1/2 Then for every c Z with gcdc, 1, the exonential sum S L c n P L e cn satisfies the bound SL c Θ L 2ord/4, where Θ 1 g 2g 11/2 g ord g < 1 Proof Since S 2L c a 0 1 a 1 0 a 0 1 a L 1 0 L 1 e ca k g k g 2L 1 k k0 L 1 e ca0 1 g 2L 1 k1 a k 0 e cak g k g 2L 1 k and S 2L1 c a 0 1 it follows that a 0 1 a 1 0 a L 0 e ca0 1 g 2L L 1 e ca L g L ca k g k g 2L k a L 0 S2Lδ c g 1g δ for all L 1 and δ 0 or 1 k0 L 1 e ca L g L L 1 k1 k1 a k 0 e cak g k g 2L k, e ca g k g 2Lδ 1 k a0 Put N ord g, and write L 1 Nm l, where m L 1/N and 0 l < N Then, using the arithmetic-geometric mean ineuality, we derive 5

6 that S2Lδ c 2 Nm 2 g 1 2 g 2l2δ e ca g k g 2Lδ 1 k k1 a0 g 1 2 g 2l2δ 1 Nm 2 e ca g k g 2Lδ 1 k Nm k1 a0 Nm T g 1 2 g 2l2δ, Nm where T Nm k1 a,b0 gnm m e ca b g k g 2Lδ 1 k a,b0 a b N e ca b g k g 2Lδ 1 k k1 Since > g, it follows that gcda b, 1 whenever a b Using Lemma 21, we therefore obtain that T gnm g 1gm 2 1/2 Conseuently, S2Lδ c 2 gnm 2g 1gm g 1 2 g 2l2δ 1/2 Nm g 1 2 g 2Nm2l2δ N 2g 1 1/2 g 1 2 g 2L2δ 2 Θ L l 1 Since # P 2Lδ g 1g Lδ 1, the result follows gn Nm Nm Nm 6

7 Lemma 32 Let 2 be an integer such that gcd, gg Then for every c Z such that c, the exonential sum S L c n P L e cn satisfies the bound S L c ex L 5 gcdc, 2 2 Proof As in the roof of Lemma 31, we have S2Lδ c L 1 g 1g δ e ca g k g 2Lδ 1 k for all L 1 and δ 0 or 1 Let k1 a0 B {1 k L 1 divides cg k g 2Lδ 1 k }, G {1 k L 1 does not divide cg k g 2Lδ 1 k } Using Lemma 22 to estimate individual terms in the receding roduct when k G, and using the trivial estimate when k B, we obtain that S2Lδ c g 1g δ# G # B ex 4 gcdc, 2 # G 2 # P 2Lδ ex 4 gcdc, 2 # G 2 Now let f / gcdc, Since does not divide c, we have f 2, and the stated condition on imlies that ord f g 2 2 Thus, if k and l both lie in B, then g 2 k g 2Lδ 1 g 2 l mod f, k l mod ord f g 2 We therefore see that and the result follows # B 1 L 2/2 L/2, # G L 1 L/2 L/2 1 2L δ 5/4, 7

8 4 Distribution of Palindromes Proosition 41 Let > g be a rime number such that ord g 3 1/2 Then for every L 10 5, the following estimate holds for all a Z: #{ n P L n a mod } # < P L 099L Proof Using the relation it follows that 1 1 { 1 if m 0 mod, e cm 0 otherwise, c0 { # n P L n a mod } 1 1 e cn a n PL c0 1 1 e ca e cn n P L c0 1 1 e cas L c, where S L c is the exonential sum considered in Lemma 31 Therefore #{ n P L n a mod } c1 1 1 SL c c1 1 Θ L 2ord/4, c1 where Θ 1 g 2g 11/2 g ord g 1 g 2g 1 3g 2g 1 3g 5 6, since g 2 Also, L 2 ord g 1/4 L 2 1/4 L/5, L

9 Conseuently, #{ n P L n a mod } # P L 1 L/5 5 6 Finally, remarking that the condition ord g 3 1/2 imlies that 11, we have L/5 5 1 < 099 L, L This comletes the roof Proosition 42 Let 2 be an integer such that gcd, gg Then for every L log, the following estimate holds for all a Z: #{ n P L n a mod } # < P L ex L 2 2 Proof Using the relation it follows that 1 1 { 1 if m 0 mod, e cm 0 otherwise, c0 # { n P L n a mod } n PL 1 1 e cn a c0 1 1 e ca e cn n P L c0 1 1 e cas L c, where S L c is the exonential sum considered in Lemma 32 If 1 c 1, then c, hence by Lemma 32 we derive the estimate: SL c L 5 gcdc, 2 ex log 2 ex log L 5 ex L, c1

10 the last ineuality following from the stated condition on L The result follows immediately Theorem 43 Let 2 be a fixed integer, and suose that there exist constants A 1 and 2/3 ξ < 1, deending only on, such that #{ n P L n a mod } A ξ L for all L 1 and a Z Then for some constant B 1 that deends only on g, the following estimate holds for all x 1 and a Z: #{ n Px n a mod } # Px # Px AB ξ log x/2 log g Proof We remark that the condition ξ 2/3 guarantees that gξ 2 is bounded below by an absolute constant greater than 1; since g 2, we have g 1 gξ 2 1 g 1 2 g For all L 1, x y > 0, and a Z, let us denote We also denote P a {n P n a mod }, P a,l {n P a g L 1 n < g L }, P a x {n P a n x}, P a y; x {n P a y < n x} Py; x {n P y < n x} In what follows, the imlied constants in the symbol O may deend on g but are absolute otherwise We recall that the notation U OV for ositive functions U and V is euivalent to U cv for some constant c Let a Z be fixed in what follows, and suose that g 2Mδ 1 x < g 2Mδ, where M is an integer and δ 0 or 1 We observe that # Px # Pg 2Mδ 1 # Pg 2Mδ 1 ; x, 1 10

11 and that # P a x # P a g 2Mδ 1 # P a g 2Mδ 1 ; x Our goal is to estimate # # Px P a x 2 # P a g 2Mδ 1 # Pg 2Mδ 1 # P a g 2Mδ 1 # Pg 2Mδ 1 ; x ; x Since the integer g 2Mδ 1 is not a alindrome a fact that is only used to simlify our notation, we have by a straightforward calculation: On the other hand, # P a g 2Mδ 1 M 1 l0 2Mδ 1 L1 # Pg 2Mδ 1 g M g Mδ # P a,l # # P 2l1 P a,2l1 # Pg 2Mδ 1 M 1 l0 M 1 l0 # P 2l1 # P a,2l1 # P a,2l1 # P 2l1 Mδ 1 l1 Mδ 1 l1 # P a,2l # # P 2l P a,2l Mδ 1 l1 # P a,2l Using the hyothesis of the theorem, it therefore follows that # P a g 2Mδ 1 # Pg 2Mδ 1 M 1 # P 2l1 A ξ 2l1 Since M 1 # P 2l1 ξ 2l1 Mδ 1 l0 # P 2l ξ 2l Mδ 1 l1 # P 2l # P 2l # P 2l A ξ 2l l0 l1 M 1 g 1g l ξ 2l1 l0 < g 1 gξ 2 1 Mδ 1 l1 g 1g l 1 ξ 2l g M ξ 2M1 g Mδ 1 ξ 2M2δ O g M ξ 2M, 11

12 we see that # P a g 2Mδ 1 # Pg 2Mδ 1 O Ag M ξ 2M 4 We now turn to the more delicate estimation of # P a g 2Mδ 1 ; x To this end, ut M K L, where K and L are ositive integers to be selected later Examining the base g reresentation of an arbitrary alindrome n in P 2Mδ, we see that n may be exressed either in the form n n 1 g Kµ n 2 g K2Lδ n 3, or the form where n n 1 g K2Lδ n 3, 1 n 1 < g K, g K 1 n 3 < g K, n 1 g K n 3 P 2K, 5 and, in the former case, n 2 P 2Lδ 2µ for some 0 µ L δ 1 The integers n 1, n 2, n 3, µ are uniuely determined by n We call n 3 the K-signature of n and write s K n n 3 The integer n 1 is uniuely determined by n 3 together with the first and third conditions of 5; we call n 1 the K-comlement of n 3 and write c K n 3 n 1 Note that the number of alindromes n P 2Mδ with a fixed K-signature s K n n 3 is recisely 1 Lδ 1 µ0 # P 2Lδ 2µ 1 Lδ 1 µ0 g 1g Lδ µ 1 g Lδ 6 Now, given x in the range g 2Mδ 1 x < g 2Mδ, let y be the alindrome in P 2Mδ defined by y y 1 g K g 2Lδ 1 g K2Lδ y 3, where x/g K2Lδ 1 if g 2Mδ 1 x < g 2Mδ 1/2, y 3 x/g K2Lδ 1 if g 2Mδ 1/2 x < g 2Mδ, 12

13 and y 1 c K y 3 If x lies in the smaller range, then x < y, while y < x if x lies in the larger range In either case, we have # Pg 2Mδ 1 ; x # Pg 2Mδ 1 ; y Og L 7 and # P a g 2Mδ 1 ; x # P a g 2Mδ 1 ; y Og L, since there are at most O1 distinct K-signatures for alindromes between x and y Conseuently, # P a g 2Mδ 1 # Pg 2Mδ 1 ; x ; x 8 # P a g 2Mδ 1 # Pg 2Mδ 1 ; y ; y OgL Now, if n Pg 2Mδ 1 ; y, then its K-signature lies in the range g K 1 s K n y 3 Thus, # Pg 2Mδ 1 ; y y 3 g K 1 1g Lδ 9 On the other hand, if n P a g 2Mδ 1 ; y with s K n n 3, then either or n n 1 g Kµ n 2 g K2Lδ n 3 a mod n n 1 g K2Lδ n 3 a mod, deending on the form of n In the latter case, there is at most one such alindrome n for each fixed K-signature n 3, while in the former case, since n 2 g K µ a c K n 3 g K2Lδ n 3 mod, the number of such alindromes n is # P b,2lδ 2µ for each 0 µ L δ 1, where b bn 3, µ g K µ a c K n 3 g K2Lδ n 3 13

14 Hence, using 6, we derive that # P a g 2Mδ 1 ; y y 3 Lδ 1 # P b,2lδ 2µ Og K y 3 n 3 g K 1 n 3 g K 1 Lδ 1 1 µ0 y 3 n 3 g K 1 # Pg 2Mδ 1 ; y µ0 # P 2Lδ 2µ Lδ 1 µ0 y 3 n 3 g K 1 # P b,2lδ 2µ Lδ 1 µ0 # P b,2lδ 2µ # P 2Lδ 2µ Og K # P 2Lδ 2µ Using the hyothesis of the theorem, it therefore follows that # P a g 2Mδ 1 # Pg 2Mδ 1 ; y ; y y 3 n 3 g K 1 y 3 n 3 g K 1 Lδ 1 µ0 Lδ 1 µ0 < Ay 3 g K 1 1 # P 2Lδ 2µ A ξ 2Lδ 2µ Og K Og K g 1g Lδ µ 1 A ξ 2Lδ 2µ Og K g 1 gξ 2 1 glδ ξ 2Lδ2 Og K, and conseuently, # P a g 2Mδ 1 # Pg 2Mδ 1 ; y ; y O Ag M ξ 2L Og K Using this estimate together with 2, 4 and 8, it follows that # # Px P a x O Ag M ξ 2L g L g K We now choose integers K M/2 O1 and L M/2 O1 such that K L M Since gξ 2 > 1 and A 1, we have max{g K, g L } Og M/2 O Ag M/2 gξ 2 M/2 O Ag M ξ M, 14

15 therefore # # Px P a x O Ag M ξ M To comlete the roof, we need only observe that ξ M O log x/2 log ξ g for x in the range g 2Mδ 1 x < g 2Mδ, and using 1, 3, 7 and 9 together with our choice of y 3, it follows that # Px g M x g M OgM/2 ; thus g M O # Px Using Theorem 43, we can now derive two immediate corollaries Corollary 44 Let > g be a rime number such that ord g 3 1/2 Then for some constant C > 0, deending only on g, the following estimate holds for all x 1 and a Z: #{ n Px n a mod } # Px # Px C 099 log x 2 log g 10 Proof Using the trivial estimate #{ n P L n a mod } for 1 L 10 6, it follows from Proosition 41 that the estimate #{ n P L n a mod } 099 L 106 holds for all L 1 and a Z The result now follows immediately from Theorem 43 Corollary 45 Let 2 be an integer such that gcd, gg Then for some constant C > 0, deending only on g, the following estimate holds for all x 1 and a Z: #{ n Px n a mod } # Px # Px C ex log x 4 2 log g 15

16 Proof Using the trivial estimate #{ n P L n a mod } for 1 L < log, it follows from Proosition 42 that the estimate #{ n P L n a mod } ex L log 2 2 holds for all L 1 and a Z The result now follows immediately from Theorem 43 5 Prime Palindromes We now come to the main result of this aer Theorem 51 As x, we have # { n Px n is rime } O # Px where the imlied constant deends only on g log log log x, log log x Proof As in the roof of Theorem 43, all imlied constants in the symbol O may deend on g but are absolute otherwise Assuming that x is sufficiently large, let h e log log log x, Let y e 1 log x 1/4h ex Q Qy g 3 < y, 1o1 log log x 4e log log log x where the roduct runs over rime numbers Note that gcd Q, gg By Mertens formula see Theorem 11 in I16 of [5], we have the estimate ϕq Q 1 1 O log y 1 log log log x O, 10 log log x g 3 < y 16

17 where ϕn is the Euler function Now, if n Px is rime, either gcdn, Q 1 or n is a rime divisor of Q We aly Brun s combinatorial sieve in the form given by Corollary 11 in I42 of [5]: # { n Px n is rime } y Q ω 2h µa, where µ is the Möbius function, ω is the number of distinct rime divisors of, and A # {n Px n 0 mod } By Corollary 45, we see that A # Px O If Q and ω 2h, then # Px ex log x 4 2 log g y 2h log x1/2 e 2h, and since the number of such divisors is bounded by y 2h, we have ex log x log x 4 2 log g e ex e4h 4h 4 log g Q ω 2h since h e log log log x Therefore, # { n Px n is rime } y # Px Q ex log log x 4h e4h 1 O, 4 log g log x µ O # Px 17 Q ω>2h 1 O # Px log x

18 Since y x o1 and x 1/2 O # Px, the first term in this estimate is negligible Also, using 10, we have # Px µ # Px 1 1 O # log log log x Px log log x Q g 3 < y Finally, we have Q ω>2h 1 Q ω>2h e ω 2h e 2h e/ ex 2h e y1 1/ y Observing that y 1 log log y1 o1 log log log x1 o1, by our choice of h it follows that # Px Q ω>2h 1 # Px ex log log log x e o1 O This comletes the roof # Px log log x 2 6 Remarks and Oen Problems Using estimates from [1], it is ossible to establish a version of Lemma 31 under the weaker assumtion that ord g log ; this yields analogues of Proosition 41 and Corollary 44, however the uniform constant 099 in those results must be relaced by a term like ex log log c for some constant c > 0 It seems natural to conjecture that the set of alindromes should behave as random integers, thus one might exect that the asymtotic relation # { n Px n is rime } C # Px log x 18

19 holds for some constant C > 0 While this uestion seems out of reach at the moment, it should be feasible to derive the uer bound { # n Px n is rime } # Px O log x using more sohisticated sieving techniues couled with better estimates for the distribution of alindromes in congruence classes It is still an oen roblem to show the existence of infinitely many rime alindromes for any fixed base g 2 References [1] T Cochrane, C Pinner and J Rosenhouse, Bounds on exonential sums and the olynomial Waring roblem mod, J London Math Soc no 2, [2] P Erdős and R Murty, On the order of a mod, Number theory Ottawa, ON, 1996, 87 97, CRM Proc Lecture Notes 19, Amer Math Soc, Providence, RI, 1999 [3] K-H Indlekofer and N Timofeev, Divisors of shifted rimes, Publ Math Debrecen no 3-4, [4] F Paalardi, On the order of finitely generated subgrous of Q mod and divisors of 1, J Number Theory no 2, [5] G Tenenbaum, Introduction to analytic and robabilistic number theory, University Press, Cambridge, UK, 1995 [6] A Weil, On some exonential sums, Proc Nat Acad Sci USA