E-SYMMETRIC NUMBERS (PUBLISHED: COLLOQ. MATH., 103(2005), NO. 1, )

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1 E-SYMMETRIC UMBERS PUBLISHED: COLLOQ. MATH., ), O., 7 25.) GAG YU Abstract A positive integer n is called E-symmetric if there exists a positive integer m such that m n = φm), φn)). n is called E-asymmetric if it is not E-symmetric. We show that there are infinitely many E-symmetric and E-asymmetric primes Mathematics Subject Classification. A4, 36.. Introduction Given any two distinct odd primes p and q, the well known quadratic reciprocity asserts ) ) p q = ) p )q ) 4. q p Among the existing proofs, a most popular one that Gauss gave cf., for example, proof of Theorem 98 in [3]) counts the lattice points inside the rectangle Rp, q) with sides parallel to the axes and two opposite vertices at the origin and p/2, q/2). In [], P. Fletcher, W. Lindgren and C. Pomerance studied the so-called symmetric primes. A prime number p is called symmetric if it is one of the two members of a symmetric pair. Two distinct odd primes p and q form a symmetric pair if the number of lattice points in Rp, q) above the main diagonal is equal to the number of lattice points below. In [], the authors characterized symmetric pairs by the condition p q = p, q ) and, from this, they showed that almost all primes are not symmetric. Generalizing the concept of symmetric pair, Fletcher calls m, n) 2 an E-symmetric pair if m n = φm), φn)), and a positive integer E-symmetric if it belongs to an E-symmetric pair, where φ ) is the Euler totient function. E-asymmetric if it is not E-symmetric. A positive integer is called At the 2002 Southeast Regional Meeting on umbers, Peter Fletcher asked: Are there infinitely many a). E-asymmetric numbers b). E-asymmetric primes c). E-symmetric primes? It is clear that p, p+2) gives an E-symmetric pair if p and p+2 are both primes. Hence, there are infinitely many E-symmetric primes if assuming the twin prime conjecture. In fact, it is fairly easy to show that there are many more E-symmetric primes than twin primes. More precisely, we prove unconditionally

2 Theorem.. There exists a constant c > 0 such that, for every sufficiently large, we have #{E symmetric primes p } c log log log log..) With the same idea, we can also show that there are infinitely many E-asymmetric primes. Theorem.2. There exists a constant c 2 > 0 such that, for every sufficiently large, we have #{E asymmetric primes p } c 2..2) log ) 50 We remark that, by making the argument tighter, it is easy to improve the lower bound.2). We shall not do so, however, since the method we use to prove the theorem doesn t seem to give the best bound that one can expect. Throughout the paper, p, q, l and r always stand for primes; As usual, for given coprime integers k and a, we denote πx; k, a) = {p x : p a mod k)} and Ex; k, a) = πx; k, a) πx) φk), Ex; k) := max Ex; k, a), a,k)= where πx) = πx;, ) is the number of primes up to x; µd) denotes the Möbius function, and vd) denotes the number of distinct prime divisors of integer d. 2. Preliminaries In this section, we give some simple sieve results that we need in the proofs of the theorems. Let A be a finite sequence of positive integers not necessarily distinct), and, for any given integer d, A d := {a A : a 0modd)}. Suppose X := A >, and for each positive integer d, A d is approximated by a main term ωd) X, where ωd) is a multiplicative function with ω) =. Let d and, for a given real number z >, R d := A d ωd) d X, SA, z) := {a A : l a l z}. 2

3 Lemma 2.. Let ξ z and τ = log ξ. Suppose there exist some constants A log z 0 > 0, 0 < δ < such that, for every prime p, then we have 0 ωp) min{ δ)p, A 0 }, 2.) SA, z) = XW z) + Oe τlog τ+) )) + θ d<ξ 2 µ 2 d)3 vd) R d, 2.2) where θ, and W z) := p<z ωp) ). p Proof. This is essentially a special case q = ) of Theorem 7. in [2]. We now make use of Lemma 2. to deduce a lower bound sieve result. Let A > 0 be a fixed number, ɛ > 0 be a constant which is sufficiently small such that, if letting C be the constant involved in the O-symbol in 2.2), C exp log 00ɛ))/00ɛ) < 2. Suppose is sufficiently large, Q, R, S log ) A, 2 < D ɛ, where Q, R and S are odd integers, pairwise coprime and 3 S. Let PQ, R, S, D, ) be the set {p : p fmod2qrs) and l p 2Q l D}, where fmod2qrs) is determined by the following three congruences: p mod2q), p 2modR), p 2modS). 2.3) Lemma 2.2. There is a constant c 0 such that, for large, Proof. Let PQ, R, S, D, ) A = AQ, R, S, D, ) := { p 2Q c QRS log log D. } : p and p fmod2qrs). Then we have PQ, R, S, D, ) = SA, D). We also note that, for any integer d, we have A d = { π; 2QRSd, g) if d, RS) = 0 if d, RS) >, where g mod 2QRSd) is the intersection of the residue classes 3 2.4)

4 Let X := A = π; 2QRS, f), and mod2qd), 2modR) and 2modS). 2.5) ωd) = { dφ2q) φ2qd) if d, RS) = 0 if d, RS) >. 2.6) It is easy to see that ωd) is multiplicative and it satisfies 2.) with A 0 = 2 and δ = 2. We also note that R d = 0 if d, RS) > and R d = E; 2QRSd, g) φ2q) E; 2QRS, f) φ2qd) E; 2QRSd) + exp κ log ) φd) 2.7) if d, RS) =, where the estimate for the term φ2q) E; 2QRS, f) in 2.7) is from the φ2qd) Siegel-Walfisz Theorem, and κ > 0 is a certain constant. Applying Lemma 2. with ξ = 5 and noticing the restriction on ɛ, we get PQ, R, S, D, ) XW D) OEQ, R, S, D, )) 2.8) 2 where the error term EQ, R, S, D, ) is given by the sum in 2.2). By the Siegel-Walfisz Theorem and Mertens Theorem, the term XW D) is 2 2 π; 2QRS, f) ) ) p p p 2Q p 2QRS p D QRS log log D. 2.9) The sum of the term exp κ log ) φd) in 2.7) over d gives a contribution O exp κ 2 log )) which is obviously negligible. Then, from Lemma 3.5 of [2], we have EQ, R, S, D, ) log ) B 2.0) for any given B > 0. We thus have proved the lemma. We also state an upper bound result here. Lemma 2.3 Let g be a natural number, a i, b i i =, 2,, g) be pairs of integers satisfying a i, b i ) =, i =, 2,, g, 4

5 and E := g i= a i i<j g For prime p, let ρp) be the number of solutions of a i b j a j b i ) 0. g a i n + b i ) 0 mod p). i= Then for any fixed constant δ 0, ) and D x δ, we have g #{n x : p i n + b i ) p > D} i=a ) ρp) g x p log g D, 2.) p E where the constant involved in the -symbol depends on δ only. Proof. This is a straightforward corollary of Theorem 2.2 in [2]. 3. Infinitude of E-symmetric primes In this section, we prove Theorem.. Suppose is sufficiently large. Let P ) = {p : l p 2 l log log ) 2 }. 3.) Then from Lemma 2.2 with Q = R = S = and D = log log ) 2 ), we have for some constant c 3 > 0. P ) c 3 log log log log We claim that, for almost all p P ), we have 3.2) φp), φp + 2)) = ) Assume p P ) and 3.3) doesn t hold, then there is a prime q log log ) 2 such that divides φp), φp + 2)), thus p + 2 is divisible by a prime l mod ). Let l = 2sq + for some s, then p + 2 = 2sq + )2tq + 3) for some t. ote that t = 0 is impossible since p mod ) and 3 p + 2).) Thus the number of such primes p P ) with q > log ) 3 is bounded by log ) 3 <q dm) st q q 2 m q 2 5 log ) P ). 3.4) 2 log

6 The number of those p P ) with D q log ) 3 and l > is bounded by, 3.5) D q log ) 3 t s 4tq 2 l 2sq+)2sq+)2tq+3) 2) l 4 for which, we appeal to Lemma 2.3 with g = 2) and have upper bound D q log ) 3 t q 2 tlog ) 2 p 2tq+3) ) p log log D log P ), 3.6) log log n where we have used the fact that, for n 3, log log n. By Brun-Titchmarsh φn) inequality cf., Theorem 3.8 in [2], for example), we have the upper bound for the number of the primes with D q log ) 3 and l : D q log ) 3 log log D log + l l mod ) q D q log ) 3 D q log ) 3 log p p mod ) p 2 mod l) l l mod ) q + 2 q l 4q q D q log ) 3 ) dπt;, ) t q log /) + q 4q ) t logt/) dt log log D log P ). 3.7) log log From 3.4)-3.7), we have shown that we thus have proved Theorem.. {p P ) : φp), φp + 2)) 2} P ) log log, 4. Infinitude of E-asymmetric primes We now devote to proving Theorem.2. Let ɛ > 0 be sufficiently small, as required in Lemma 2.2. Suppose is sufficiently large. We let q log ) 4 be a fixed prime. From 6

7 Heath-Brown s work [4], we can further fix two distinct primes l, l 2 satisfying l, l 2 mod ) and l, l 2 q 5.5. Let P 2 ) be the set of primes p satisfying and p mod), p 2modl ), p 2modl 2 ), 4.) Directly from Lemma 2.2, we have l p P 2 ) l ɛ. 4.2) c 4 ql l 2 log ) 2 4.3) for some constant c 4 > 0. We shall show that almost all p P 2 ) are E-asymmetric. p P 2 ), amely, for almost all φp), φp ± 2k)) 2k 4.4) for all 2k p. We discuss this case by case. k = ote that φp), φp ± 2)), thus φp), φp ± 2)) 2. k = q If p P 2 ) satisfies φp + ), then p + is divisible by a prime lq) mod). ). lq) = p + In this case, p+)p ) is free of prime divisors ɛ, and p fq) mod l l 2 ). Thus the number of such primes is bounded by the number of integers n such that each such n satisfies the congruences 4.) and nn+)n ) f l l 2, f is free of prime divisors ɛ. This is equal to the number of integers k ] l l 2 such that l l 2 k+f)l l 2 k++f)l l 2 k+ f ) is free of prime divisors ɛ. By Lemma 2.3, the number of such integers k is bounded by ql l 2 log ) + 3 φ + ) P 2). 4.5) log 2). lq) < p + 7

8 Then there is an integer a > 0 such that p + = lq)aq + ), the number of such p P 2 ) is then bounded by. 4.6) aq+)bq+) aq+)bq+) fq)+ mod l l 2 ) ote fq) + is obviously prime to l l 2, this sum is a q aq 2 l l 2 ql l 2 log ) P 2) 3 log. 4.7) Therefore, {p P 2 ) : φp), φp + )) = } P 2) log. 4.8) Similarly, we can show that {p P 2 ) : φp), φp )) = } P 2) log. 4.9) k > q Suppose there is a prime l > ɛ, such that φp), φp + 2ml)) = 2ml, where m satisfies I). if q m, then m = or r m r ɛ ; q q II). if q m, then m = or r m r ɛ. We estimate the number of such primes p P 2 ) in accordance with whether p + 2ml is a prime. ). p + 2ml is a prime The two cases q m or q m) can be treated similarly. When q m, the corresponding prime p P 2 ) satisfies the following conditions i). p = st +, r st r ɛ ; ii). iii). st + s + is a prime; st gq)modl l 2 ) for some s, t) 2, where gq) = fq). 8

9 The number of such primes is bounded by. 4.0) st, s>ɛ st gq) mod l l 2 ) r stst+)st+s+) r ɛ From Lemma 2.3, this sum is bounded by t ɛ t,l l 2 )= r t r ɛ t ɛ t,l l 2 )= r t r ɛ ql l 2 log ) 3 s t s gq) t mod l l 2 ) r sst+)st+s+) r ɛ ql l 2 tlog ) 3 t ɛ r t r ɛ l ql l 2 3 ) l 3<l tt+) 2 ) l 3 l t + φtt + )). 4.) Taking by log log )2 as an upper bound of, by Mertens Theorem, this sum is bounded tt+) φtt+)) log log ) 2 ql l 2 log ) 3 t r t r ɛ t log log )2 ql l 2 log ) 3 log log )2 ql l 2 log ) 3 ɛ r + ) r P 2) log. 4.2) We can similarly deal with the case when q m and get the same upper bound. 2). p + 2ml is not a prime Then in this case, p + 2ml is strictly) divisible by a prime rl) modl). This is similar to the case that when p + is divisible by a prime rq) mod). With the same argument as 4.6)-4.7), the number of these p P 2 ) is bounded by O ɛ ). We thus have shown that Similarly, we have {p P 2 ) : φp), φp + 2k) = 2k for some k > q} P 2) log. 4.3) {p P 2 ) : φp), φp 2k) = 2k for some k > q} P 2) log. 4.4) 9

10 Collecting all these estimates together, we have shown that almost all primes in P 2 ) are E-asymmetric, and thus Theorem.2 follows. References [] P. Fletcher, W. Lindgren and C. Pomerance, Symmetric and asymmetric primes, Journal of umber Theory, 58996), [2] H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, 974. [3] G. H. Hardy and E. M. Wright, An Introduction to the Theory of umbers, 4th ed., Oxford Univ. Press, London, 968. [4] D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions and the least prime in an arithmetic progression, Proc. London Math. Soc. 3), 64992), no. 2, Gang Yu, Department of Mathematics, LeConte College, 523 Greene Street, University of South Carolina, Columbia, SC 29208, USA yu@math.sc.edu 0