Carmichael numbers with a totient of the form a 2 + nb 2

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1 Carmichael numbers with a totient of the form a 2 + nb 2 William D. Banks Department of Mathematics University of Missouri Columbia, MO USA bankswd@missouri.edu Abstract Let ϕ be the Euler function. Fix l N, and let P be an arbitrary set of primes of positive lower natural density. Using a variant of the Alford-Granville-Pomerance construction, we show that there are infinitely many Carmichael numbers N with a totient of the form ϕ(n) = m l m, where m, m N and m is a nonempty product of primes from the set P. In particular, for any fixed natural number n, there are infinitely many Carmichael numbers N such that ϕ(n) = a 2 +nb 2 for some positive integers a and b. AMS Classification Numbers: 11N25 (11A25) Keywords: Carmichael numbers, totient, smooth shifted primes.

2 1 Introduction For any prime number N, Fermat s little theorem asserts that a N a (mod N) for all a Z. (1) Around 1910, Carmichael began an in-depth study of composite numbers N with this property, which are now known as Carmichael numbers. In 1994 the existence of infinitely many Carmichael numbers was established by Alford, Granville and Pomerance [1]. More recently, Grantham [7, Theorem 4.3] has shown that for any fixed algebraic number field K, there are infinitely many Carmichael numbers which are composed entirely of primes that split completely in K. Since prime numbers and Carmichael numbers are linked by the common property (1), it is natural to ask whether certain questions about primes can be settled for Carmichael numbers; see [2, 3]. For example, it is well known that for every natural number n, there are infinitely many primes of the form a 2 + nb 2 with a, b N (cf. Cox [4]). Using the theorem of Grantham one sees that the same result holds for Carmichael numbers; that is, for any fixed natural number n, there are infinitely many Carmichael numbers of the form a 2 + nb 2 with a, b N. Indeed, let S n = {a 2 + nb 2 : a, b N}, and let K n be the ring class field associated to the order Z[ n ] in the imaginary quadratic field Q( n ). According to [4, Theorem 9.4], if p is an odd prime not dividing n, then p splits completely in K n if and only if p S n. Applying Grantham s theorem with K = K n, we see that there are infinitely Carmichael numbers N composed solely of primes in p S n, and as S n is closed under multiplication, every such N lies in S n. Similarly, it is well known that for every natural number n, there are infinitely many primes p of the form p = a 2 + nb (cf. [11, 14]), so it is reasonable to ask: Question 1. Is it true that for any fixed n N there are infinitely many Carmichael numbers N of the form N = a 2 + nb for some a, b N? At present, this problem appears to be intractable as there is no known method for constructing infinitely many Carmichael numbers N for which the shifted Carmichael number N 1 has prescribed multiplicative features. On the other hand, noting that for every prime p one has p = a 2 + nb if and only if ϕ(p) = a 2 + nb 2, where ϕ is the Euler function, it is also natural to consider the following variant of Question 1. 2

3 Question 2. Is it true that for any fixed n N there are infinitely many Carmichael numbers N such that ϕ(n) = a 2 + nb 2 for some a, b N? In the present note we show that Question 2 has an affirmative answer by establishing the following result. Theorem 1. Fix l N, and let P be an arbitrary set of primes for which the lower natural density # { primes p x : p P } δ P = lim inf x π(x) is positive. Then, there are infinitely many Carmichael numbers N with a totient of the form ϕ(n) = m l m, where m, m N and m is a nonempty product of primes from the set P. A quantitative version of this result can be obtained by combining Theorem 2 and Lemma 3 (see 4). For fixed n N, let S n and K n be defined as before, and let P be the set of primes in S n. Since P is precisely the set of primes p n that split completely in the number field K n, the Chebotarev density theorem [15] guarantees that δ P > 0. Applying Theorem 1 with l = 2, we see that there are infinitely many Carmichael numbers N with a totient of the form ϕ(n) = m 2 m, where m lies in S n, i.e., m = ã 2 +n b 2 for some ã, b N. Then ϕ(n) = a 2 + nb 2 with a = mã and b = m b. Therefore, as a consequence of Theorem 1 we have: Corollary 1. For any fixed n N there are infinitely many Carmichael numbers N such that ϕ(n) = a 2 + nb 2 for some a, b N. An essential feature of the construction of Carmichael numbers in [1] is the use of primes p for which p 1 has only small prime divisors, and the smoothness parameter E is taken to be large in order to produce as many Carmichael numbers as possible. In our proof of Theorem 1, which is based on that of [1, Theorem 4.1], we use primes q for which the shifted prime q 1 is only slightly smooth; that is, our smoothness parameter E is a positive real number which is close to zero. Although we produce far fewer Carmichael numbers than in [1], this approach allows us to construct Carmichael numbers whose totients have the desired form regardless of the choices of l and P. Although we have not done so here for the sake of brevity, it is worth noting that one can combine the methods of this paper with those of [7] to 3

4 produce Carmichael numbers N for which the multiplicative structures of N and ϕ(n) are controlled simultaneously. For example, for any fixed m, n N one can show that there are infinitely many Carmichael numbers N such that N = a 2 + mb 2 and ϕ(n) = c 2 + nd 2 for some a, b, c, d N. 2 Notation Below, the letters p and q always denote prime numbers. We denote by π(x) the number of primes p x and by π(x; d, a) the number of those primes in the arithmetic progression a mod d. Following [1], we denote by B the set of numbers B (0, 1) for which there exist constants x 1 (B) > 0 and D B N such that whenever x x 1 (B), gcd(a, d) = 1, and 1 d min{x B, y/x 1 B }, the inequality π(y; d, a) π(y) 2 ϕ(d) holds provided that d is not divisible by any member of D B (x), a set of at most D B integers, each of which exceeds log x. By [1, Section 2] it is known that (0, 5 ) B (see also [9, 10]). 12 In what follows, any constants implied by the symbols O and are absolute unless specified otherwise. We recall that the notations f g and f = O(g) are equivalent to the inequality f c g for some constant c > 0. 3 Slightly smooth shifted primes For any integer n > 1, let P(n) denote its largest prime divisor, and set P(1) = 0. Let P(x, y) = { p x : P(p 1) y }, and put π(x, y) = #P(x, y). The goal of this section is to show that P(x, x 1 ε ) contains all but O(ε π(x)) primes p x. The following statement is an easy application of the Selberg upper bound method in sieve theory; see the book [8] by Halberstam and Richert. 4

5 Lemma 1. Uniformly for m N we have # { p y : mp + 1 is prime } m y (4 + o(1)) ϕ(m) log 2 y (y ). (2) Proof. For an even number m the stated inequality follows immediately from [8, Theorem 3.12] in view of the crude bound ( ) 1 p 1 1 (p 1) 2 p 2 < p p 1 = m 2 ϕ(m). p>2 2<p m 2<p m When m is odd the result is trivial since the left side of (2) is 0 or 1. We also need the rough estimate ζ(3) = log y + O(1). (3) ϕ(n) 2π 4 n y Note that the more explicit estimate ( ζ(3) = log y + γ ϕ(n) 2π 4 p n y ) log p + O p 2 p + 1 ( ) log y is a well known result of Landau [12, Equation (10)]; see also [13]. Lemma 2. For every ε > 0 there is a number x 2 (ε) > 0 such that π(x, x 1 ε ) (1 10 ε) π(x) (x x 2 (ε)). Proof. We can assume that ε < 1. Observe that every prime q x which 10 does not lie in P(x, x 1 ε ) can be uniquely expressed in the form q = mp + 1, where p is a prime exceeding x 1 ε, and m x ε. Consequently, π(x) π(x, x 1 ε ) = m x ε # { p (x 1)/m : mp + 1 is prime }. Applying Lemma 1 with y = (x 1)/m, and taking into account the fact that log(x/m) (1 ε) log x 9 log x for all m 10 xε, we see that ( ) 400 x π(x) π(x, x 1 ε ) 81 + o(1) 1 log 2 (x ). x ϕ(m) m x ε Using (3) with y = x ε we have ( ) 7000 ζ(3) x π(x) π(x, x 1 ε ) ε + o(1) 9 π 4 log x y (x ). Noting that 7000 ζ(3)/(9 π 4 ) = < 10, we finish the proof. 5

6 4 Construction of Carmichael numbers Following [1], we denote by E the set of numbers E (0, 1) for which there exist constants x 3 (E), c(e) > 0 such that π(x, x 1 E ) c(e) π(x) (x x 3 (E)). (4) Friedlander [6] showed that any positive number less than 1 (2 e) 1 lies in E, and Erdős [5] has conjectured that E = (0, 1). Let B be defined as in 2, and let C(x) be the number of Carmichael numbers up to x. The main result of [1] is the following: Theorem. (Alford-Granville-Pomerance) For each E E and B B there is a number x 4 (E, B) such that C(x) x EB for all x x 4 (E, B). Now fix l N, and let P be an arbitrary set of primes for which the lower natural density π P (x) δ P = lim inf x π(x) is positive, where π P (x) = # { p x : p P }, and put π P (x, y) = # ( P(x, y) P ) = # { p x : P(p 1) y and p P }. Let E P be the set of numbers E (0, 1) for which there exist numbers x 3 (E, P), c(e, P) > 0 such that π P (x, x 1 E ) c(e, P) π(x) (x x 3 (E, P)). (5) Following the proof of [1, Theorem 4.1], for each E E P we put θ = (1 E) 1 and let L P be the product of the primes q (y θ / log y, y θ ] P for which q 1 is free of prime factors exceeding y. Using the bound (5) instead of (4) as needed, and replacing the group G = (Z/L Z) at [1, p. 718] with G = (Z/L P Z) (Z/l Z) +, the Alford-Granville-Pomerance construction is easily modified to produce Carmichael numbers N that are composed of primes of the form p = dk + 1 with d L P (and k fixed) and for which the number of prime factors ω(n) is a multiple of l. Writing N = (d j k + 1), ω(n) j=1 6

7 we see that ϕ(n) = k ω(n) d 1 d ω(n), and therefore, ϕ(n) = m l m with m = k ω(n)/l and m = d 1 d ω(n). In this way we derive the following quantitative version of Theorem 1. Theorem 2. Let P be a set of primes with δ P > 0. For each E E P and B B there is a number x 0 (E, B, P) such that for all x x 0 (E, B, P) there are at least x EB Carmichael numbers N x with a totient of the form ϕ(n) = m l m, where m, m N and m is a nonempty product of primes from the set P. Theorem 2 is not vacuous in light of the next result. Lemma 3. Let P be a set of primes with δ P > 0. Then, the interval (0, 1 10 δ P) is contained in E P. Proof. Let E (0, 1 10 δ P), and put θ = 5E δ P. Since θ < δ P we have π P (x) θ π(x) for all sufficiently large x (depending on E and P). Applying Lemma 2 with ε = E, we also have π(x, x 1 E ) (1 10E)π(x) if x x 2 (E). Hence, for some number x 3 (E, P) we have whenever x x 3 (E, P); that is, π(x) π(x, x 1 E ) + π P (x) π P (x, x 1 E ) (1 10E + θ)π(x) π P (x, x 1 E ) π P (x, x 1 E ) (θ 10E)π(x) = ( 1 2 δ P 5E)π(x). Since 1 2 δ P 5E > 0, it follows that E E P. References [1] W. Alford, A. Granville, and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), [2] W. Banks, Carmichael numbers with a square totient, Canad. Math. Bull. 52 (1) (2009), no. 1, 3 8. [3] W. Banks and C. Pomerance, On Carmichael numbers in arithmetic progressions, J. Aust. Math. Soc. 88 (2010),

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