Carmichael numbers and the sieve

Transcription

1 June 9, 2015

2 Dedicated to Carl Pomerance in honor of his 70th birthday

3 Carmichael numbers Fermat s little theorem asserts that for any prime n one has a n a (mod n) (a Z)

4 Carmichael numbers Fermat s little theorem asserts that for any prime n one has a n a (mod n) (a Z) A composite number n N with the same property is called a Carmichael number.

5 Carmichael numbers If n is composite, the following are equivalent: n is a Carmichael number

6 Carmichael numbers If n is composite, the following are equivalent: n is a Carmichael number λ(n) n 1, where λ(n) is the Carmichael function

7 Carmichael numbers If n is composite, the following are equivalent: n is a Carmichael number λ(n) n 1, where λ(n) is the Carmichael function Korselt s criterion: n is squarefree, and p 1 n 1 for every prime p n

8 Carmichael numbers If n is composite, the following are equivalent: n is a Carmichael number λ(n) n 1, where λ(n) is the Carmichael function Korselt s criterion: n is squarefree, and p 1 n 1 for every prime p n Examples 561 (found by Carmichael)

9 Carmichael numbers If n is composite, the following are equivalent: n is a Carmichael number λ(n) n 1, where λ(n) is the Carmichael function Korselt s criterion: n is squarefree, and p 1 n 1 for every prime p n Examples 561 (found by Carmichael) 1729 (taxicab number)

10 Carmichael numbers If n is composite, the following are equivalent: n is a Carmichael number λ(n) n 1, where λ(n) is the Carmichael function Korselt s criterion: n is squarefree, and p 1 n 1 for every prime p n Examples 561 (found by Carmichael) 1729 (taxicab number) (largest prime factor = 41)

11 Carmichael numbers If n is composite, the following are equivalent: n is a Carmichael number λ(n) n 1, where λ(n) is the Carmichael function Korselt s criterion: n is squarefree, and p 1 n 1 for every prime p n Examples 561 (found by Carmichael) 1729 (taxicab number) (largest prime factor = 41) (smallest prime factor = 41)

12 Carmichael numbers how many? Let C(x) = # { n x : n is a Carmichael number }

13 Carmichael numbers how many? Let C(x) = # { n x : n is a Carmichael number } Erdős conjectured that C(x) = x 1+o(1) as x.

14 Carmichael numbers how many? Let C(x) = # { n x : n is a Carmichael number } Erdős conjectured that C(x) = x 1+o(1) as x. Pomerance, Selfridge and Wagstaff showed that ( C(x) x exp (1 + o(1)) log x log 3 x ) log 2 x

15 Carmichael numbers how many? Let C(x) = # { n x : n is a Carmichael number } Erdős conjectured that C(x) = x 1+o(1) as x. Pomerance, Selfridge and Wagstaff showed that ( C(x) x exp (1 + o(1)) log x log 3 x ) log 2 x and they conjectured that ( C(x) x exp (2 + o(1)) log x log 3 x ) log 2 x

16 Infinitely many Carmichael numbers The existence of infinitely many Carmichael numbers was first proved in 1994 by Alford, Granville, and Pomerance.

17 Infinitely many Carmichael numbers The existence of infinitely many Carmichael numbers was first proved in 1994 by Alford, Granville, and Pomerance. They established the unconditional lower bound C(x) ε x β ε where 5 β= = > 7 2 e ( )

18 Designer Carmichael numbers Since primes and Carmichael numbers are linked by the relation a n a (mod n) (a Z) it is natural to ask whether certain properties of primes also hold for Carmichael numbers.

19 Designer Carmichael numbers Since primes and Carmichael numbers are linked by the relation a n a (mod n) (a Z) it is natural to ask whether certain properties of primes also hold for Carmichael numbers. This leads to the study of Carmichael numbers with prescribed arithmetic features: designer Carmichael numbers.

20 Carmichael + Dirichlet For example, is there an analogue of Dirichlet s theorem for Carmichael numbers?

21 Carmichael + Dirichlet For example, is there an analogue of Dirichlet s theorem for Carmichael numbers? Theorem (Matomäki; Wright) There are infinitely many Carmichael numbers in any arithmetic progression a mod m with gcd(a, m) = 1.

22 Carmichael + Dirichlet For example, is there an analogue of Dirichlet s theorem for Carmichael numbers? Theorem (Matomäki; Wright) There are infinitely many Carmichael numbers in any arithmetic progression a mod m with gcd(a, m) = 1. Formulated as a conjecture by B. & Pomerance (2010); they gave a conditional proof

23 Carmichael + Dirichlet For example, is there an analogue of Dirichlet s theorem for Carmichael numbers? Theorem (Matomäki; Wright) There are infinitely many Carmichael numbers in any arithmetic progression a mod m with gcd(a, m) = 1. Formulated as a conjecture by B. & Pomerance (2010); they gave a conditional proof Matomäki (2013) proved the special case in which a is a quadratic residue modulo m

24 Carmichael + Dirichlet For example, is there an analogue of Dirichlet s theorem for Carmichael numbers? Theorem (Matomäki; Wright) There are infinitely many Carmichael numbers in any arithmetic progression a mod m with gcd(a, m) = 1. Formulated as a conjecture by B. & Pomerance (2010); they gave a conditional proof Matomäki (2013) proved the special case in which a is a quadratic residue modulo m Wright (2013) proved the theorem in full generality

25 Carmichael + Dirichlet For example, is there an analogue of Dirichlet s theorem for Carmichael numbers? Theorem (Matomäki; Wright) There are infinitely many Carmichael numbers in any arithmetic progression a mod m with gcd(a, m) = 1. Formulated as a conjecture by B. & Pomerance (2010); they gave a conditional proof Matomäki (2013) proved the special case in which a is a quadratic residue modulo m Wright (2013) proved the theorem in full generality Open: Can one remove the restriction gcd(a, m) = 1? For example, are there infinitely many Carmichael numbers divisible by three?

26 Carmichael + Schinzel Let φ be the Euler function.

27 Carmichael + Schinzel Let φ be the Euler function. For any k 2 it is unknown whether there are infinitely many primes p such that p = m k + 1 ( m N)

28 Carmichael + Schinzel Let φ be the Euler function. For any k 2 it is unknown whether there are infinitely many primes p such that p = m k + 1 ( m N) Equivalently, φ(p) is a perfect k th power.

29 Carmichael + Schinzel Let φ be the Euler function. For any k 2 it is unknown whether there are infinitely many primes p such that p = m k + 1 ( m N) Equivalently, φ(p) is a perfect k th power. Theorem (B.) For every integer k 2 there are infinitely many Carmichael numbers n for which φ(n) is a perfect k th power.

30 Carmichael + Chebotarev Theorem (B., Güloğlu & Yeager) Let K /Q be a finite Galois extension, and let C be a fixed conjugacy class in Gal(K /Q). There are infinitely many Carmichael numbers composed of primes for which the associated class of Frobenius automorphisms is the class C.

31 Carmichael + Chebotarev Theorem (B., Güloğlu & Yeager) Let K /Q be a finite Galois extension, and let C be a fixed conjugacy class in Gal(K /Q). There are infinitely many Carmichael numbers composed of primes for which the associated class of Frobenius automorphisms is the class C. Corollary For every fixed integer N 1 there are infinitely many Carmichael numbers of the form a 2 + N b 2 with a, b Z.

32 Carmichael + Beatty Theorem (B. & Yeager) Let α, β R with α > 1, and suppose that α is irrational and of finite type. Then, there are infinitely many Carmichael numbers composed of primes of the form p = αn + β with n N.

33 Carmichael + Piatetski-Shapiro Theorem (Baker, B., Brudern, Shparlinski & Weingartner) For every fixed c ( 1, ) there are infinitely many Carmichael numbers composed of primes of the form p = n c with n N.

34 No time like the present

35 Theorem (B. & Freiberg) There are infinitely many Carmichael numbers whose prime factors all have the form p = 1 + a 2 + b 2 with a, b Z. Moreover, there is a positive constant C for which the number of such Carmichael numbers up to x is at least x C provided that x is sufficiently large in terms of C.

36 Theorem (B. & Freiberg) There are infinitely many Carmichael numbers whose prime factors all have the form p = 1 + a 2 + b 2 with a, b Z. Moreover, there is a positive constant C for which the number of such Carmichael numbers up to x is at least x C provided that x is sufficiently large in terms of C. Up to 10 8, there are seven such Carmichael numbers, namely 561, , , , , , whereas there are 255 ordinary Carmichael numbers 10 8.

37 Elements of the AGP proof P is the set of primes.

38 Elements of the AGP proof P is the set of primes. E, B, ε are small positive parameters, with 0 < ε < EB.

39 Elements of the AGP proof P is the set of primes. E, B, ε are small positive parameters, with 0 < ε < EB. θ = (1 E) 1 and δ = εθ/(4b).

40 Elements of the AGP proof P is the set of primes. E, B, ε are small positive parameters, with 0 < ε < EB. θ = (1 E) 1 and δ = εθ/(4b). y 2 is a large real parameter, and x = exp(y 1+δ ).

41 Elements of the AGP proof P is the set of primes. E, B, ε are small positive parameters, with 0 < ε < EB. θ = (1 E) 1 and δ = εθ/(4b). y 2 is a large real parameter, and x = exp(y 1+δ ). Put Q = {q P (y θ / log y, y θ ] : P + (q 1) y}

42 Elements of the AGP proof P is the set of primes. E, B, ε are small positive parameters, with 0 < ε < EB. θ = (1 E) 1 and δ = εθ/(4b). y 2 is a large real parameter, and x = exp(y 1+δ ). Put Q = {q P (y θ / log y, y θ ] : P + (q 1) y} L = q Q q

43 Elements of the AGP proof P is the set of primes. E, B, ε are small positive parameters, with 0 < ε < EB. θ = (1 E) 1 and δ = εθ/(4b). y 2 is a large real parameter, and x = exp(y 1+δ ). Put Q = {q P (y θ / log y, y θ ] : P + (q 1) y} L = q Q q G = (Z/LZ)

44 Elements of the AGP proof The length n(g) of the longest sequence in G for which no nonempty subsequence has product the identity satisfies n(g) e 3θy

45 Elements of the AGP proof The length n(g) of the longest sequence in G for which no nonempty subsequence has product the identity satisfies n(g) e 3θy In particular, any set of primes whose cardinality greatly exceeds e 3θy has many subsets S for which p 1 mod L p S

46 Elements of the AGP proof There exists k x 1 B with gcd(k, L) = 1 such that d L dk+1 x 1 P (dk + 1) #{d L : d x B } log x

47 Elements of the AGP proof There exists k x 1 B with gcd(k, L) = 1 such that d L dk+1 x 1 P (dk + 1) #{d L : d x B } log x Thus, one has available a large set P of primes of the form p = dk + 1 with d L

48 Elements of the AGP proof The set P has many subsets S for which n S = p S p 1 mod L

49 Elements of the AGP proof The set P has many subsets S for which n S = p S p 1 mod L Since gcd(k, L) = 1 and p 1 mod k for each p n S, we have n S 1 mod Lk

50 Elements of the AGP proof The set P has many subsets S for which n S = p S p 1 mod L Since gcd(k, L) = 1 and p 1 mod k for each p n S, we have n S 1 mod Lk For each p n S we have p = dk + 1 with d L, and p 1 = dk Lk n S 1

51 Elements of the AGP proof The set P has many subsets S for which n S = p S p 1 mod L Since gcd(k, L) = 1 and p 1 mod k for each p n S, we have n S 1 mod Lk For each p n S we have p = dk + 1 with d L, and p 1 = dk Lk n S 1 Thus n S is a Carmichael number by Korselt s criterion.

52 Theorem (B. & Freiberg) There are infinitely many Carmichael numbers whose prime factors all have the form p = 1 + a 2 + b 2 with a, b Z.

53 New elements Let P 4,1 denote the set of primes congruent to 1 mod 4. Let B = {1, 5, 13, 17, 25,...} be the multiplicative semigroup generated by P 4,1.

54 New elements Let P 4,1 denote the set of primes congruent to 1 mod 4. Let B = {1, 5, 13, 17, 25,...} be the multiplicative semigroup generated by P 4,1. Put P = P (1 + 2B) Then P is a set of primes of the form p = 1 + a 2 + b 2.

55 New elements Let P 4,1 denote the set of primes congruent to 1 mod 4. Let B = {1, 5, 13, 17, 25,...} be the multiplicative semigroup generated by P 4,1. Put P = P (1 + 2B) Then P is a set of primes of the form p = 1 + a 2 + b 2. To prove the theorem, we show that P is well-distributed over arithmetic progressions with large, smooth moduli. Then, following the AGP method, we construct Carmichael numbers from the primes in P.

56 New elements We work with the set of primes P 4,1.

57 New elements We work with the set of primes P 4,1. E, B, ε are small positive parameters, with 0 < ε < EB.

58 New elements We work with the set of primes P 4,1. E, B, ε are small positive parameters, with 0 < ε < EB. θ = (1 E) 1 and δ = εθ/(4b).

59 New elements We work with the set of primes P 4,1. E, B, ε are small positive parameters, with 0 < ε < EB. θ = (1 E) 1 and δ = εθ/(4b). y 2 is a large real parameter, and x = exp(y 1+δ ).

60 New elements We work with the set of primes P 4,1. E, B, ε are small positive parameters, with 0 < ε < EB. θ = (1 E) 1 and δ = εθ/(4b). y 2 is a large real parameter, and x = exp(y 1+δ ). For a certain integer l x B we put Q l = {q P 4,1 (y θ / log y, y θ ] : P + (q 1) y, q l}

61 New elements We work with the set of primes P 4,1. E, B, ε are small positive parameters, with 0 < ε < EB. θ = (1 E) 1 and δ = εθ/(4b). y 2 is a large real parameter, and x = exp(y 1+δ ). For a certain integer l x B we put Q l = {q P 4,1 (y θ / log y, y θ ] : P + (q 1) y, q l} L l = q (note that d L l = d B) q Q l

62 New elements We work with the set of primes P 4,1. E, B, ε are small positive parameters, with 0 < ε < EB. θ = (1 E) 1 and δ = εθ/(4b). y 2 is a large real parameter, and x = exp(y 1+δ ). For a certain integer l x B we put Q l = {q P 4,1 (y θ / log y, y θ ] : P + (q 1) y, q l} L l = q (note that d L l = d B) q Q l G l = (Z/L l Z)

63 New elements Using a result of Matomäki, we again derive the bound n(g l ) e 3θy and we show that, for carefully chosen l, there exists k x 1 B with gcd(k, L l ) = 1 and d L l 2dk+1 x 1 B (k) 1 P (2dk + 1) #{d L l : d x B } (log x) 3/2

64 New elements Thus, for some integer k B with k x 1 B and gcd(k, L l ) = 1, one has available a large set P of primes of the form p = 2dk + 1 with d L l.

65 New elements Thus, for some integer k B with k x 1 B and gcd(k, L l ) = 1, one has available a large set P of primes of the form p = 2dk + 1 with d L l. The set P has many subsets S for which n S = p S p 1 mod L l and each n S is a Carmichael number as before.

66 New elements Thus, for some integer k B with k x 1 B and gcd(k, L l ) = 1, one has available a large set P of primes of the form p = 2dk + 1 with d L l. The set P has many subsets S for which n S = p S p 1 mod L l and each n S is a Carmichael number as before. Since P 1 + 2B, all such Carmichael numbers are composed of primes of the form p = 1 + a 2 + b 2.

67 Semilinear sieve Let P 4,3 denote the set of primes congruent to 3 mod 4.

68 Semilinear sieve Let P 4,3 denote the set of primes congruent to 3 mod 4. For a given divisor d L l we need to carefully estimate π (x; d, 1) = { p x : p 1 + 2B, p 1 (d) } To do so, we apply the semilinear sieve to the set A = { (p 1)/2 : p x, p 1 (d), p 3 (8) }, sieving A by the primes in P 4,3.

69 Semilinear sieve Let P 4,3 denote the set of primes congruent to 3 mod 4. For a given divisor d L l we need to carefully estimate π (x; d, 1) = { p x : p 1 + 2B, p 1 (d) } To do so, we apply the semilinear sieve to the set A = { (p 1)/2 : p x, p 1 (d), p 3 (8) }, sieving A by the primes in P 4,3. Complication: we have to do this for all d L l with d x B, thus we have to consider the possible effect of exceptional zeros of certain Dirichlet L-functions.

70 Exceptional zeros Lemma (B., Freiberg & Maynard) Let T 3. Among all primitive Dirichlet characters χ mod l with modulus l T and P + (l) T 1/ log 2 T, there is at most one for which the associated L-function L(s, χ) has a zero in the region R(s) > 1 c log 2 T / log T, I(s) exp ( log T / log2 T ), where c > 0 is a certain absolute constant. If such a character χ mod l exists, then χ is real and L(s, χ) has just one such zero, which is real and simple, and we put l(t ) = l; otherwise, we set l(t ) = 1.

71 Modified Bombieri-Vinogradov theorem Our estimate of π (x; d, 1) = { p x : p 1 + 2B, p 1 (d) } via the semilinear sieve relies on the following result. 1 Theorem (B. & Freiberg) Fix B > 0. Let x 3 1/B, and let d [1, x B ] be a squarefree integer such that P + (d) x B/ log 2 x and gcd(d, l) = 1, where l = l(x B ) as in the Lemma. If B = B(A, δ) is sufficiently small in terms of any fixed A > 0 and δ (0, 1/2), then max m gcd(a,dm)=1 x/x δ π(x) π(x; dm, a) φ(dm) δ,a x φ(d)(log x) A. 1 This extends Theorem 4.1 in B., Freiberg and Maynard, On limit points of the sequence of normalized primes gaps (arxiv: ).

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