aa r X i v : . [ m a t h . N T ] J a n A NOTE ON CARMICHAEL NUMBERSIN RESIDUE CLASSES
CARL POMERANCE
Abstract.
Improving on some recent results of Matom¨aki and ofWright, we show that the number of Carmichael numbers to X ina coprime residue class exceeds X / (6 log log log X ) for all sufficientlylarge X depending on the modulus of the residue class. In memory of Ron Graham (1935–2020)and Richard Guy (1916–2020) Introduction
The “little theorem” of Fermat asserts that when p is a prime num-ber, we have b p ≡ b (mod p ) for all integers b . Given two integers b, p with p > b >
0, it is computationally easy to check this congruence,taking O (log p ) arithmetic operations in Z /p Z . So, if the congruence ischecked and we find that b p b (mod p ) we immediately deduce that p is composite. Unfortunately there are easily found examples where n is composite and the Fermat congruence holds for a particular b . Forexample it always holds when b = 1. It holds when b = 2 and n = 341,and another example is b = 3, n = 91. There are even compositenumbers n where b n ≡ b (mod n ) holds for all b , the least examplebeing n = 561. These are the Carmichael numbers , named after R. D.Carmichael who published the first few examples in 1910, see [4]. (In-terestingly, ˇSimerka published the first few examples 25 years earlier,see [8].)We now know that there are infinitely many Carmichael numbers,see [1], the number of them at most X exceeding X c for a fixed c > X sufficiently large.A natural question is if a given residue class contains infinitely manyCarmichael numbers. After work of Matom¨aki [7] and Wright [9], wenow know there are infinitely many in a coprime residue class. More Date : January 26, 2021.2000
Mathematics Subject Classification.
Key words and phrases.
Carmichael number. precisely, we have the following two theorems. Let C a,M ( X ) = { n ≤ X : n is a Carmichael number , n ≡ a (mod M ) } . Theorem M (Matom¨aki).
Suppose that a, M are positive coprimeintegers and that a is a quadratic residue mod M . Then C a,M ≥ X / for X sufficiently large depending on the choice of M . Theorem W (Wright).
Suppose that a, M are positive coprime in-tegers. There are positive numbers K M , X M depending on the choice of M such that C a,M ( X ) ≥ X K M / (log log log X ) for all X ≥ X M . Thus, Wright was able to remove the quadratic residue conditionin Matom¨aki’s theorem but at the cost of lowering the count to anexpression that is of the form X o (1) . The main contribution of thisnote is to somewhat strengthen Wright’s bound. Theorem 1.
Suppose that a, M are positive coprime integers. Then C a,M ( X ) ≥ X / (6 log log log X ) for all sufficiently large X depending on thechoice of M . That is, we reduce the power of log log log X to the first power and weremove the dependence on M in the bound, though there still remainsthe condition that X must be sufficiently large depending on M . (It’sclear though that such a condition is necessary since if M > X and a = 1, then there are no Carmichael numbers n ≤ X in the residueclass a mod M .)Our proof largely follows Wright’s proof of Theorem W, but with afew differences.Unlike with primes, it is conceivable that a non-coprime residue classcontains infinitely many Carmichael numbers, e.g., there may be in-finitely many that are divisible by 3. This is unknown, but seemslikely. In fact, it is conjectured in [3] that if gcd( g, ϕ ( g )) = 1, where g = gcd( a, M ), then there are infinitely many Carmichael numbers n ≡ a (mod M ). Though we don’t know this for any example with g >
1, the old heuristic of Erd˝os [5] suggests that C a,M ( X ) ≥ X − o (1) as X → ∞ . 2. Proof of Theorem 1
There is an elementary and easily-proved criterion for Carmichaelnumbers: a composite number n is one if and only if it is squarefreeand p − | n − p dividing n . This is due to Korselt,and perhaps others, and is over a century old. In our construction wewill have a number L composed of many primes, a number k coprimeto L that is not much larger than L , and primes p of the form dk + 1 NOTE ON CARMICHAEL NUMBERS IN RESIDUE CLASSES 3 where d | L . We will show there are many n ≡ a (mod M ) that aresquarefree products of the p ’s and are 1 (mod kL ). Such n , if theyinvolve more than a single p , will satisfy Korselt’s criterion and so aretherefore Carmichael numbers.We may assume that M ≥
2. Let µ = ϕ (4 M ), so that 4 | µ . Let y be an independent variable; our other quantities will depend on it. Fora positive integer n let P ( n ) denote the largest prime factor of n (with P (1) = 1), and let ω ( n ) denote the number of distinct prime factorsof n .Let Q = { q prime : y < q ≤ y log y, q ≡ − µ ) , P ( q − ≤ y } . If q ≤ y log y and P ( q − > y , then q is of the form mr + 1, where m < log y and r is prime. By Brun’s sieve (see [6, (6.1)]), the numberof such primes q is at most X m< log y X r prime mr ≤ y log yrm +1 prime ≪ X m< log y y log yϕ ( m ) log y ≪ y log log y. Also, the number of primes q ≤ y log y with q ≡ − µ ) is ∼ ϕ ( µ ) y log y as y → ∞ by the prime number theorem for residue classes.We conclude that(1) Q ∼ ϕ ( µ ) y log y and Y q ∈Q q = exp (cid:16) o (1) ϕ ( µ ) y log y (cid:17) , y → ∞ . We also record that(2) X q ∈Q q = o (1) , y → ∞ , since this holds for all of the primes in the interval ( y, y log y ].Fix 0 < B < /
12; we shall choose a numerical value for B near to5 /
12 at the end of the argument. Let(3) x = M /B Y q ∈Q q /B . It follows from [1, (0.3)] that there is an absolute constant D and a set D ( x ) of at most D integers greater than log x , such that if n ≤ x B , n isnot divisible by any member of D ( x ), b is coprime to n , and z ≥ nx − B ,then the number of primes p ≤ z with p ≡ b (mod n ) is > π ( z ) /ϕ ( n ).For each number in D ( x ) we choose a prime factor and remove thisprime from Q if it happens to be there. Let L be the product of the CARL POMERANCE primes in the remaining set Q , so that L is not divisible by any memberof D ( x ), and Q satisfies (1) and (2). In particular,(4) L = exp (cid:16) o (1) ϕ ( µ ) y log y (cid:17) , ω ( L ) ∼ ϕ ( µ ) y log y, and X q | L q = o (1) as y → ∞ . In addition, we have
M L ≤ x B .For each d | L and each quadratic residue b (mod L/d ) we considerthe primes • p ≤ dx − B , • p ≡ a (mod M ), • p ≡ d ), • p ≡ b (mod L/d ).Since M is coprime to L , the congruences may be glued to a singlecongruence modulo M L , and the number of such primes p is > π ( dx − B )2 ϕ ( M L ) > dx − B ϕ ( M L ) log x for y sufficiently large.We add these inequalities over the various choices of b , the numberof which is ϕ ( L/d ) / ω ( L/d ) , so the number of primes p correspondingto d | L is > dx − B ω ( d ) · ω ( L ) ϕ ( M d ) log x .
We wish to impose an additional restriction on these primes p , namelythat gcd(( p − /d, L ) = 1. For a given prime q | L the number of primes p just counted and for which q | ( p − /d is, via the Brun–Titchmarshinequality, ≪ dx − B ω ( d ) ω ( L ) qϕ ( M d ) log( x/ ( qM L )) ≪ dx − B ω ( d ) ω ( L ) qϕ ( M d ) log x .
Summing this over all q | L and using that P q | L /q = o (1), theseprimes p are seen to be negligible. It follows that for y sufficientlylarge, there are > dx − B ω ( d ) ω ( L )+2 ϕ ( M d ) log x > x − B ω ( d ) ω ( L )+2 ϕ ( M ) log x primes p ≤ dx − B with p ≡ d ), gcd(( p − /d, L ) = 1, p ≡ a (mod M ), and p is a quadratic residue (mod L ) (noting that 1 (mod d )is a quadratic residue (mod d )). NOTE ON CARMICHAEL NUMBERS IN RESIDUE CLASSES 5
For each pair p, d as above, we map it to ( p − /d which is an integer ≤ x − B coprime to L . The number of pairs p, d is > x − B ω ( L )+2 ϕ ( M ) log x X d | L ω ( d ) = x − B ω ( L ) ω ( L )+2 ϕ ( M ) log x . We conclude that there is a number k ≤ x − B coprime to L which has > (3 / ω ( L ) / (4 ϕ ( M ) log x ) representations as ( p − /d . Let P be theset of primes p = dk + 1 that arise in this way. Then(5) P > (3 / ω ( L ) ϕ ( M ) log x . For a finite abelian group G , let n ( G ) denote Davenport’s constant,the least number such that in any sequence of group elements of length n ( G ) there is a non-empty subsequence with product the group identity.It is easy to see that n ( G ) ≥ λ ( G ) (the universal exponent for G ), andin general it is not much larger: n ( G ) ≤ λ ( G )(1 + log( G )). Thisresult is essentially due to van Emde Boas–Kruyswijk and Meshulam,see [1].Let G be the subgroup of ( Z /kM L Z ) ∗ of residues ≡ k ). Wehave G ≤ M L . Also, λ ( G ) ≤ M λ ( L ). (Note that, as usual, wedenote λ (( Z /L Z ) ∗ ) by λ ( L ). It is the lcm of q − q | L ,using that L is squarefree.) Each prime dividing λ ( L ) is at most y andeach prime power dividing λ ( L ) is at most y log y , so that λ ( L ) ≤ ( y log y ) π ( y ) . Thus, for large y , using (4),(6) n ( G ) ≤ M ( y log y ) π ( y ) log( M L ) ≤ e y . For a sequence A of elements in a finite abelian group G , let A ∗ denote the set of nonempty subsequence products of A . In Baker–Schmidt [2, Proposition 1] it is shown that there is a number s ( G ) suchthat if A ≥ s ( G ), then G has a nontrivial subgroup H such that( A ∩ H ) ∗ = H . Further, s ( G ) ≤ λ ( G ) Ω( G ) log(3 λ ( G )Ω( G )) , where Ω( m ) is the number of prime factors of m counted with multi-plicity. Thus, with G the group considered above, we have s ( G ) ≤ e y for y sufficiently large.It is this theorem that Matom¨aki and Wright use in their papers onCarmichael numbers. The role of the sequence A is played by P , the CARL POMERANCE set of primes constructed above of the form dk + 1 where d | L . So,if P > s ( G ) we are guaranteed that every member of a nontrivialsubgroup H of G is represented by a subset product of P ∩ H .We don’t know precisely what this subgroup H is, but we do knowthat it is nontrivial and that it is generated by members of P . Well,suppose p is in P ∩ H . Then p m ∈ H for every integer m . Notethat by construction, gcd( λ ( L ) / , ϕ ( M )) = 1, so there is an integer m ≡ ϕ ( M )) and m ≡ λ ( L ) / p is aquadratic residue (mod L ), it follows that p λ ( L ) / ≡ L ). Thus, p m ≡ L ) and p m ≡ a (mod M ) (since m ≡ ϕ ( M )).Thus, there is a subsequence product n of P that is 1 (mod kL ) and a (mod M ). (Note that every member of G is 1 (mod k .) Further, n issquarefree and for each prime factor p of n we have p − | kL . Since n ≡ kL ) we have p − | n −
1. Thus, n ≡ a (mod M ) is eithera prime or a Carmichael number.We actually have many subsequence products n of P that satisfythese conditions, and P has at most one element that is 1 (mod L ),so we do not need to worry about the case that n is prime. We let t = ⌈ e y ⌉ , so that t ≥ s ( G ). As shown in [7], [9], the Baker–Schmidtresult implies that P has at least N := (cid:18) P − n ( G ) t − n ( G ) (cid:19).(cid:18) P − n ( G ) n ( G ) (cid:19) subsequence products n of length at most t which are Carmichael num-bers in the residue class a (mod M ). Thus, N > (cid:18)
P − n ( G ) t − n ( G ) (cid:19) t − n ( G ) ( P ) − n ( G ) > (cid:18) P t (cid:19) t − n ( G ) ( P ) − n ( G ) > ( P ) t − n ( G ) t − t . Let X = x t . Since each p ∈ P has p ≤ x , it follows that all of theCarmichael numbers constructed above are at most X . Using (1), (3),and (6), we have X = exp (cid:16) /B + o (1) ϕ ( µ ) ty log y (cid:17) , NOTE ON CARMICHAEL NUMBERS IN RESIDUE CLASSES 7 and using (5) and (4) gives N ≥ exp (cid:16) log(3 /
2) + o (1) ϕ ( µ ) ty log y − t log t (cid:17) = exp (cid:16) log(3 /
2) + o (1) ϕ ( µ ) ty log y (cid:17) . Thus, N ≥ X ( B log(3 / o (1)) / log y . Now,log X ∼ Bϕ ( µ ) ty log y, G so that using t = ⌈ e y ⌉ ,log log X = 3 y + O (log y ) , log log log X = log y + O (1) . We thus have N ≥ X ( B log(3 / o (1)) / log log log X . The number B < / /
12 and since (5 /
12) log(3 / > / References [1] W. R. Alford, A. Granville, and C. Pomerance, There are infinitely manyCarmichael numbers,
Ann. of Math. (2) (1994), 703–722.[2] R. C. Baker and W. M. Schmidt, Diophantine problems in variables restrictedto the values 0 and 1,
J. Number Theory (1980), 460–486.[3] W. D. Banks and C. Pomerance, On Carmichael numbers in arithmetic pro-gressions, J. Australian Math. Soc. (2010), 313–321.[4] R. D. Carmichael, A new number-theoretic function, Bull. Amer. Math. Soc. (1910), 232–238.[5] P. Erd˝os, On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen (1956), 201–206.[6] H. Halberstam and H.-E. Richert, Sieve Methods, London Mathematical So-ciety Monographs, No. 4. Academic Press [A subsidiary of Harcourt BraceJovanovich, Publishers], London – New York, 1974.[7] K. Matom¨aki, Carmichael numbers in arithmetic progressions, J. AustralianMath. Soc. (2013), 268–275.[8] V. ˇSimerka, Zbytky z arithmetick´e posloupnosti. (Czech) [On reminders fromarithmetical sequence]. ˇCasopis pro pˇestov´an´ı mathematiky a fysiky, (1885),221–225.[9] T. Wright, Infinitely many Carmichael numbers in arithmetic progressions, Bull. London Math. Soc. (2013), 943–952. Mathematics Department, Dartmouth College, Hanover, NH 03784
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