Carmichael Numbers in Arithmetic progressions

Abstract

We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x 1/5 when x is large enough (depending on m).

Sierpiński and Carmichael numbers

“…Using recent results of Matomäki [20] and Wright [29] coupled with an extensive computer search, we prove the following result in §3.…”

“…In 1994, Alford, Granville and Pomerance [1] proved the existence of infinitely many Carmichael numbers. Since prime numbers and Carmichael numbers share the property (1), it is natural to ask whether certain results for primes can also be established for Carmichael numbers; see, for example, [2,3,5,14,20,29] and the references contained therein. Our work in this paper originated with the question as to whether there exist Sierpiński numbers k such that 2 n k + 1 is not a Carmichael number for any n ∈ N. Since there are many Sierpiński numbers and only a few Carmichael numbers, it is natural to expect there are many such k. However, because the parameter n can take any positive integer value, the problem is both difficult and interesting.…”

There are infinitely many elliptic Carmichael numbers

“…Here, we introduce a theorem of Baker and Schmidt [, Proposition 1] that was reinterpreted in terms of pseudoprimes by Matomäki . To begin, for an abelian group , is defined to be the smallest number such that a collection of at least elements must contain some subset whose product is the identity.…”

“…Much of the proof follows a similar outline to that of the previous paper by the current author , which is itself an alteration of the methods of [, ]. To begin, we create an with many prime factors , where all of the will be −1 mod 4.…”

“…We then try to find an integer which is relatively prime to such that the set is fairly large; note that is quadratic nonresidue for any of the chosen (since (mod 4)), so the two requirements on do not contradict one another. Having found a which fits our required criteria, we use a combinatorial theorem first used in a paper of Baker and Schmidt and applied to Carmichael numbers by Matomäki to prove a sufficiently large density of our chosen pseudoprimes.…”