By Fermat’s little theorem \(b^p \equiv b \pmod{p}\) for every prime \(p\) and every integer \(b\). Composite numbers \(n\) that satisfy \(b^n \equiv b \pmod{n}\) for every integer \(b\) are called Carmichael numbers.
In a groundbreaking work, W. R. Alford et al. [Ann. Math. (2) 139, No. 3, 703–722 (1994; Zbl 0816.11005)] proved that there are infinitely many Carmichael numbers. Nowadays it is known by work of G. Harman [Int. J. Number Theory 4, No. 2, 241–248 (2008; Zbl 1221.11194)] that, for any sufficiently large \(x\), there are more than \(x^{1/3}\) Carmichael numbers up to \(x\).
Later the reviewer [J. Aust. Math. Soc. 94, No. 2, 268–275 (2013; Zbl 1368.11106)] showed that whenever \(M\) and \(a\) are co-prime integers, \(a\) is a quadratic residue \(\pmod{M}\), and \(X\) is sufficiently large in terms of \(M\), there exist at least \(X^{1/5}\) Carmichael numbers \(n \leq X\) such that \(n \equiv a \pmod{M}\).
Matomäki’s result was extended to the case when \(a\) is quadratic non-residue by T. Wright [Bull. Lond. Math. Soc. 45, No. 5, 943–952 (2013; Zbl 1319.11065)] but with a weaker lower bound \(X^{K_M/(\log \log \log X)^2}\) for some constant \(K_M\) depending on \(M\).
The purpose of the paper under review is to improve Wright’s lower bound to \(X^{1/(6 \log \log \log X)}\). The proof largely follows Wright’s proof but introduces some refinements.