Arithmetic progressions of Carmichael numbers in a reduced residue class (Q2043469)

The author shows that, under the assumption of prime \(k\)-tuple conjecture, there are arbitrarily long arithmetic progressions of Carmichael numbers situated in the residue class \(a\) mod \(q\) where \(a\) and \(q\) are coprime. Carmichael numbers are composite numbers \(n\) for which \(x^{n} - x\) divides \(n\) for each natural \(x\). \par One of the main tools of the proof is Lemma 2 which provides sufficient conditions for a number to be Carmichael. The prime \(k\)-tuple conjecture is invoked to ensure the criteria involving primes of this lemma are met in its application.\par Towards the end of the paper there is a minor technical gap in the proof as the validity of (15) becomes unclear if \(p | B\), or more precisely, if \( p|b_{i}\) and \(z = b_{j}\) for \(i \neq j\). The gap is only of technical nature though, as it would be possible to guarantee (15) in this case by imposing some extra conditions on \(b\) when \(b\) is introduced.