Factors of Carmichael numbers and a weak \(k\)-tuples conjecture (Q2811987)

In [Ann. Math. (2) 139, No. 3, 703--722 (1994; Zbl 0816.11005)], \textit{W. R. Alford} et al. proved the long standing conjecture that there are infinitely many Carmichael numbers (that is composite integers \(n\) such that \(a^n\equiv a\mod n\) for all integers \(a\)). There remain many other conjectures about these numbers and the current paper concerns the number \(C_R(X)\) of Carmichael numbers up to \(x\) with exactly \(R\) prime factors. Let \(D= \{a_1z+ b_1, a_2z+ b_2,\dots, a_kz+ b_k\}\) denote a set of \(k\) linear forms with each \(a_i>0\) and such that for each prime \(p\) there corresponds \(z\) with NEWLINE\[NEWLINEp\nmid \prod^k_{i=1} (a_iz+ b_i).NEWLINE\]NEWLINE The author assumes a weak version of the \(k\)-tuple conjecture which states that there exists a fixed \(T\geq 1\) such that for any \(m\geq 2\) whenever \(k\geq m^T\) then \(m\) of the forms in \(D\) are prime for infinitely many \(z\). He then establishes that there exists \(R\geq 3\) such that \(C_R(x)\to\infty\) as \(x\to\infty\), and deduces that there are in fact infinitely many such \(R\). The proof builds on the ideas introduced in the paper quoted above. The author remarks that the method does not determine an exact value of \(R\). When \(T=1\) in the conjecture we obtain Dickson's \(k\)-tuple conjecture itself which if true could be used to search for Carmichael numbers.