On the size of solutions of the inequality \(\varphi(ax-b)<\varphi(ax)\) (Q2765014)

If \(a,b,c,d\) are nonnegative integers with \(ad\neq bc\), then the inequality \(\varphi(an+b)<\varphi(cn+d)\) has infinitely many solutions \(n\in \mathbb{N}\), but \(\varphi(30n+1) <\varphi(30n)\) has no solutions \(n<2\cdot 10^7\) [\textit{D. J. Newman}, Am. Math. Mon. 104, 256-257 (1997; Zbl 0873.11008)]. \textit{G. Martin} [Am. Math. Mon. 106, 449-451 (1999; Zbl 0986.11002)] found the smallest solution in this special case, a number \(n\) with 1116 decimal digits. The author generalizes the methods of Newman and Martin for the case \(\varphi(an+b) <\varphi (an)\) with \((a,b)=1\). He estimates the size of a solution \(n\) and expects that this is near the smallest solution.NEWLINENEWLINEFor the entire collection see [Zbl 0976.00054].