Proof of the \(P\)-integer conjecture of Pomerance (Q401988)

An integer \(k\) is a \(P\)-integer if the first \(\varphi(k)\) primes coprime to \(k\) form a reduced residue system modulo \(k\). In [J. Number Theory 12, 218--223 (1980; Zbl 0436.10020)] \textit{C. Pomerance} proved the finiteness of the set of \(P\)-integers, and conjectured that the \(P\)-integers are given by \(2,4,6,12,18,30\). Later, in [Acta Arith. 155, No. 2, 175--184 (2012; Zbl 1311.11093)] the reviewer, \textit{N. Saradha} and \textit{R. Tijdeman} proved that all \(P\)-integers \(k\) satisfy \(k<10^{3500}\).NEWLINENEWLINEIn the present paper, combining ideas of [Zbl 1311.11093] with sharp estimates of \textit{P. Dusart} concerning the Chebyshev function \(\theta(x)\) [``Estimates of some functions over primes without R.H.'', Preprint, \url{arXiv:1002.0442}], the authors verify the conjecture of Pomerance.