On a conjecture of Pomerance (Q2919678)

The authors study the so called P-integers; recalling that (the Euler function) NEWLINE\[NEWLINE \varphi(n):=|\{ j\leq n, (j,n)=1\}| NEWLINE\]NEWLINE is the number of positive integers up to \(n\), coprime to \(n\), then we say that \(k\) is a P-integer, by definition, if the first \(\varphi(k)\) primes coprime to \(k\) form a reduced residue system modulo \(k\). \textit{C. Pomerance} proved in [J. Number Theory 12, 218--223 (1980; Zbl 0436.10020)] the finiteness of P-integers and made theNEWLINENEWLINENEWLINEConjecture. The largest P-integer is 30.NEWLINENEWLINENEWLINEUnder RH, Riemann Hypothesis, i.e., all the complex zeros of the Riemann \(\zeta\) function, defined in \(\text{Re}(s)>1\) as NEWLINE\[NEWLINE \zeta(s):=\sum_{n=1}^{\infty}n^{-s}, NEWLINE\]NEWLINE have real part equal to \(1/2\), the authors prove Pomerance's Conjecture (and even more, see Theorem 1.2). Also, relying on previous properties of P-integers proved by two of them (see quoted publications in the paper), They are able to find some numerical ranges in which they prove there are no P-integers (esp., all of them have to be less than \(10^{3500}\), see Theorem 1.1).NEWLINENEWLINEAs the authors point out, ``Pomerance's conjecture is closely related to the classical problem about the least primes in arithmetic progressions''.NEWLINENEWLINEFor the proofs, as they say ``We use the fact that the primitive residues modulo \(k\) between \(0\) and \(k\) are symmetric around \(k=2\). Our arguments are based on results about the zeros of the Riemann zeta function and estimates for the number of primes in intervals.''NEWLINENEWLINEActually, they need very precise constants in the remainders and this is performed both by a careful estimate applying well-known results and by computer aided calculations (by Maple).