On a conjecture of Pomerance

TL;DR

This paper proves Pomerance's conjecture that 30 is the largest P-integer, assuming the Riemann Hypothesis, and extends the non-existence of P-integers beyond 10^11 up to 10^3500.

Contribution

It confirms Pomerance's conjecture under the Riemann Hypothesis and establishes bounds on the non-existence of P-integers.

Findings

Proves 30 is the largest P-integer assuming RH.

No P-integers between 30 and 10^11.

No P-integers above 10^3500.

Abstract

We say that k is a P-integer if the first phi(k) primes coprime to k form a reduced residue system modulo k. In 1980 Pomerance proved the finiteness of the set of P-integers and conjectured that 30 is the largest P-integer. We prove the conjecture assuming the Riemann Hypothesis. We further prove that there is no P-integer between 30 and 10^11 and none above 10^3500.