The least modulus for which consecutive polynomial values are distinct

TL;DR

This paper investigates the least modulus for which consecutive polynomial values are distinct, establishing a connection with prime numbers in specific residue classes and proposing a conjecture related to prime pairs and quadratic residues.

Contribution

It provides a new characterization of the least prime in a residue class related to polynomial value distinctness and introduces a conjecture linking minimal moduli to twin primes.

Findings

For large n, the minimal prime p in a residue class is characterized by pairwise distinct polynomial values modulo m.

Established bounds for n > 24310 for certain degrees d, ensuring the minimal prime p meets specific inequalities.

Proposed a conjecture relating minimal moduli with twin primes and quadratic residues for all n > 4.

Abstract

Let $d\ge4$ and $c\in(-d,d)$ be relatively prime integers. We show that for any sufficiently large integer $n$ (in particular $n>24310$ suffices for $4\le d\le 36$), the smallest prime $p\equiv c\pmod d$ with $p\ge(2dn-c)/(d-1)$ is the least positive integer $m$ with $2r(d)k(dk-c)\ (k=1,\ldots,n)$ pairwise distinct modulo $m$, where $r(d)$ is the radical of $d$. We also conjecture that for any integer $n>4$ the least positive integer $m$ such that $|\{k(k-1)/2\ \mbox{mod}\ m:\ k=1,\ldots,n\}|= |\{k(k-1)/2\ \mbox{mod}\ m+2:\ k=1,\ldots,n\}|=n$ is the least prime $p\ge 2n-1$ with $p+2$ also prime.