number theory | Yet Another Mathblog

Let $P$ denote the set of all primes and, for $p \in P$ , let $(*/p)$ denote the Legendre quadratic residue symbol mod p. Let ${\mathbb N}=\{1,2,\dots\}$ denote the set of natural numbers and let

$L: {\mathbb N}\to \{-1,0,1\}^P,$

denote the mapping $L(n)=( (n/2), (n/3), (n/5), \dots),$ so the $k$ th component of $L(n)$ is $L(n)_k=(n/p_k)$ where $p_k$ denotes the $k$ th prime in $P$ . The following result is “well-known” (or, as the joke goes, it’s well-known to those who know it well:-).

Theorem: The restriction of $L$ to the subset ${\mathbb S}$ of square-free integers is an injection.

In fact, one can be a little more precise. Let $P_{\leq M}$ denote the set of the first $M$ primes, let ${\mathbb S}_N$ denote the set of square-free integers $\leq N$ , and let