For any prime p, let $y(p)$ denote the smallest integer y such that every reduced residue class (mod p) is represented by the product of some subset of $\{1,\dots ,y\}$ . It is easy to see that $y(p)$ is at least as large as the smallest quadratic nonresidue (mod p); we prove that $y(p) \ll _\varepsilon p^{1/(4 \sqrt e)+\varepsilon }$ , thus strengthening Burgess’ classical result. This result is of intermediate strength between two other results, namely Burthe’s proof that the multiplicative group (mod p) is generated by the integers up to $O_\varepsilon (p^{1/(4 \sqrt e)+\varepsilon })$ , and Munsch and Shparlinski’s result that every reduced residue class (mod p) is represented by the product of some subset of the primes up to $O_\varepsilon (p^{1/(4 \sqrt e)+\varepsilon })$ . Unlike the latter result, our proof is elementary and similar in structure to Burgess’ proof for the least quadratic nonresidue.

对于任何素数 p，让 $y(p)$ 表示最小整数y，这样每个减少的残差类 (mod p ) 都由 $\{1,\dots ,y\}$ 的某个子集的乘积表示 。很容易看出 $y(p)$ 至少与最小的二次非残差 (mod p ) 一样大；我们证明了 $y(p) \ll _\varepsilon p^{1/(4 \sqrt e)+\varepsilon }$ ，从而加强了 Burgess 的经典结果。该结果在其他两个结果之间具有中等强度，即 Burthe 证明乘法群 (mod p ) 由不超过的整数生成 $O_\varepsilon (p^{1/(4 \sqrt e)+\varepsilon })$ ，以及 Munsch 和 Shparlinski 的结果，即每个约化残差类 (modp) 都由质数的某个子集的乘积表示 $O_\varepsilon (p^{1/(4 \sqrt e)+\varepsilon })$ 。与后一个结果不同，我们的证明是基本的，并且在结构上类似于 Burgess 对最小二次非残差的证明。