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Products of quadratic residues and related identities

Volume 167 / 2022

Hai-Liang Wu, Li-Yuan Wang Colloquium Mathematicum 167 (2022), 197-206 MSC: Primary 11A15; Secondary 11R11, 11R18. DOI: 10.4064/cm8437-2-2021 Published online: 28 May 2021


We study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $p\equiv 5\,({\rm mod}\, 8)$, then $$\prod _{0 \lt x \lt p/2,\,(\frac {x}{p})=1}x\equiv (-1)^{1+r}\,({\rm mod}\,p),$$ where $\bigl (\frac {\cdot }{p}\big )$ is the Legendre symbol and $r$ is the number of $4$th power residues modulo $p$ in the interval $(0,p/2)$. Our work involves the class number formula, quartic Gauss sums, Stickelberger’s congruence and values of Dirichlet L-series at negative integers.


  • Hai-Liang WuSchool of Science
    Nanjing University of Posts
    and Telecommunications
    Nanjing 210023
    People’s Republic of China
  • Li-Yuan WangSchool of Physical
    and Mathematical Sciences
    Nanjing Tech University
    Nanjing 211816
    People’s Republic of China

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