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Absolutely small solutions to a hyperelliptic congruence

Volume 166 / 2021

Mariusz Skałba Colloquium Mathematicum 166 (2021), 335-340 MSC: Primary 11A05, 11A07, 11A15. DOI: 10.4064/cm8340-3-2021 Published online: 11 October 2021

Abstract

We investigate the hyperelliptic congruence $$y^2\equiv (x+1)(x+2)\cdot \ldots \cdot (x+l)\pmod {m}$$ and ask whether there exists a constant $C(l)$ such that for any $m$ there is a solution $x,y$ satisfying $0\leq x\leq C(l)$. We prove that such a $C(l)$ does exist if we allow only prime modules $m$ but does not exist for general $m$.

Authors

  • Mariusz SkałbaInstitute of Mathematics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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