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Small solutions of the congruencea 1 x 21 +a 2 x 22 +a 3 x 23 +a 4 x 24 ≡c(modp)

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Abstract

For any integersa 1,a 2,a 3,a 4 andc witha 1 a 2 a 3 a 4≢0(modp), this paper shows that there exists a solutionX=(x 1,x 2,x 3,x 4) ∈Z 4 of the congruencea 1 x 21 +a 2 x 22 +a 3 x 23 +a 4 x 24 ≡c(modp) such that

$$\left\| X \right\| = \max \left\{ {\left| {x_1 } \right|,\left| {x_2 } \right|,\left| {x_3 } \right|,\left| {x_4 } \right|,} \right\} \ll \sqrt p \log p.$$

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Authors and Affiliations

  1. Department of Mathematics, Kansas State University, 66506, Manhattan, KS, USA

    Todd Cochrane

  2. Department of Mathematics, Zhongshan University, 510275, Guangzhou, China

    Zheng Zhiyong

Authors
  1. Todd Cochrane
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  2. Zheng Zhiyong
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Additional information

Research of Zheng Zhiyong is supported by NNSF Grant of China. He would also like to thank the first author and the Mathematics Department of Kansas, State University for their hospitality and support.

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Cochrane, T., Zhiyong, Z. Small solutions of the congruencea 1 x 21 +a 2 x 22 +a 3 x 23 +a 4 x 24 ≡c(modp). Acta Mathematica Sinica 14, 175–182 (1998). https://doi.org/10.1007/BF02560204

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  • DOI: https://doi.org/10.1007/BF02560204

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