The number of solutions of a certain quadratic congruence related to the class number of $\textbf {Q}(\sqrt {p})$
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- by Mao Hua Le PDF
- Proc. Amer. Math. Soc. 117 (1993), 1-3 Request permission
Abstract:
Let $p$ be an odd prime, and let $k$ be a positive integer with $i \leqslant k \leqslant (p - 1)/2$. In this note we give a formula for the number of solutions $({x_1}, \ldots ,{x_k})$ of the congruence $x_1^2 + \cdots + x_k^2 \equiv 0\;(\bmod p)$, $1 \leqslant {x_1} < \cdots < {x_k} \leqslant (p - 1)/2$.References
- Takashi Agoh, A note on unit and class number of real quadratic fields, Acta Math. Sinica (N.S.) 5 (1989), no.ย 3, 281โ288. MR 1019628, DOI 10.1007/BF02107554
- Rudolf Lidl and Harald Niederreiter, Finite fields, Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. With a foreword by P. M. Cohn. MR 746963
- Qi Sun, The number of solutions to the congruence $\sum ^k_{i=1}x^2_i\equiv 0\pmod p$ and class numbers of quadratic fields $\textbf {Q}(\sqrt {p})$, Sichuan Daxue Xuebao 27 (1990), no.ย 3, 260โ264 (Chinese, with English summary). MR 1077801 โ, On the number of solutions of $\sum \nolimits _{i = 1}^k {x_i^2 \equiv 0\,(\bmod p)(1 \leqslant {x_1} < \cdots < {x_k} \leqslant (p - 1)/2)}$, Adv. in Math. (Beijing) 19 (1990), 501-502.
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 117 (1993), 1-3
- MSC: Primary 11D79; Secondary 11R11, 11R29
- DOI: https://doi.org/10.1090/S0002-9939-1993-1110547-7
- MathSciNet review: 1110547