A recursive method to calculate the number of solutions of quadratic equations over finite fields

Author:
Kenichi Iyanaga

Journal:
Math. Comp. **64** (1995), 1319-1331

MSC:
Primary 11T30; Secondary 11D79, 11R29, 11Y16

DOI:
https://doi.org/10.1090/S0025-5718-1995-1297472-6

MathSciNet review:
1297472

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Abstract | References | Similar Articles | Additional Information

Abstract: The number ${S_m}(\alpha )$ of solutions of the quadratic equation \[ x_1^2 + x_2^2 + \cdots + x_m^2 = \alpha \quad (x_i^2 \ne \pm x_j^2\quad {\text {for}}\;i \ne j)\] for given *m*, with $\alpha$ and ${x_i}$ belonging to a finite field, is studied and a recursive method to compute ${S_m}(\alpha )$ is established.

- Takashi Agoh,
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*The number of solutions to the congruence $\sum ^k_{i=1}x^2_i\equiv 0\pmod p$ and class numbers of quadratic fields ${\bf Q}(\sqrt {p})$*, Sichuan Daxue Xuebao**27**(1990), no. 3, 260โ264 (Chinese, with English summary). MR**1077801**
---, - Kenneth S. Williams,
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*On the number of solutions of*$\sum \nolimits _{i = 1}^k {x_i^2 \equiv 0\; \pmod p (1 \leq {x_1} < \cdots < {x_k} \leq (p - 1)/2)}$, Adv. in Math. (Beijing)

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Additional Information

Keywords:
Quadratic equations over a finite field,
number of solutions,
algorithm

Article copyright:
© Copyright 1995
American Mathematical Society