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

HTML articles powered by AMS MathViewer

- by Kenichi Iyanaga PDF
- Math. Comp.
**64**(1995), 1319-1331 Request permission

## 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.

## 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 - Kenneth F. Ireland and Michael I. Rosen,
*A classical introduction to modern number theory*, Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York-Berlin, 1982. Revised edition of*Elements of number theory*. MR**661047** - Mao Hua Le,
*The number of solutions of a certain quadratic congruence related to the class number of $\textbf {Q}(\sqrt {p})$*, Proc. Amer. Math. Soc.**117**(1993), no.ย 1, 1โ3. MR**1110547**, DOI 10.1090/S0002-9939-1993-1110547-7 - 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, - Kenneth S. Williams,
*The quadratic character of $2$ $\textrm {mod}\ p$*, Math. Mag.**49**(1976), no.ย 2, 89โ90. MR**392791**, DOI 10.2307/2689440

*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)

**19**(1990), 501-502.

## Additional Information

- © Copyright 1995 American Mathematical Society
- 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