二次互反律
高斯和勒让德的叙述 > 高斯的叙述

如果 q\equiv 1{\pmod {4}} 那么  x^{2}\equiv p{\pmod {q}} 可解当且仅当 x^{2}\equiv q{\pmod {p}}可解。
如果 q\equiv 3{\pmod {4}} 那么  x^{2}\equiv p{\pmod {q}} 可解当且仅当 x^{2}\equiv -q{\pmod {p}}可解。
借助于以下变量： q^{*}=(-1)^{\frac {q-1}{2}}q，命题可以简化为：
在高斯的叙述中已经可以见到“互反”的体现，即将 x^{2}\equiv p{\pmod {q}} 的可解性与 x^{2}\equiv q{\pmod {p}}的可解性联系起来。在下表中可以看出，这表现了一种对称性（反对称性）。
下表列明了质数之间相互是否为二次剩余的情况。方格内为R表示对应的 q（横列元素）为对应的 p（竖列元素）的二次剩余，N则表示相反情况（此表示法由高斯创造）。可以看到白格内的元素是关于对角线对称的，黄格内则关于对角线反对称。可以说黄格代表了一种“特殊情况”。
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q |
|  |  | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | 97 |
| p | 3 |   | N | R | N | R | N | R | N | N | R | R | N | R | N | N | N | R | R | N | R | R | N | N | R |
| p | 5 | N |   | N | R | N | N | R | N | R | R | N | R | N | N | N | R | R | N | R | N | R | N | R | N |
| p | 7 | N | N |   | R | N | N | N | R | R | N | R | N | R | N | R | N | N | R | R | N | R | N | N | N |
| p | 11 | R | R | N |   | N | N | N | R | N | R | R | N | N | R | R | R | N | R | R | N | N | N | R | R |
| p | 13 | R | N | N | N |   | R | N | R | R | N | N | N | R | N | R | N | R | N | N | N | R | N | N | N |
| p | 17 | N | N | N | N | R |   | R | N | N | N | N | N | R | R | R | R | N | R | N | N | N | R | R | N |
| p | 19 | N | R | R | R | N | R |   | R | N | N | N | N | R | R | N | N | R | N | N | R | N | R | N | N |
| p | 23 | R | N | N | N | R | N | N |   | R | R | N | R | N | R | N | R | N | N | R | R | N | N | N | N |
| p | 29 | N | R | R | N | R | N | N | R |   | N | N | N | N | N | R | R | N | R | R | N | N | R | N | N |
| p | 31 | N | R | R | N | N | N | R | N | N |   | N | R | N | R | N | R | N | R | R | N | N | N | N | R |
| p | 37 | R | N | R | R | N | N | N | N | N | N |   | R | N | R | R | N | N | R | R | R | N | R | N | N |
| p | 41 | N | R | N | N | N | N | N | R | N | R | R |   | R | N | N | R | R | N | N | R | N | R | N | N |
| p | 43 | N | N | N | R | R | R | N | R | N | R | N | R |   | R | R | R | N | R | N | N | R | R | N | R |
| p | 47 | R | N | R | N | N | R | N | N | N | N | R | N | N |   | R | R | R | N | R | N | R | R | R | R |
| p | 53 | N | N | R | R | R | R | N | N | R | N | R | N | R | R |   | R | N | N | N | N | N | N | R | R |
| p | 59 | R | R | R | N | N | R | R | N | R | N | N | R | N | N | R |   | N | N | R | N | R | N | N | N |
| p | 61 | R | R | N | N | R | N | R | N | N | N | N | R | N | R | N | N |   | N | N | R | N | R | N | R |
| p | 67 | N | N | N | N | N | R | R | R | R | N | R | N | N | R | N | R | N |   | R | R | N | R | R | N |
| p | 71 | R | R | N | N | N | N | R | N | R | N | R | N | R | N | N | N | N | N |   | R | R | R | R | N |
| p | 73 | R | N | N | N | N | N | R | R | N | N | R | R | N | N | N | N | R | R | R |   | R | N | R | R |
| p | 79 | N | R | N | R | R | N | R | R | N | R | N | N | N | N | N | N | N | R | N | R |   | R | R | R |
| p | 83 | R | N | R | R | N | R | N | R | R | R | R | R | N | N | N | R | R | N | N | N | N |   | N | N |
| p | 89 | N | R | N | R | N | R | N | N | N | N | N | N | N | R | R | N | N | R | R | R | R | N |   | R |
| p | 97 | R | N | N | R | N | N | N | N | N | R | N | N | R | R | R | N | R | N | N | R | R | N | R |   |