判断下列同余式是否有解：11x^2=-6(mod91)

Question

判断下列同余式是否有解：11x^2 =-6( mod91 )

Answer 1

由中国剩余定理.
11x² = -6 (mod 91)有解,当且仅当11x² = -6 (mod 7)和11x² = -6 (mod 13)都有解.
11x² = -6 (mod 7)可化为(2x)² = 1 (mod 7),有解x = ±3 (mod 7).
11x² = -6 (mod 13)可化为-2x² = -6 (mod 13)也即x² = 3 (mod 13),有解x = ±4 (mod 13).
因此原方程有解.
如果要具体求出解来,就解以下四组线性同余方程组(其实只需解前两个,另两个取负号就行):
x = 3 (mod 7),x = 4 (mod 13);
x = 3 (mod 7),x = -4 (mod 13);
x = -3 (mod 7),x = 4 (mod 13);
x = -3 (mod 7),x = -4 (mod 13).
得x = ±17,±4 (mod 91).
如果数比较大,而且不用求出解来,也可用二次互反律来计算.
例如(3/13) = (13/3) = (1/3) = 1,即x² = 3 (mod 13)有解.
当然本题数比较小,其实没必要.