Primitive Zeros of Quadratic Formsmod P^2

Ali H. Hakami

《Tamkang Journal of Mathematics》 46卷3期 (2015/09) Pp. 349-364

https://doi.org/10.5556/j.tkjm.46.2015.1745

並列摘要

Let Q(x) = Q(x_1,x_2, . . . ,x_n) be a quadratic form with integer coefficients, p be an odd prime and ∥x∥ = max_i |x_i |. A solution of the congruenceQ(x) ≡ 0 ( mod p^2) is said to be a primitive solution if p □ x_i for some i. In this paper, we seek to obtain primitive solutions of this congruence in small rectangular boxes of the type B = {(The equation is abbreviated)} where for 1 ≤ i ≤ l we have M_i ≤ p, while for i ＞ l we have Mi ＞ p. In particular, we show that if n ≥ 4, n even, (The equation is abbreviated), and Q is nonsingular (mod p), then there exists a primitive solution with xi = (The equation is abbreviated).