p-2-3q-2-11907-p-q-Z-find-p-amp-q-

Question Number 84461 by Tony Lin last updated on 13/Mar/20

p^{2} + 3 q^{2} = 11907, p, q \in Z, f i n d p & q

Commented by mr W last updated on 13/Mar/20

i g o t o n l y t w o s o l u t i o n s :

p = 0, q = \pm 63

p = \pm 108, q = \pm 9

Commented by Tony Lin last updated on 13/Mar/20

p^{2} + 3 q^{2} = 3^{5} \times 7^{2}

\Rightarrow p^{2} = 3 (63 + q) (63 - q), q ⩽ 63

\Rightarrow p^{2} = 3^{5} (7 + \frac{q}{9}) (7 - \frac{q}{9})

∵ p & q \in Z

∴ 9 ∣ q

\to c h e c k t h a t (7 + \frac{q}{9}) (7 - \frac{q}{9}) = 3 \times n^{2}, n \in Z

(7 + 1) (7 - 1) = 3 \times 4^{2}

(7 + 7) (7 - 7) = 3 \times 0^{2}

\Rightarrow {\begin{cases} p = 0, q = \pm 63 \\ p = \pm 108, q = \pm 9 \end{cases}