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Question Number 112575 by Aina Samuel Temidayo last updated on 08/Sep/20

	Answered by Rasheed.Sindhi last updated on 09/Sep/20

	$x^{2} - 3 y^{2} = 1376$ $∵ 4 ∣ 1376 ∴ 4 ∣ x^{2} - 3 y^{2}$ $c a s e 1 : x, y \in O$ $x = 2 m + 1, y = 2 n + 1$ $x^{2} - 3 y^{2} = {(2 m + 1)}^{2} - 3 {(2 n^{2} + 1)}^{2}$ $= 4 m^{2} + 4 m + 1 - 12 n^{2} - 12 n - 3$ $= 4 (m^{2} + m - 3 n^{2} - 3 n) - 2$ $∴ 4 ∤ x^{2} - 3 y^{2}$ $c a s e 2 : x \in O \land y \in E o r x \in E \land y \in O$ $S i m i l a r l o g i c p r o v e s t h a t$ $4 ∤ x^{2} - 3 y^{2}$ $c a s e 1 : x, y \in E$ $S i m i l a r l o g i c p r o v e s t h a t$ $4 ∣ x^{2} - 3 y^{2}$ $L e t x = 2 m, y = 2 n$ $x^{2} - 3 y^{2} = 1376$ $\Rightarrow {(2 m)}^{2} - 3 {(2 n)}^{2} = 1376$ $\Rightarrow 4 m^{2} - 12 n^{2} = 1376$ $\Rightarrow m^{2} - 3 n^{2} = 344$ $A g a i n 4 ∣ 344 \Rightarrow 4 ∣ m^{2} - 3 n^{2}$ $B y s i m i l a r p r o c e s s$ $L e t m = 2 u, n = 2 v$ $u^{2} - 3 v^{2} = 86$ $u = 2 p + 1, v = 2 q + 1$ $u^{2} - 3 v^{2} = 86$ $u =$ ${(2 p + 1)}^{2} - 3 {(2 q + 1)}^{2} = 86$ $4 p^{2} + 4 p - 12 q^{2} - 12 q - 2 = 86$ $4 p^{2} + 4 p - 12 q^{2} - 12 q = 88$ $p^{2} + p - 3 (q^{2} + q) = 22$ $p (p + 1) - 3 q (q + 1) = 22$ $p (p + 1) \in E \land q (q + 1) \in E$ $p (p + 1) = 2 a, q (q + 1) = 2 b ◂$ $\Rightarrow 2 a - 3 (2 b) = 22$ $a - 3 b = 11$ $a = 3 b + 11$ $(a, b) = (3 b + 11, b)$ $C o n t i n u e$
	Commented by Aina Samuel Temidayo last updated on 09/Sep/20

	$There are no solutions .$
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