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Question Number 196117 by sonukgindia last updated on 18/Aug/23

	Answered by Rasheed.Sindhi last updated on 18/Aug/23

	$m^{2} (1 + 3 n^{2}) - 10 (3 n^{2} + 1) = 517 - 10$ $(m^{2} - 10) (3 n^{2} + 1) = 507$ $(m^{2} - 10) (3 n^{2} + 1) = 3 \times 13^{2}$ $. . . .$
	Answered by gatocomcirrose last updated on 18/Aug/23
	$m^2 +3m^2 n^2 −30n^2 =517 m^2 (1+3n^2 )−10(1+3n^2 )+10=517 (m^2 −10)(3n^2 +1)=507=3.13^2 { ((m^2 −10=1, 3n^2 +1=507 )),((m^2 −10=3, 3n^2 +1=169)),((m^2 −10=13, 3n^2 +1=39)),((m^2 −10=39, 3n^2 +1=13)),((m^2 −10=169, 3n^2 +1=3)),((m^2 −10=507, 3n^2 +1=1)) :} ⇒(m,n)={(7,2), (7,−2), (−7,2), (−7,−2)}$
	$m^{2} + {3 m}^{2} n^{2} - {30 n}^{2} = 517$ $m^{2} (1 + {3 n}^{2}) - 10 (1 + {3 n}^{2}) + 10 = 517$ $(m^{2} - 10) ({3 n}^{2} + 1) = 507 = 3 . 13^{2}$ ${\begin{cases} m^{2} - 10 = 1, {3 n}^{2} + 1 = 507 \\ m^{2} - 10 = 3, {3 n}^{2} + 1 = 169 \\ m^{2} - 10 = 13, {3 n}^{2} + 1 = 39 \\ m^{2} - 10 = 39, {3 n}^{2} + 1 = 13 \\ m^{2} - 10 = 169, {3 n}^{2} + 1 = 3 \\ m^{2} - 10 = 507, {3 n}^{2} + 1 = 1 \end{cases}$ $\Rightarrow (m, n) = {(7, 2), (7, - 2), (- 7, 2), (- 7, - 2)}$
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