Solve for natural numbers the equation: 3x^2 + 15y^2 = 5z^2 + t^2


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Question Number 146201 by mathdanisur last updated on 11/Jul/21

$S o l v e f o r n a t u r a l n u m b e r s t h e e q u a t i o n :$ $3 x^{2} + 15 y^{2} = 5 z^{2} + t^{2}$
	Answered by mindispower last updated on 13/Jul/21

	$3 x^{2} - t^{2} = 5 (z^{2} - 3 y^{2}) \equiv 0 [5] . . . . (1)$ $t = (0, 1, 2, 3, 4) [5] \Rightarrow t^{2} = (0, 1, - 1)$ $3 x^{2} - t^{2} = 0 (5) \Rightarrow x = t = 0 [5] o n l y p o s s i b i l i t y$ $\Rightarrow x = 5 k, t = 5 m$ $1 \Rightarrow$ $3.25 k^{2} + 15 y^{2} = 5 z^{2} + 25 m^{2}$ $15 k^{2} + 3 y^{2} = z^{2} + 5 m^{2} . . . . 1$ $l e t (x, m, z, k), (a, b, c . d) s o l u t i o n o f 1$ $u s i n g t h e r e s u l t b e f o r$ $\Rightarrow 5 ∣ y, 5 ∣ z$ $b y r e c c u r s i o n$ $i f (a, b, c, d)$ $s o l u t i o n o f 3 x^{2} + 15 y^{2} = 5 z^{2} + t^{2}$ $\Rightarrow \forall n \in N (\frac{x}{5^{n}}, \frac{y}{5^{n}}, \frac{z}{5^{n}}, \frac{t}{5^{n}}) s o l u t i o n$ $\forall n \in N 5^{n} ∣ a \Rightarrow a = 0$ $\Rightarrow (x, y, z, t) = (0, 0, 0, 0) t h e o n l y s o l u t i o n$
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