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Question Number 31595 by Nayon.Sm last updated on 11/Mar/18

Commented by Rasheed.Sindhi last updated on 11/Mar/18

$a = 0, b = 353$
Commented by Joel578 last updated on 11/Mar/18

$if a \neq b \neq 0, is there any solutions ?$
	Answered by Rasheed.Sindhi last updated on 11/Mar/18

	$It can be proved that$ $m^{2} + n^{2}, m^{2} - n^{2} & 2 mn is a pathagorean$ $triplet for m, n \in N & m > n$ $m^{2} + n^{2} = 353$ $a = m^{2} - n^{2}$ $b = 2 mn$ $n^{2} = 353 - m^{2}$ $For what values of m 353 - m^{2} is perfect$ $square ? Recall$ $that 1 ⩽ m^{2} ⩽ 353 \Rightarrow 1 ⩽ m ⩽ \sqrt{353} = 18$ $So testing 353 - 1^{2}, 353 - 2^{2}, . . ., 353 - 18^{2}$ $we^{'} ll find that m = 17, n = 8 .$ $m^{2} - n^{2} = 17^{2} - 8^{2} = 225$ $2 mn = 2 \times 17 \times 8 = 272$ $272^{2} + 225^{2} = 353^{2}$
	Answered by naka3546 last updated on 11/Mar/18

	Commented by Rasheed.Sindhi last updated on 11/Mar/18

	$(a + b + 353) (a + b - 353) = 2 a b$ $T h e p o s s i b i l i t y o f a + b + 353 b e i n g o n l y$ $2 a, 2 b o r a b i s o n l y w h e n a & b$ $b e p r i m e s . B u t {t h e r e}^{'} s n o g u a r a n t y$ $o f b e i n g p r i m e o f a a n d b .$
	Commented by Rasheed.Sindhi last updated on 11/Mar/18

	$G \infty D t r i c k t o t r o n s f o r m i n t o f a c t o r f o r m :$ $353^{2} = a^{2} + b^{2}$ $\to (a + b + 353) (a + b - 353) = 2 a b$
	Commented by mrW2 last updated on 11/Mar/18

	$v e r y u s e f u l t r i c k, s e e a l s o$ $Q 27936$
	Commented by Rasheed.Sindhi last updated on 11/Mar/18
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