Question-31595

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}, \dots, 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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