Find-out-pairs-of-numbers-a-b-as-many-as-you-can-such-that-a-b-a-b-a-2-b-2-P-

Question Number 159925 by Rasheed.Sindhi last updated on 22/Nov/21

Find out pairs of numbers (a, b) (as

many as you can) such that :

\sqrt{a} + \sqrt{b}, a + b, a^{2} + b^{2} \in P

Answered by nikif99 last updated on 23/Nov/21

P a i r s (a, b) u p t o 121 :

(4, 9), (4, 25), (9, 64), (25, 64),

(36, 121), (49, 100), (81, 100) .

Commented by Rasheed.Sindhi last updated on 23/Nov/21

T han k s sir!

Commented by Rasheed.Sindhi last updated on 23/Nov/21

{What}^{'} s the maximum value of n for

which a + b, a^{2} + b^{2}, a^{3} + b^{3}, \dots, a^{n} + b^{n} \in P

Commented by Rasheed.Sindhi last updated on 23/Nov/21

Or if m < n, what is the maximum value

of n - m for which

a^{m} + b^{m}, a^{m + 1} + b^{m + 1}, a^{m + 2} + b^{m + 2}, \dots, a^{n} + b^{n} \in P

Commented by nikif99 last updated on 23/Nov/21

I f y o u o m i t t h e d e m a n d \sqrt{a} + \sqrt{b} \in P,

t h e n u m b e r o f a c c e p t a b l e p a i r s

i n c r e a s e e n o r m o u s l y .

N e v e r t h e l e s s, m a x (n) s e e m s t o b e 2 .

P . S . A n o t h e r p a i r t i l l 200 : (100, 169) .

Commented by Rasheed.Sindhi last updated on 23/Nov/21

T h a n k s a l o t s i r! {Y o u}^{'} r e r i g h t b e c a u s e

a^{3} + b^{3} i s a l w a y s c o m p o s i t e .

T h e c l a i m s h o u l d b e :

a + b, a^{2} + b^{2}, a^{4} + b^{4}, a^{8} + b^{8}, \dots \in P

Y o u r p r o v i d e d e x a m p l e s a r e

s u c c e s s f u l u p t o 3 l e v e l s . F o r e x a m p l e

I f i n s t e a d o f (100, 169) i f w e

c o n s i d e r (a, b) = (\sqrt{100}, \sqrt{169}) = (10, 13)

10 + 13 = 23 \in P

10^{2} + 13^{2} = 269 \in P

10^{4} + 13^{4} = 38561 \in P

T han k s for attention!

Commented by nikif99 last updated on 23/Nov/21

E x c e l l e n t p o s t! T h a n k y o u t o o .

Commented by Rasheed.Sindhi last updated on 24/Nov/21

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