Question-193886

Question Number 193886 by Rupesh123 last updated on 22/Jun/23

Answered by MM42 last updated on 22/Jun/23

a = 521 (521^{n} - 521^{n - 1} + 1) = 521 m

“ 521^{″} i s p r i m e n u m b e r . t h e r e f o r e “ m^{″}

m u s t b e a m u l t i p l e “ 521^{″} .

w h i c h i s v a l i d o n l y f o r “ n = 1^{″}

Commented by Rupesh123 last updated on 22/Jun/23

Perfect ��

Answered by Subhi last updated on 22/Jun/23

521 ((521 - 1) 521^{n - 1} + 1)

521 (521^{n} - 521^{n - 1} + 1)

t o b e a p e r f e c t s q u a r e

521^{n} - 521^{n - 1} + 1 = 521^{2 m + 1}, w h e r e m ⩾ 0

n ⩾ 1

o r 521^{n} - 521^{n - 1} + 1 = 521^{2 k - 1}, w h e r e k ⩽ 0

n ⩾ 1

521^{2 m + 1} - 521^{n} + 521^{n - 1} = 1

521 (521^{2 m} - 521^{n - 1} + 521^{n - 2}) = 1

521^{2 m} - 521^{n - 1} + 521^{n - 2} = \frac{1}{521} = {(521)}^{- 1}

∴ n - 2 ⩽ 0 ⇛ n ⩽ 2

n - 1 ⩽ 0 ⇛ n ⩽ 1

m ⩽ 0

∴ m = 0, n = 1

521^{2 k - 1} + 521^{n - 1} - 521^{n} = 1

521 (521^{2 k - 2} + 521^{n - 2} - 521^{n - 1}) = 1

521^{2 k - 2} + 521^{n - 2} - 521^{n - 1} = \frac{1}{521}

n - 1 ⩽ 0 ⇛ n ⩽ 1 ⇛ n = 1 ⇛ k = 1

Commented by Rupesh123 last updated on 23/Jun/23

Perfect ��