Question

the value of n for which 2^100 -2^192 . 31 +2^n is a perfect square

Answer 1

2^100 -2^192.31 + 2ⁿ

= 2^192( 2^8 - 31)+2^n

= 2^192(256-31) + 2ⁿ

= 2^192( 225) + 2ⁿ

= 2^192{ 225 + 2^( n -192)}

2^192{ 225 + 2^(n -192)} this will be perfect square only when ,( 225 + 2^( n -192) be perfect square .

if we take n = 198
then,

{225 + 2^(198-192) } = 125 + 2^6

= 225 + 64 = 289 = 17²
hence,
n = 198 ( answer )