Tricky Squares

Number Theory Level pending

Find n n , if 2 200 2 192 31 + 2 n 2^{200} -2^{192}\cdot 31+2^n is a perfect square.


The answer is 198.

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1 solution

Lakshya Singh
May 24, 2016

2^200-(2^192.2^5-1)+2^n

=2^200-(2^197-2^192)+2^n

=2^192*(2^8-2^5+1)+2^n

=2^192*(225)+2^n

=2^192[225+2^(n-192)]

therefore 225 + 2^(n-192) is a perfect square.

nearest square is 289.

225 + 2^(n-192)=289

= 2^(n-192)=64

= 2^(n-192)=2^6

n-192=6

therefore n=198

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