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2 Power Crossed Perfect Power Puzzle (Posted on 2023-12-23)

Given that:

2²⁰²² - 31*2²⁰¹⁴ + 2ⁿ is a perfect square for certain positive integer n

Find the value of n

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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solution

Comment 4 of 4 |

Factor: 2^2014 * (2^8 - 1) + 2^n = 2^2014 * (225) + 2^n = K^2

Rearrange: 2^n = (K + 15*2^1007) * (K - 15*2^1007)

Each factor is a power of 2, say 2^a and 2^b where n = a+b.

Set equivalent and subtract:

2^a - 2^b = 30*(2^1007) = 15*(2^1008) = (2^4 - 1)*(2^1008) = 2^1012 - 2^1008

and n = 1012+1008 = 2020.

Posted by xdog on 2023-12-23 18:55:01

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