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Some Sum of 2 Powers is a Perfect Square (Posted on 2024-02-26)

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

Find the value of n.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Quick solution

Comment 2 of 2 |

Simplify to 225*2^2014 + 2^n = x^2

The easy option is to assume n>2014, then some factoring gets us to (2^1007)^2 * (225+2^(n-2014)) = x^2

Then all we need is 225+2^(n-2014) to be a perfect square. This happens at n-2014=6, from 225+2^6=17^2

Then the answer is n=2020.

Posted by Brian Smith on 2024-02-26 10:50:15

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