Given that:
22022 - 31*22014 + 2n is a perfect square for certain positive integer n
Find the value of n
2^2022 - 31*2^2014 + 2^n
2^2022 - 31*2^2014 - 1*2^2014 + 1*2^2014 + 2^n
2^2022 - 32*2^2014 + 2^2014 + 2^n
2^2022 - 2^2019 + 2^2014 + 2^n
8*2^2019 - 2^2019 + 2^2014 + 2^n
7*2^2019 + 2^2014 + 2^n
7*32*2^2014 + 2^2014 + 2^n
224*2^2014 + 2^2014 + 2^n
225*2^2014 + 2^n
225*2^2014 is itself a perfect square, its square root being 15*2^1007
Find X s.t. 225 + X is a perfect square, and X is a power of 2
225 + 64 = 289
15^2 + 8^2 = 17^2
225*2^2014 + 64*2^2014 = 289*2^2014
= (17*2^1007)^2
64*2^2014 = 2^2020
n = 2020
[Edit: corrected my arithmetic error, as per Charlie's note]
Edited on December 23, 2023, 6:57 pm
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Posted by Larry
on 2023-12-23 11:29:55 |
Analytic solution
