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 April 20th, 2017, 09:40 AM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 224 Thanks: 26 Cardinality How many k satisfy $\displaystyle \;$$\displaystyle 2^n -n=k^2 Example , \displaystyle 2^7 -7=11^2 \displaystyle \;$$\displaystyle n,k \in N$ Last edited by idontknow; April 20th, 2017 at 09:44 AM.
 April 20th, 2017, 11:22 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 1,930 Thanks: 998 I can't prove it yet but it certainly looks like $(n,k)=(7,11)$ is the only integer pair such that $2^n-n=k^2$
 April 21st, 2017, 12:42 AM #3 Senior Member   Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT What about (n,k) = (1,1) or (n,k) = (0,1) [if you allow n = 0]?

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