Problems & Puzzles: Puzzles

Puzzle 490. 2^n-2^k+1, a square Sebastian Martin Ruiz  sent the following puzzle: For all even n exists k that 2^n-2^k+1 is a square. Is trivial k=n/2+1 then 2^n-2^(n/2+1)+1=(2^(n/2)-1)^2. But for odd n ? I have obtained only three solutions: 2^5-2^3+1=5^2 2^7-2^3+1=11^2 2^15-2^3+1= 181^2 Q1. Can you find solutions other than 5, 7 & 15? Q2. Is always k=3?

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