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between two perfect squares

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March 12, 2018
number theoryprime numbersPerfect Squares

Problem Statement

Let's call a positive integer nn special, if there exist two nonnegativ integers (a,ba, b), such that n=2a×3bn=2^a\times 3^b.
Prove that if kk is a positive integer, then there are at most two special numbers greater then k2k^2 and less than k2+2k+1k^2+2k+1.
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