Suppose you have a very large k. By observation (thousands of digits), 5 power k in a binary form tends to have half 1's, and half 0's. Is this observation easy to prove?
2026-03-25 06:05:00.1774418700
5 power k in binary form 1's and 0's density for large k.
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In general, very little is known about the fraction of $1$'s in binary expansions of expressions such as $5^k$. One would expect that these fractions should approach $1/2$ unless there's a good reason for them not to do so, but proving anything is going to be extremely difficult.
The numbers of $0$'s and $1$'s in this case are OEIS sequences A118737 and A118738.