4 ms·
Got an example? From my exploration, the distance between a power of 3 and the nearest power of 3 tends to grow.
by dankearney 4y ago
Got an example? From my exploration, the distance between a power of 3 and the nearest power of 3 tends to grow.
- LegionMammal978 4y agoDid you not see the n = 1000000001, e = 10439860591, o = 6586818670 example in my earlier post? (I edited it in a while after initially posting it.)
- eru 4y agoSlight tangent: I wonder if there's a limit to how close they can get in an absolute sense (instead of in a logarithmic sense)?
- LegionMammal978 4y agoYes; see, e.g., the answer at https://mathoverflow.net/a/116852 https://mathoverflow.net/a/116852, adjacent to the one cited by the parent. This gives a hard lower bound of |2^x - 3^y| ≥ max(2^x, 3^y)/((e max(x, y))^821013301). (But it's not sufficient for the parent's argument, since (3 + 1/n₁)^o = 3^o·(1 + 1/(3n₁))^o grows larger than 3^o by an exponential factor, so we can always increase (e, o) until the interval is wide enough.)