Power coincedences

I’ve always been tickled by the fact that 2^10 (1024) is very close to 10^3, and use that fact to very quickly estimate large powers of two, even if they’re unfamilar (e.g. 2^23 == (2^10)^2 * 2^3 ~= (10^3)^2 * 2^3 -> it’s roughly 8 million).

I just realised that there probably are more pairs triples of numbers with this property, and how they might be found. The properties are:

• a, b and k which are all natural numbers
• a != b
• a^b is “very close” to b^k. I think we should define “very close” to mean within 2.5%, since that way our earlier example of a, b and k (2, 10, 3) can fit :)
  • which number to take 2.5% of? uhhh whichever one makes it work, or it has to work with both if we want to be strict about it (either restriction works with our (2, 10, 3) triple)

I want to keep this short so I’m not going to write any actual code to search for these triples, or even do a basic search on the internet for existing research into this seemingly obvious “puzzle”.