[ethanwillis]

when dealing with only even and odd they are not finite in base 2^k.

if we marked sequences of integers with 3 options. even, odd, other. then these lists are not finite in bases of 3^k.

for four options. even, odd, other, another. then these lists are not finite in bases of 4^k.

there is an intersection in the infinite lists where the base is equivalent to the power of an earlier base.

so infinite lists for 2^k would overlap a subset of the infinite lists for 2^2^k=4^k

all prime bases, p, p^k would admit infinite lists that cover all the infinite lists for some composite base, c, c^k.

[ethanwillis]

there is another similar problem about the largest number where all digits are prime numbers. which afaik has only been proven in base 10.

similarly there the largest number with all prime digits actually differs if you ask the question in different bases.

and there is also a pattern that exists to predict what the number will be in a given base.

[RheingoldRiver]

> the largest number where all digits are prime numbers

do you mean the largest prime number where all digits are prime numbers?