Is this sort of toying with numbers just for fun, or does it in fact reveal something interesting about primes or mathematics in general?

Sounds like just a bit of fun to me. For one, emirp-hood depends on the number base, which means it’s not intrinsic to the number, but depends on some arbitrary choice of representation. Generally, things like that don’t indicate anything deep.

What's so special about base 10 other than that it's the number of fingers that humans (usually) have?

Well, it's the base in which we label all other bases. In base twelve, 'base 12'(decimal) is written 'base 10'!

:D

It isn't, so OEIS [1] correctly tags this sequence as base-dependent (keyword "base") and has a similar sequence in base 2 [2].

[1] https://oeis.org/A006567

[2] https://oeis.org/A080790

I do also find the amount of recreational math focused on base 10 to be disproportionate to the interestingness of base 10.

Come on, mate, don't tease me with that and not offer your suggestions!

Fractional bases, negative bases, negative fractional bases, base e (supposed to be optimal IIRC), perhaps complex bases, summat weirder?

The one true number base: https://en.m.wikipedia.org/wiki/Quater-imaginary_base

Nothing.

https://en.wikipedia.org/wiki/Decimal

Is there a better way to check for an emirp other than checking if the number is prime, and then checking if its reverse is also prime?

I can think of a couple of things off the top of my head:

- You can reject numbers that start with 2,4,5,6 or 8 since their reverse won't be prime.

- If you're searching through a range of numbers for emirps avoid checking for the same pair twice. For example if you start at 1 and check for emirps, by the time you're checking 311 you've already checked 113 so you don't need to check 311.

Proposal to define "primemirp" as a prime that is also a palindrome.

Looks like those are already called "palprimes" [0].

[0] https://en.wikipedia.org/wiki/Palindromic_prime

Plural: sprimemirps.

See also: http://www.cadaeic.net/silopolis.htm