[openasocket]

There are some patterns that emerge when you look at them in other bases. For example, in base 6, the final "digit" (hexit?) is either 1 or 5, (with the exception of the primes 2 and 3). In base 4, the final "digit" is either 1 or 3 (with the exception of the prime number 2). Of course mathematicians generally don't talk about this in terms of their base representation, they usually talk about the primes modulo 6 or the primes modulo 4.

And some of those representations actually do reveal some patterns. For example, an odd prime (so any prime other than 2) p can be written as the sum of two squares p = x^2 + y^2 if and only if p = 1 (mod 4). So those primes that end in 1 in the base 4 representation can be written as the sum of two squares, but the ones that end in 3 cannot. This is called Fermat's theorem on sums of two squares: https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_... .

My guess is that there's a number of different theorems about prime numbers that are phrased in terms of modulo arithmetic or whatever that can be converted into statements about the base representations of primes.

If I had to guess, though, I would guess there isn't a base where the pattern suddenly looks regular. That's very much a guess, but I have a couple data points to support that. The first is Dirichlet's prime number theorem: https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arith... . For any coprime integers a and b, the sequence a, a + b, a + 2b, a + 3b, ... contains infinite primes. This seems to imply that primes are, in some sense, evenly distributed across the different possible last "digits" of any base-b representation. There's also the Green-Tao theorem (https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem ) which says that there exist arbitrarily long arithmetic progressions. So for any integer k, there exists some a and b such that a, a + b, a + 2b, ... a + (k-1)b, and a + k*b are all prime! I don't have a good formal argument, but that seems like it would introduce arbitrary "noise" into any proposed pattern of "digits".

Finally, there's the Riemann Hypothesis. This is both my strongest data point and also my weakest data point. There's a deep relationship between the number of primes less than a given number and the zeroes of the Riemann Zeta function. Any pattern on the base-n representation of primes would imply some pattern on the number of primes less than a given number, which would in turn imply some pattern on the zeroes of the Riemann zeta functions. But the Riemann Hypothesis remains unsolved after over 150 years, despite being one of the most-studied problems in number theory. It seems like any pattern in the base-n representation would have meant some pattern in the zeroes of the zeta function, which means we would have made some progress on the Riemann Hypothesis after all this time. I consider this argument both very convincing and not convincing at all, because on one hand I'm relying on the lack of progress of so many people on this problem, which seems convincing, but also maybe it's basically just a logical fallacy, like an appeal to authority.