Is there an infinite amount of primes in base n made from an equal amount of even and odd digits.

Question

Is there an infinite amount of primes in base n made from an equal amount of even and odd digits?

A list of primes that have this property is this sequence

$$23,29,41,43,47,61,67,83,89,1009,1021,1049,1061,\small\dots$$

I think this is true because if you pick a big random number the number of digits in that number with an n or an m is about equal. but I'm sure most numbers don't have an exact number of 2's as 1's just really close to equal. So I'm guessing that the number of primes like this is either less and less frequent or more and more frequent.

So my second question is what is the percentage of primes that have this property. is it almost 0% as the number of primes goes to infinity or does it go to 50%?

Answer 1

As twnly already mentioned in their answer, research by Mauduit and Rivat on a related problem proved (a generalized version of the result) that asymptotically half the primes have an even/odd sum of digits. I want to point out that this seemingly simple statement was an open problem for over 40 years, was proved less than a decade ago, and the proof was published in arguably the most prestigious mathematics journal in the world.

Questions about digits of primes are very hard!

The probability that a randomly chosen $2n$-digit integer has an equal number of odd and even digits is asymptotically $1/\sqrt{\pi n}$ (from asymptotics for central binomial coefficients). The probability that a randomly chosen $2n$-digit integer is prime is asymptotically $1/(2n\ln10)$ (from the prime number theorem).

The natural conjecture would be that these two events are asymptotically independent, so that the probability that a randomly chosen $2n$-digit integer both is prime and has the same number of odd/even digits should be asymptotically $1/(n^{3/2}\sqrt\pi\ln10)$. In particular, the probability that a randomly chosen $2n$-digit prime has the same number of odd/even digits should also be asymptotically $1/\sqrt{\pi n}$, which in particular tends to $0$ as $n\to\infty$. But also, in particular, there should be infinitely many primes with this property—this heuristic predicts that the number of such primes is rather larger than the number of twin primes, for example.

Answer 2

This is not an exact answer, but here are some results about digits of prime numbers that might be of interest.

In Christian Mauduit and Joël Rivat. Sur un problème de Gelfond : la somme des chiffres des nombres premiers, they show that asymptotically, 50% of the primes have an even sum of digits and 50% have an odd sum of digits in base-10.

Also there is a result of Bourgain, where you can specify arbitrary fixed positions of binary digits and still get primes with the prescribed digits. And Maynard has shown that there are infinitely many primes with no digit 7.

All this is written up in a survey by Maynard that can be found here.