Twin stars and twin primes

Are there more twin stars or twin primes?

If the twin prime conjecture is true, there are an infinite number of twin primes, and that would settle the question.

We don’t know whether there are infinitely many twin primes, and it’s a little challenging to find any results on how many twin primes we’re sure exist.

According to OEIS, we know there are 808,675,888,577,436 pairs of twin primes less than 1018.

There are perhaps 1024 stars in the observable universe. If so, there are certainly less than 1024 pairs of binary stars in the observable universe. In our galaxy about 2/3 of stars are isolated and about 1/3 come in pairs (or larger clusters). If that holds in every galaxy, then the number of binary stars is within an order of magnitude of the total number of stars.

Do we know for certain there are at least 1024 twin primes? It doesn’t seem anybody is interested in that question. There is more interest in finding larger and larger twin primes. The largest pair currently know have 388,342 digits.

The Hardy-Littlewood conjecture speculates that π2(x), the number of twin prime pairs less than x, is asymptotically equal to the following

\pi_2(x) \sim C_2 \int_2^x \frac{dt}{(\log t)^2}

where C2 is a constant, the product of (1 − 1/(p − 1)²) over all odd primes. Numerically C2 = 0.6601618….

When x = 1018 the right hand side of the Hardy Littlewood conjecture agrees with the actual number to at least six decimal places. If the integral gives an estimate of π2(x) within an order of magntude of being correct for x up to 1028, then there are more twin primes than twin stars.

It’s interesting that our knowledge of both twin stars and twin primes is empirical, though in different ways. We haven’t counted the number of stars in the universe or, as far as I know, the number of twin primes less than 1028, but we have evidence that gives us reason to believe estimates of both.