Comment by jackhalford
1 day ago
Seems like the constant is another representation of the set of all prime numbers. I wonder of there is a branch of math that formalizes these different representations, there is the infinite series here that defines the constants, but how does that relate to the set of primes? And what are the other representations?
One such representation is a polynomial with integer coefficients in 26 variables given in
https://www.tandfonline.com/doi/abs/10.1080/00029890.1976.11...
which has the curious property that as you substitute nonnegative integers for the variables, the positive values of the polynomial are exactly the set of prime numbers. (The polynomial also yields negative values.)
When put like this, it sounds like the polynomial must reveal something deep about the primes... but it's another cool magic trick. The MRDP theorem (famous for solving Hilbert's 10th problem negatively) implies that this kind of multivariate polynomial exists for exactly those sets of natural numbers that are computably enumerable, so the polynomials could be seen as a really esoteric programming language for set-enumeration algorithms.
More tricks: https://en.wikipedia.org/wiki/Formula_for_primes