I call an H-machine a machine that can be connected to turing machines and that takes as input a natural integer n and instantly returns the n'th digit of the mathematical constant H.

Is there a proof/disproof/(or any kind of argument of what we should expect) that access to any combination of the e,Pi,sqrt(2),zeta(3) or euler-macheroni-machines would help us solve any open conjectures in number-theory, such as the twin-prime conjecture?

Would any other problems such as factoring large integers gain any speedup?