Thread: Not the largest prime View Single Post
2020-05-03, 01:46   #6
Dr Sardonicus

Feb 2017
Nowhere

66478 Posts

Quote:
 Originally Posted by Tomws And secondly given the well known proof that there are an infinite number of primes, how many digits are there in the answer if all primes less than or equal to 'my' prime are multiplied together?
Believe it or not, this question runs you smack-dab into the Prime Number Theorem (PNT)!

Alas, the answer is not known as precisely as one might wish, but still...

What you want is a "reasonable" estimate for

log(2) + log(3) + ... + log(p),

where p is the largest prime <= X, X some "large" positive number. In PNT-related literature, it is the natural log, log to the base e, or ln, that is used. And a statement equivalent to PNT is

$\sum_{p\le X}\ln(p) \;\sim \; X$

(ratio of RHS to LHS approaches 1 as X increases without bound) so that the number of decimal digits in the product of the primes up to X is something like X/ln(10).