Let n be some arbitrarily huge number; call each prime between 1 and n a 'GIRL'; call each even semiprime between 1 and n a 'BOY'.
Prove that there are two GIRLs for every BOY.
(An even semiprime is a composite number one of whose two prime factors is 2.)
(In reply to I don't think it's true. Maybe I do.
The prime number theorem tells us that the number of primes under a given number x approaches x/ln(x). The question is then whether (x/ln(x))/(x/2)/(ln(x/2)) approaches 2.
(x/ln(x))/(x/2)/(ln(x/2)) = 2 ln(x/2) / ln(x)
= 2 (ln(x)-ln(2)) / ln(x)
which does indeed approach 2 as ln(2) becomes insignificant compared to ln(x).
Posted by Charlie
on 2011-09-27 16:55:12