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Surf City Semiprimes (Posted on 2011-09-27) Difficulty: 2 of 5

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.)

See The Solution Submitted by broll    
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Solution proof | Comment 3 of 4 |
(In reply to I don't think it's true. Maybe I do. by Jer)

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
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