Conference rusure::math

700.0. "Goldbach's Conjecture Revisited." by CHOVAX::YOUNG (Back from the Shadows Again,) Fri May 08 1987 03:28

    Goldbach's Conjecture is that for any even positive integer N > 4
    there exists at least one pair of prime numbers whose sum is N.
    
    This has never been proved or disproved, though it has been verified
    as true up to some huge number (fill me in here, guys).
    
    Questions are:
    
    	1.  What is the probability that for any N as described above, 
    	there would exist such a pair of primes?  Call this probabilty
    	G(N).
    
    	3.  What is the limit of G(N) as N approaches infinity?
    
    	4.  Based on #1, what is the probability of Goldbach's Conjecture
    	being true for all cases <= N?  Call this C(N).
    
    	5.  What is the limit of C(N) as N approaches infinity?
    
    	6.  If Goldbach's Conjecture is not true, how large an N would
    	it take before we would expect to see failure?
    
    
    --  Barry

The number of primes less than N has been found (by Legendre and Gauss, 
independently) to be asymptotic to 

	Li(N) = integral (dt/ln(t)), t={0,N},

the *Logarithmic Integral* of N. The error in this approximation is only
1701 for N = 10**9, where the number of primes < N is 50,847,534.

As N increases, the number of *combinations* of pairs of candidate primes
also increases and can be calculated roughly from Li(N), and from that you 
can determine the probabilities. For 'small' N (<10**9) I think the 
probability is near 1, but I have not calculated it.

Lynn Yarbrough 

    >Goldbach's Conjecture is that for any even positive integer N > 4
    >there exists at least one pair of prime numbers whose sum is N.
    
    why N>4 ?
    I think that should be N>=4.
    
    /nasser