## Re: [PrimeNumbers] Re:Goldbach: Probabilistic argument

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• Let k be an arbitrary positive integer 4. Let t be the number of primes strictly between 2 and k Let s be the number of primes strictly between k and 2k-2
Message 1 of 1 , Dec 21, 2005
Let k be an arbitrary positive integer > 4.

Let t be the number of primes strictly between 2 and k

Let s be the number of primes strictly between k and 2k-2

Probability that odd p < k is prime is t/(k-3)

p < 2k - p < 2 k - 2

p < k
p + k < 2 k
k < 2 k - p

probability that 2k - p is prime is s/(k-3)

Probability that p and 2k - p are prime is

( t/[ k-3 ] ) ( s/ [k - 3 ] ) = ts / [ k-3]^2

So the expected number of prime pairs (p, 2k-p) is t s / [ k-3 ].

Does this expected number of pairs increase or decrease as k increases?

If it increases, as k increases, then the Goldbach conjecture is almost
certainly true.

If it decreases as k increases, then the Goldback conjecture is in doubt.

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