Re: [PrimeNumbers] Re:Goldbach: Probabilistic argument
- 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
If it decreases as k increases, then the Goldback conjecture is in doubt.
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