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• ## Re: [PrimeNumbers] Goldbach conjecture [better explanation-deeper]

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• ... Ok, so this is the main inequality that you seem to want to prove, where a_n is the n th prime. So how is your proof affected by the fact that a_n is
Message 1 of 3 , Sep 13, 2003
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> to simplify inequality we it is allowed to say:
> (n-1)^2>a_n....................(Z)

Ok, so this is the main inequality that you seem to want to prove, where
a_n is the n'th prime. So how is your proof affected by the fact that
a_n is asymptotic to n*log n? I presume you know this fact...?

> We have to determine number of numbers that can be sum of k pairs
> of different primes. For example 24=13+11=17+7=23+1.the number of
> that combinations is n*(n+1)/2
> when we sub it from n^2 we get that of n primes could be a least
> formed n(n-1)/2
> now let sub of an, n+1 (because there are n+1 prime numbers in
> interval a_n) and then divide it by 2 to eliminate odd numbers and
> using (Z)
> ((n-1)^2-n-1)/2>(a_n-n-1)/2
> (n^2-2n+1-n-1)/2>a_n
> (n^2-n)/2=n(n-1)/2>a_n..............(R)
> where (a_n-n-1)/2 is actually number od even numbers in interval
> (0,an] so the GoldBach conjecture is proven.

Is it? So the number of combinations of two different primes, taken from
the first n primes, is obviously n(n-1)/2. And so you go on. How does
(R) prove Goldbach? Forgive me if I'm being stupid, but your explanation
still isn't clear. Your inequality (Z) is trivial to prove, so the only
sticking point is these last few lines that I've included. So yes, there
are many more pairs of prime numbers than there are integers which they
could sum to, but how do you know that every integer is represented by
such a pair?

Andy
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