Hi,

We can set much tighter bounds than prime(n+1) and prime(2n+1).

Chebyshev said it and I'l say it again:There's always a twin between n and fn.

Well, for sufficiently large n if f is small. Of course a proof would require the truth of the twin prime conjecture.

One can then whittle away at f = 1.5n,1.1n,1.01n,...fn and still find twins between n and fn for sufficiently large n. For example using pari,

gpi2(n,f) = twinpi(f*n)-twinpi(n)

twinpi(n) = \\ Count twin primes less than and including n { local(c,x); c=0;forprime(x=3,n,if(ispseudoprime(x+2),c++));c }

No matter how small we make f, there is an n such that for alln after that there is a twin between n and fn.

Certainly, f = 3 appears to cover all numbers. f=2 covers all n > 5, f=1.5 covers all n > 19, f=1.01 covers all n > 30000 etc

This will be difficult to prove in general even if we assume the TPC.

Enjoy,

Cino Hilliard

To:

crivera@...;

primenumbers@yahoogroups.comFrom:

sebi_sebi@...: Sun, 30 Nov 2008 09:24:25 -0800Subject: [PrimeNumbers] Twin Primes Conjecture

Hello all:Twin Primes ConjectureFor all n positive integer exists always a couple of twin primes between Prime[n+1] and Prime[2n+1]n=1 Prime[2]=3 , Prime[3]=5 3,5n=2 Prime[3]=5, Prime[5]=11 5,7n=3 Prime[4]=7,Prime[7]=17 11,13I have it checked for n=1 to 4000.SincerelySebastián Martín Ruiz[Non-text portions of this message have been removed]

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