"prime-connectedness"

[spados at katamail.com]

Ignore my previous messagge, there's no need to verify anything since what I'm 
asking is true.


"Given a, b in {1,2,...,n} there always exists a sequence x_{1}, x_{2},..., x_{k} in 
{1,2,...n}, such that a+x_{1}, x_{1}+x_{2},...,x_{k-1}+x_{k}, x_{k}+b are all 
primes." (call x_{1}, x_{2},...x_{k} a "path")

PROOF: By Bertrand's Postulate there is a prime p_{1} between a and 2a, so there 
exists x_{1}<a such that a+x_{1}=p_{1}. Now x_{1}<a so 2x_{1}<p_{1}. So 
there exists a prime  p_{2} between x_{1} and 2x_{1} and so an integer 
x_{2}<x_{1} such that x_{1}+x_{2}=p_{2}... Since the sequence x_{i} decreases 
and we are dealing with positive integers it will stop at 1.
This shows that there is a path P from a to 1. The same argument shows that there is 
a path P' from b to 1. Then the union of P and P' is a path from a to b. qed

Cheers,
Santi












__________________________________________


Fai i tuoi acquisti su www.kwshopping.it