Erdos Conjecture

Erdos Conjecture
Andre Willers
30 Sep 2011

Synopsis :
A trivial special case of Riemann's Zeta function and Kantor's Infinities .

Discussion :
Erdos conjecture :
 It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions.

This sounds difficult , but is actually very simple .
Arbitrary here means even two . An arithmetic progression means that the difference between any two term anywhere in the sequence forces the sequence into divergence (ie the sum will go to infinity)

If Z= 1/(s1)^n + 1/(s2)^n + …
where Z =0 if and only if n=2 (Discussed previously) Riemann when the irrational component is zero .
If any two terms s1 ,s2 in the infinite sequence are in arithmetical sequence (ie there is a finite difference between them ) , the difference is of the form 1/s^t , where t>=n-1 .
1. Multiply each side of the equation by an Aleph(0) infinity and the right-hand side becomes infinite ,while the left hand side remains zero if n>=2
2. There are some nice little fractals in between .

Physical significance :
Any logical system is unstable . Any stable system has infinitesimals forced on it .
Real systems fluctuate between them in chaotic attractor fashion .

Bah !
Things are supposed to get simpler .

Andre

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