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Re: A sequence of primes with a fixedpoint property based on a subsequence of 1/n
Posted:
Feb 3, 2014 4:24 AM


On 2/3/2014 1:46 AM, Ross A. Finlayson wrote:
<snip>
> > Not having carried the arithmetic for knowing the theme, > > "However, this might need to be a multiplicity > of 22 for large primes. >
This is built on so many little pieces of finite geometry that it is stunning. In the end, however, my understanding of the 11 as a multiplier resolves to a property of edge lengths in parallelepipeds defined in terms of Pythagorean triples. All such parallelepipeds have an edge divisible by 11.
But, I did find this today,
http://mathworld.wolfram.com/MarionsTheorem.html
Most of the significant numbers involved in the finite geometries I use are in those two theorems. The 11 appears in Marion's theorem. One might look at that as a base.
Morgan's theorem yields a "march to infinity" on the basis of odd numbers. For every odd prime, say p, one has that 4p is representable as the sum of 4 odd integral squares.
So, Morgan's theorem directly introduces the relationship between primes and quadratics. And, the numbers associated with that theorem and Marion's theorem are all associated with significant finite geometries.
The number 22 asserts itself in the Matheiu groups which have a role in the calculus I used in formulating many ideas. The large Mathieu groups are M_22, M_23, and M_24. But, there is another Mathieu group, M_11.
http://en.wikipedia.org/wiki/Mathieu_group_M11
http://en.wikipedia.org/wiki/Mathieu_group_M22
If these correspond to the edge length of the parallelepipeds, that multiplier could become even greater. I am actually betting on the particular sphere packing associated with
http://www.math.rwthaachen.de/~Gabriele.Nebe/LATTICES/ANABASIC_11.html#GRAM
I figure that numbers appearing to be related to 11 dimensions might reflect to have 22 components.
I admit to not always understanding my constructs. Without more data, I can only make a guess based on what influenced the constructions.
... > Secondly, it is in the definition of a modified > Fort space that a set with two "special points" > can be given a topology in which every infinite > sequence converges to the pair of special points:" >
Any complement of open sets containing either of the two points is finite. So, by the second definition of convergence in
http://www.proofwiki.org/wiki/Definition:Convergent_Sequence_%28Topology%29
And, each of the special points is in the closure of any open set containing the other special point. So, no sequence can converge to one without converging to the other. The pair form a quasicompoenent.
> If you would discuss on that I'd thank you. > > "We are accustomed to thinking about the density of primes continuously > decreasing, but if we think about perfect squares as being the > 'backbone' of the number line  if we count primes in relation to these > regular markers  we can turn this conception on its head and state: > The number of primes is increasing with successive quadratic intervals >  and at about the same rate as the quadratic intervals. (Refer to the > chart and data if you doubt this.) " > > Oh, then where is that as for primes, the exponents, or here as > squares to exponents? > > Seems as marking for arcs the edge/compass construction. > >
chuckle
It is consistent that the parallel postulate is false because the state of nonintersecting, equidistant lines at infinity cannot be known.
On the other hand, find a person who does not operate in the world assuming that the objects of experience do not translate rigidly.
You should not throw away that compass and straightedge just yet. It works fine at every cocktail party to which physicists are not invited. There is an inherent statistical bias associated with area because it is nonlinear. In other words, you cannot make the equations work because of "incommensurables". Numbers, however, can be compared with linear segments. And, Pythagorean triples relate lengths through quadratics rather than areas. So, if a physicist does show up to a party, put the compass away.
:)



