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Topic: A sequence of primes with a fixed-point property based on a subsequence
of 1/n

Replies: 2   Last Post: Feb 3, 2014 4:24 AM

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fom

Posts: 1,968
Registered: 12/4/12
Re: A sequence of primes with a fixed-point property based on a subsequence
of 1/n

Posted: Feb 3, 2014 4:24 AM
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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.rwth-aachen.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 non-intersecting,
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 non-linear.
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.

:-)






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