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Topic: Proof that Twin-Primes is an infinite set; Polignac Conjecture proof
#1548 Correcting Math

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 plutonium.archimedes@gmail.com Posts: 8,734 Registered: 3/31/08
Proof that Twin-Primes is an infinite set; Polignac Conjecture proof
#1548 Correcting Math

Posted: Feb 12, 2014 11:51 PM
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Proof that Twin-Primes is an infinite set; Polignac Conjecture proof #1548 Correcting Math

I did these proofs several years back, if my memory is correct, after I discovered that infinity borderline is 1*10^603 and the technique I used was a gauge measure rod technique.

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Proof of the Infinitude of Twin Primes
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When mathematics is honest about its definitions of finite versus infinite, it seeks a borderline between the two concepts, otherwise they are just one concept. From several proofs of regular polyhedra and of the tractrix versus circle area we find the borderline to be 1*10^603. That causes a measuring rod to use for all questions of sets as to whether they are finite or infinite. The Naturals are infinite because there are exactly 1*10^603 inside of 1*10^603 (not counting 0).
The algebraic-closure of numbers is 1*10^1206 which forms a density measure. So are the twin-primes finite or infinite? Well, are there 1*10^603 twin-primes between 0 and 1*10^1206? The question is related to the minimal infinitude set of Perfect Squares. The Perfect-Squares {1, 4, 9, 16, 25, . .} are the minimal infinite set. The perfect-squares are a special set since they are "minimal infinite" since there are exactly 1*10^603 of them from 0 to 1*10^1206.

How about Twin-Primes? Well, what we do is make a induction count.

There are 15 twin primes from 0 to 100.
There are 14 more twin primes or 29 twin primes from 0 to 200.
There are 8 more twin primes or 37 twin primes from 0 to 300.

So the series progression for Twin-Primes from 0 to 300 is that of:

15, 29, 37, . .

There are 10 Perfect Squares from 0 to 100.
There are 4 more Perfect Squares or 14 Perfect Squares from 0 to 200.
There are 3 more Perfect Squares or 17 Perfect Squares from 0 to 300.

So the series progression for Perfect Squares from 0 to 300 is that of:

10, 14, 17, . .

Obviously the Twin Primes are always ahead of the Perfect Squares, and hence the Twin Primes are an infinite set.

QED

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Proof of the Polignac Conjecture up to a special number
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How about quad-primes, those separated by 4 units, then those separated by 6 units, then those separated by 8 units, etc etc? This is called the Polignac Conjecture.

Well, the proof is the same as Twin Primes, except, however, at some moment or point between 0 and 1*10^1206, the Polignac Primes will be a finite set, since the confinement of the borderline of infinity impacts the set of Polignac primes. So here is the question? What prime separation distance do the Polignac Primes turn from being an infinite set to that of a finite set?

For Polignac primes, we do a induction count.

There are 13 quad primes from 0 to 100.
There are 10 more quad primes or 23 quad primes from 0 to 200.
There are 8 more quad primes or 31 quad primes from 0 to 300.

So the series progression for Quad-Primes from 0 to 300 is that of:

13, 23, 31, . .

Now for Polignac primes separated by 6, then by 8 then by 10 etc etc, may start out slow from 0 to 100 then from 0 to 1000 then from 0 to 10,000 etc etc but they all come in more numerous than their counterpart-- the Perfect Squares. This happens up until some special separation distance where the distance is so large that the number of them falls below the count of the Perfect Squares. So the Polignac Conjecture is true up to this special number, that they are infinite sets until that special number is reached.

There are 10 Perfect Squares from 0 to 100.
There are 4 more Perfect Squares or 14 Perfect Squares from 0 to 200.
There are 3 more Perfect Squares or 17 Perfect Squares from 0 to 300.

So the series progression for Perfect Squares from 0 to 300 is that of:

10, 14, 17, . .

Obviously the Quad Primes are always ahead of the Perfect Squares, and hence the Quad Primes are an infinite set.

The Polignac Conjecture is true up until that special separation distance is reached since infinity borderline is 1*10^603 and there must be 1*10^603 of an entity between 0 to 1*10^1206 in order to be an infinite set.

QED

--

Recently I re-opened the old newsgroup of 1990s and there one can read my recent posts without the hassle of mockers and hatemongers.

https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe

Archimedes Plutonium

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