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From: Archimedes Plutonium on 11 Aug 2010 03:56
Let me talk about the rational way of going about discovering the
proof of the Infinitude of Twin Primes. In my own case, I noticed a
symmetry circa 1991 that W+1 and W-1 in multiply the lot and add or
subtract 1 is a twin primes. So that steered me to the proof.

And then it was the reasoning that all I needed was that W+1 was
"necessarily prime".

But that meant that nearly every proof attempt of Euclid's venerated
ancient proof Indirect
Method was wrong. That most every professor of mathematics could not
do a valid Infinitude
of Primes in the Euclid style. By Euclid style, I mean "multiply the
lot and add or subtract 1.

That was the rational way of going about proving the Infinitude of
Twin Primes.

But now a new set of questions looms over this feat. Now that we have
a proof of the Infinitude of Twin Primes, of Mersenne Primes, and of
most all the other infinitude proofs
using this indirect method, the question looms as to the Algebra of
the solution. What I mean
is that the Regular Primes is a more general set than the Twin Primes,
Polignac primes, perfect numbers primes, Mersenne primes, etc etc. So
the big Algebra question is whether
mathematics is so patterned that if a proof of Regular primes accrues
from the Euclid style of
proof, that Algebra should bust into the arena by saying that due to
Algebra that the Euclid
style of proof technique must work on these subset classes of
infinitude proof.

So if Euclid's method works for the general set of Regular Primes, it
must work, given a few
tinkerings, it must work for Twin, Polignac, Mersenne, perfect
numbers, etc etc.

So what I thought was the ultra rational approach of making a few
changes to Euclid's Regular Primes of using W-1 in addition to W+1 and
making them necessarily prime in the steps of the
proof. That really, I was not going down far enough into the
foundation of mathematics itself. If I had gone deeper, I should have
realized that if a general set has a proof via Euclid style that the
same technique insures a proof of Twin Primes. Now maybe Galois
Algebra has a fancy name for this concept.

And making a survey of mathematics proofs overall, it is true in the
majority of cases where a proof of a general set uses the same
technique to prove the less general subsets. Normally these are called
corollaries to the theorem. So that Twin primes would be a corollary
of Regular Primes.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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