First published June 26, 2014 with major revisions and additions through July 29th.
Special thanks to Shafiq U. Ahmed for his encouragement and advice.
I know, I know: Statement 1 appears to be banal, and Statement 2 seems to be wishfulthinking.
What if I added to this a third statement that can be shown empirically to be true for any known twinpair prime?
To prove Statement 2, begin
with the following elaboration of Statement 1:
I will now provide a proof by
induction that of the infinite set of natural numbers
X1 and X+1 (which can be compositecomposite, compositeprime, primecomposite, or primeprime) there is an infinite subset of X1 and X+1 that are primeprime:
Therefore, "There are
infinitely many primes p such that p + 2 is also
prime".
To prove Statement 3, begin
with the following rules for an algorithm:
I admit that it does not look like much. However, this algorithm can find twin primes very fast indeed and without using a primality test:
 The outer loop takes care of Premises 1 and 2.
 The inner loop takes care of Premise 3.
Note that the use of an efficient primality test, such as MillerRabin, may prove to be useful when the smallest prime factor of sqrt(nn) is not relatively trivial and exceeds a certain magnitude. I cannot characterize this threshold yet.
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The base case of the proof  the
elaborated Statement 1  says that the
X1 and X+1 factors can be one of the following
pairs (and in no particular order):
X1 is
composite, X+1 is prime
X1 is prime, X+1 is prime
X1 is prime, X+1 is composite
X1 is composite, X+1 is composite
The following four successive values
of X^21 illustrate the four possible
combinations of composite and prime, with red
being prime numbers:
(X^2)1

(X1)^2

(X+1)^2

(X^2)1
 (X1)^2

(X+1)^2  (X^2)1

X1

X+1

783

729

841

54

58

27

29

899

841

900

58

60

29

31

1023

961

1089

62

64

31

33

1155

1089

1369

66

68

33

35

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.
To step back, correctly
understood, Statement 1 is saying by
implication that the twinprime conjecture is a
verisimilitude of the Infinitude of Primes and
the Fundamental Theorem of Arithmetic.
That the
twinprime observation ever acquired the
sobriquet conjecture is, actually, something of
a mystery unto itself. Why it became established as
"one of the great open questions in number theory" most likely lies in
its very late "discovery", in the mid19th century.
One can imagine that Euclid 
if he was the first person to realize the Infinitude of Primes  as a geometer
saw that twin primes are an artifact of square numbers and would not have thought
it an observation worth making. If he had made the observation then, it would not have been the "mountain out of a molehill" that
modern mathematicians have made of it.
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Now what I've told you does
not seem to be all that earthshattering, does
it?
However, it turns out that
this is the whole story as to where you'll find
twin primes. The frequency is subject to the
rapid expansion of square numbers and the
gradual thinning of the prime numbers. The
distribution is subject to the coincident points
where the X1 and X+1 factors are also prime
numbers.
 We know that the perfect
squares are growing exponentially.
and
 We know that the frequency of the primes is
diminishing asymptotically.
We infer that these two
artifacts of the number line can cause the
appearance of an apparently chaotic coincidence
of the prime factors for square numbers minus 1.
This distribution suggests to those who assume
complexity without testing their premises that
proving the twinprime conjecture is difficult 
when it is, essentially, selfevident.
I suggest that solving the Rubik's cube is a
much harder proposition than this problem could
be in a multitude of arithmetic universes. Simple
things can appear complicated, and complicated
things can appear simple. This is most
certainly in the former category.
Your comments, suggestions,
objections, and counterarguments are most
welcome on Quora.
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.
There is one very nice
practical result of this proof that is
selfevidently correct and that also empirically
proves it. This is simply that you can
infinitely traverse the number line in
polynomial time (using only the operations of addition, subtraction, multiplication, and nonnegative integer exponents) to find an infinite number of
twinned primes (as illustrated in the Excel VBA code example).
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There is a ton of empirical
evidence on this site, since 2007, that
illustrates the general distribution of twin
primes  and twinned primes  is predicated on
and predicted by the geometric expansion of
perfect squares: the "quadratic interval".
In truth, the number line is
neither a line nor is it straight. Twinprime
and twinnedprime distribution is unbounded.
The architecture of numbers does not enter a
parallel universe, understood only by the
selfselected few, when they reach a certain
outlandish magnitude. We do not require our
mathematicians to count "how many angels can
dance on the point of a needle".
Let's begin with a thought
experiment...
“Go
forth and multiply”
.
  
     
     
     
     
   
Monty Hall Problem for Dummies
No other mathematical puzzle
has produced more heated arguments and more
misunderstanding than the Monty Hall Problem.
It's tempting to call it deceptively simple, but
the truth is it's deceptively complex. In fact,
I can simplify its essence to one question: If
you were handed two decks of cards, one with
2/3 aces and the other with 2/3 jokers, and
were asked to pick an ace, from which pack
would you draw the card? If
that is not a sufficient clue, you need to
read this.
Factor
Families
A factor family is a set of
composites that share common factors. The family
begins with a semiprime, and all members of the
set share the two prime factors of the
semiprime. The family can be extended
indefinitely. For example, for the first 100,000
odd composites the factor family of 4619
(149*31) looks like this:
There are a few things to note
about this representation: The prime factors (in
red) are distinct, so, for example, 3 can be 3*3
or 3*3*3, and so on. The complete factors
(nonprime and prime in blue) of each composite
all have three common factors: the semiprime and
its prime factors. By definition, a semiprime
cannot have nonprime factors.
Download Excel
worksheet: Factor
Families (1.5
MB)
Modal Distribution of
Distinct Prime Factors:
It’s Not Necessarily What You Might Think
Yes, you can find the answers
to life's most persistent questions here! What
is the cumulative distribution of distinct prime
factors? The answer is both what you were
expecting and more than you were expecting. As
expected, the cumulative frequency of unique
prime factors diminishes with size, so that 2 is
the most frequent factor, 3 is the secondmost
frequent factor, 5 the third, and so on for all
prime factors. However, it turns out that the
cumulative frequency of the largest
prime factors  the biggest factor of each
composite  has a modal distribution that is increasing
at a slow rate approximating the cube root of N.
Up to 10,000, this mode is 19. Up to 1,000,000,
this mode is 73. It's fairly safe to assume that
if we graph any given set of numbers, there will
be a distinctive peak near P^3 = N.
Distribution of the
largest prime factor for each composite less
than 10,000. (The difference
between the even and odd curves is accounted
for entirely by the exclusion of prime
numbers.)
Download Excel
worksheet: Distinct Factor Analysis
(macroenabled) (30KB)
Paired,
Squared, Cubed, and Primed...?
If we pick a square or a cube
and count up to the nearest prime number, what
are the odds that a prime number will also exist
if we count up by the same amount from its root?
The answer, it turns out, is surprisingly high 
and even higher for cubes than for squares. If
the question sounds confusing, it's easy to
illustrate:
Squared example:
Square 
Prime 
Offset 
Root 
Prime 
Offset 
3025 
3037 
+12 
55 
67 
+12 
Cubed example:
Cube 
Prime 
Offset 
Root 
Prime 
Offset 
79507 
79531 
+24 
43 
67 
+24 
The results are:
 For squared numbers up to 1
million, this relationship holds true 41.0%
of the time (410 out of 1,000).
 For cubed numbers up to 1
billion, this relationship holds true 59.8%
of the time (598 out of 1,000).
Download CSV files: Squared1000
(30KB) and Cubed1000
(35KB)
 For squared numbers up to
100 million, this relationship holds true 31.4%
of the time (3,143 out of 10,000).
 For cubed numbers up to 1
trillion, this relationship holds true 44.7%
of the time (4,472 out of 10,000).
Download CSV files: Squared10000
(351KB) and Cubed10000
(425KB)
What has 6
got to do with it?
Here is a little observation:
The difference between the first
and fourth prime number of a
proximateprime polynomial is
ALWAYS A MULTIPLE OF 6.
For example:
Prox. Prime Poly.

1st Term

4th Term

4t1t

7th Term

7t1t

10th Term

10t1t

13th Term

13t1t

n^2 + n
+ 10157

10159

10177

18

10213

54

10267

108

10339

180

n^2  n
+ 10331

10331

10343

12

10373

42

10421

90

10487

156

2n^2 
2n + 10627

10627

10651

24

10711

84

10807

180

10939

312

n^2
 n + 11777

11777

11789

12

11819

42

11867

90

11933

156

n^2  n
+ 12107

12107

12119

12

12149

42

12197

90

12263

156

2n^2 
2n + 12277

12277

12301

24

12361

84

12457

180

12589

312

2n^2 
2n + 12409

12409

12433

24

12493

84

12589

180

12721

312

3n^2 
3n + 12653

12653

12689

36

12779

126

12923

270

13121

468

n^2 + 5n
+ 12785

12791

12821

30

12869

78

12935

144

13019

228

n^2 + n
+ 12887

12889

12907

18

12943

54

12997

108

13069

180

Surprisingly, this divisibility
by 6 does not stop with the fourth term. It
recurs with the polynomial's 7th term, 10th
term, 13th term, and so on ad infinitum.
Lots more data available: PPPs
< 50000, t1  t15 ("T" denotes each t
value divisible by 6) Download (115KB)
Do
you know why?
Factoring in Polynomial
Time: A Pronic Solution...
Sometimes the best things
in life are free  well, almost free... and
very simple... and blindingly obvious. A single
GCD
calculation using the closest pronic
number to N will produce a factor for
onethird of composites not divisible by 2 or
5 up to any size. For example, the
nearest pronic to 898097881 is 898110992, and
these numbers share a GCD of 1873  a prime
factor of both numbers.
An analysis of N <
10,000,000 shows that 35.8% of the nonobvious
composites are factorable with a single GCD
calculation. What is the common characteristic
of this huge class of numbers? They appear to
conform to rational
angles in the Sacks number spiral. Such
numbers can be generated with RadiusTest
using the Lines option and produce polynomials
with a third coefficient of 0.
Expanding the GCD calculation to
pronic numbers within 6 quadratic intervals of N
provides a nearly instantaneous factorization
test for more than twothirds of composites
ending in 1, 3, 7, or 9 regardless of magnitude.
Here is an analysis of composites less than 10^{7}
with GCDs calculated for pronics from 6
quadratic intervals less than N through 6
quadratic intervals greater than N. It shows a
74.4% success rate.
Analysis
for N<10^{7} Download
(11MB)
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