A proximate-prime polynomial
is simply a quadratic equation - a finite polynomial of
the second degree - that is derived from four successive
(proximate, or neighboring) primes. Proximate-prime
polynomials are interesting because they exhibit much
greater prime densities than other polynomials.
When you graph
primes against an X-axis that treats the expanding
interval between successive perfect squares as a constant
unit subdivided into equal parts, you produce a
distinctive wave form for primes and prime factors.
For every composite number that is
not itself a perfect square there exists a pair of
nonconsecutive perfect squares whose difference is equal
to the composite. Even before we get to the subject of
factorization, the consequences of this observation are
fascinating and far-reaching.
It began with an
exploration of biquadratic paired primes: 2
primes separated by the equivalent of exactly 2
quadratic intervals.... Then the investigation took the
logical next level by asking the question: Are there
prime pairs that are separated by other, greater
multiples of the quadratic interval? And if there are,
what are the frequency characteristics by interval size
and perfect square offset? The results are in, with
charts, an Excel visualization, and masses of
Find examples throughout this site that demonstrate using
VBA code with worksheets and graphing - including
generating primes, perfect squares, and composites, doing
modular arithmetic, calculating GCDs, and more....
First published June 26, 2014 with major revisions and additions through August 12.
Special thanks to Shafiq U. Ahmed for his encouragement and advice.
A logical proof of the Twin-Prime
"There are infinitely many
primes p such that p + 2 is also prime"
A natural number N of form X^2-1 where X>3
can be expressed as the product of two factors:
X-1 and X+1.
Of the infinite set of Ns of a form X^2-1, there
is an infinite subset for which X-1 and X+1 are
The first argument is an
arithmetic property that does not require
elaboration. It states that there is an infinite
set of square numbers minus one that are the
product of two factors, composite or prime.
The second argument states that there is an
infinite subset of square numbers minus one that
are the product of two prime factors (semiprime).
The second argument is correct using the following
logic: Of the set of X^2-1 there is a subset for
which both factors are prime. (To be clear, X-1
and X+1 can also be composite - one or the other
or both. Although a deterministic proof of
Statement 2 cannot be provided yet, it is proved
by the Infinitude of Primes and the Fundamental
Theorem of Arithmetic. Using axiomatic set theory,
both the set and the subset must be infinite.
I know, I know: Statement 1 appears to be banal, and Statement 2 seems to be wishful-thinking.
What if I added a third statement just to be perfectly clear about how simple the distribution of twin primes is?
All Ps and P+2s are of the form - and only of the form - described in Statement 1.
However, I have learned that academic mathematicians need more proof than commonsense logic can
provide. This is because they are concerned about infinity. I will therefore attempt to prove in my naive fashion Statement 2 by induction and Statement 3 by algorithm.
The algorithms are interesting because they reveal the obvious truth that finding twin primes, by virtue of their multiplicative products (semiprimes) having a single position in relation to perfect squares and the quadratic interval, is subpolynomial (nearly linear actually).
This is followed by what I like to think is my best attempt so far: a proof in the style of Euclid's proof of the infinitude of primes. I do this in the forlorn hope that it will spur a truly mathematical mind to find the simple solution to this problem, rather than the most complicated and obscure!
I will now provide a proof by
induction that of the infinite set of natural numbers
X-1 and X+1 (which can be composite-composite, composite-prime, prime-composite, or prime-prime) there is an infinite subset of X-1 and X+1 that are prime-prime:
If a natural number N of the form X^2-1 is an odd number, there
exists an even-number difference with the
previous square number ("perfect square").
There is an infinitude of such Ns.
All prime numbers can be expressed as half an
All prime numbers can be expressed as half the
difference between an odd N of the form X^2-1 and the previous perfect
There are an infinitude of primes such that P =
X^2 +Y^2 (Fermat's theorem on the sums of two
squares), and there is an infinitude of primes
such that P = X^2 +Y^2 (Pythagorean primes)
where X = Y-1.
The Ns of Step IV are of the same form as the
subset of the Pythagorean primes where X=Y-1.
It follows that there are an infinitude of
primes of which Step IV and Step V are true.
Thus, there must be an infinitude of Ns of
the form X^2-1 for which the factors X-1 and X+1
are both prime.
Therefore, "There are
infinitely many primes p such that p + 2 is also
To step back, correctly
understood, Statement 1 is saying by
implication that the twin-prime conjecture is a
verisimilitude of the Infinitude of Primes and
the Fundamental Theorem of Arithmetic.
twin-prime 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 mid-19th 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.
In the following I will attempt to state the infinitude of twin primes as Euclid might have stated it (or at least in the same spirit).
Let's first agree on what the Proof of the Infinitude of Primes says using, according to the Prime Pages, wording "closer to that which Euclid wrote, but still using our modern concepts of numbers and proof". (I have changed P to X for clarity.)
There are more primes than found in any finite list of primes.
Call the primes in our finite list p1, p2, ..., pr. Let X be any common multiple of these primes plus one (for example, X =p1p2...pr+1). Now X is either prime or it is not. If it is prime, then X is a prime that was not in our list. If X is not prime, then it is divisible by some prime, call it p. Notice p cannot be any of p1, p2, ..., pr, otherwise p would divide 1, which is impossible. So this prime p is some prime that was not in our original list. Either way, the original list was incomplete.
Allow me now to make an equivalent statement for the Infinitude of Twin Primes:
There are more twin primes than found in any finite list of twin primes.
Call the primes in our finite list p1, p2, ..., pr of which there are at least two of the form p and p+2. Let X be any number of the form N^2-1 that is the product of one or more of these primes. Now X is either the semiprime of p and p+2, or it is not. If X is not semiprime, then it is divisible by some factor, call it f, that is the square root of the next odd square number. When all the primes less than p and p+2 are exhausted there will be another N^2-1 that is p*p+2 where p+ 2 is the prime square root of the next odd perfect square. Either way, the original list of primes that are twinned was incomplete.
I will now state a Euclidean-type full Proof of the Twin Prime Conjecture as follows:
Consider any finite list of square numbers ("perfect squares").
By the method of common differences we know that this growth is quadratic with a 2nd degree common difference.
Thus, a polynomial describing this growth contains as its first term a square X2.
Therefore, the factors of the perfect squares [a] must also be growing by 2.
Thus, we see that the highest factor of the composite immediately preceding an even perfect square is the square root of the next odd perfect square.
Further, we know that these common factors, prime or composite, are an infinite set whose members are every odd number.
Because we know that every odd number is a member of this set and because the common difference of quadratic growth is 2, for every composite of the form X2-1 the two largest factors (prime or composite) must be X-1 and X+1. [b]
Therefore, when the next square (X + 1)2 is semiprime, its prime root is one prime factor of the previous square X2-1. [b]
Thus, when all the composites less than X2-1 are factored, X2-1 is the semiprime for which said prime root is one factor.
Therefore, this X2-1 is the product of two prime factors, one less than X, which is X-1, and one more than X, which is X+1.
Further, Steps VIII and IX must recur with the infinite occurrence of prime square roots [c] to produce an infinite set of semiprimes X^2-1.
Therefore, "There are infinitely many primes p such that p + 2 is also prime".
[a] The members of the set of odd natural numbers from 3 to the X of the given X2.
[b] As proved by the Fundamental Theorem of Arithmetic
[c] As proved by the Infinitude of Primes
The point of this is to illustrate the knot of numbers around the perfect squares that show the factors and their products (prime-semiprime, red; nonprime-composite, blue) that happen to be twinned (prime or composite). This is the basis for my algorithms.
Whereas a mathematician might think about the twin primes first, giving them primacy (after all, they are primes), I look at it from the perspective of the products first, using basic arithmetic and the unique factorization of every composite number. This is not an unreasonable point of view. Many fundamental number theorems, such as the granddaddy, Fermat's Little Theorem, exploit the properties of squares. Squares have properties that "square roots" do not have.
Now what I've told you does
not seem to be all that earth-shattering, does
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 X-1 and X+1 factors are also prime
- We know that the perfect
squares are growing exponentially.
- We know that the frequency of the primes is
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 twin-prime conjecture is difficult -
when it is, essentially, self-evident.
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.
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. Twin-prime
and twinned-prime distribution is unbounded.
The architecture of numbers does not enter a
parallel universe, understood only by the
self-selected 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".
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
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.
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 second-most
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
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
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).
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
calculation using the closest pronic
number to N will produce a factor for
one-third 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 two-thirds of composites
ending in 1, 3, 7, or 9 regardless of magnitude.
Here is an analysis of composites less than 107
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.
Desktop program and complete project source code
for implementing the gold standard in primality
testing. A fast and reliable test for numbers up
to 1027-1 (that's 1 with 26 9s - a
(The project illustrates how to use a legacy
language, VB6, not designed for big integers. It
includes modular exponentiation code by DI Management Cryptography
"Fermatic" is a made-up word: Fermat + Automatic.
This tool takes Fermat's great theorem
to the limit, with some experiments to weed out
pesky pseudoprimes. Rapidly generate prime,
pseudoprime, and composite data.
Enter 3 or 4 numbers in a sequence and find out
what the next 10, next 1,000, or next 10,000
values are. QTest lets you derive a quadratic
equation from the values you input (and solve the
equation's roots). Then you can use this
polynomial to generate and analyze long number
sequences for primality. (See Robert Sacks' method
for quadratic derivation, used
Calculate the products of infinite series using
almost any inputs you can think of. Generates real
number zeta series, demonstrating the calculation
of many important constants - including e, pi, and
phi, the Basel equality and Apéry's constant.
Tools that are digitally signed
are published by: Wordwise Solutions
site is dedicated to exploring
proposes that the universe is the way it is
neither by accident nor by design. The axioms
of mathematics, the laws of physics are what
they are innately and without reference to
percepted existent reality. The intelligence
inheres in and is nonreferentially existent
without regard to the observer. Undesigned
intelligence is uncreated.