Pull back the curtain on

exploring the undesigned
intelligence of the numberverse

lists/generators: Perfect Squares, Composites, Primes


Larger Paired-Prime Gaps

I'm not sure why you need a team of mathematicians "to close the gap". There is a great deal I do not understand about Yitang Zhang's work; I have the sense it's a problem of interpretation when explained to others. However, I used to be interested in the subject of large paired-prime gaps: "Exhibit A" and "Exhibit B"

Another Landau Problem

Legendre's conjecture:
Deceptively easy or difficult?

"...out all day long.
All day long,
it goes on and on"

The First 1,000,000...

Lists of Perfect Squares

Lists of Composites

Lists of Primes

Generate Your Own Up to 1027...

Perfect Square Generator

Prime Factors Generator

Prime Generator

Experimental Calculators

ZetaTest: Infinite series

MinusSquare: Factoring

FermaticTest: Probable prime

QTest: Quadratic Series

RadiusTest: Spiral lines and curves

PrimeTest: Analysis

Miller-Rabin: Demonstration

Secrets of the Spiral

Explore the Sacks Number Spiral


Proximate-Prime Polynomials

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.

The high primality of prime-derived quadratic sequences

What is a "perfect prime polynomial"?

What is polynomial pronic alignment?

Use an Excel spreadsheet to explore polynomial primes!


Catch the Wave...

The Wave

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.

Try out an ingenious Excel worksheet to see the wave!

Counting primes by quadratic interval


Fixing Fermat’s Factorization Method

How do you make Fermat's method of factorization faster than trial division...?


The Magic Square of Subtraction...

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.

Part 1 - A 'Classic Discovery' 
Part 2 - A Deterministic Test


Q-Paired Primes...

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 half-digested data...!

Read about the latest findings

What are Biquadratic Paired Primes?

Two "new" rules of perfect squares


How I learned not to be afraid of big numbers

It's a question of magnitude...


Using Excel as a tool for number theory

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....

June 26 - August 12, 2014

A logical proof of the Twin-Prime Conjecture

"There are infinitely many primes p such that p + 2 is also prime."

Or, you could say it this way:

"There are infinitely many semiprimes of the form X2-1."

I provide the following "starter proofs", with the knowledge that they are not formal or complete.

And, last but not least,

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All these attempted proofs share a statement of arithmetic fact, as follows:

If X2 - 1 is semiprime then X - 1 and X + 1 are its prime factors.

To unpack the meaning of this:

If X2 - 1 is semiprime

(12 * 12) - 1 = 143

then X - 1 and X + 1 are its prime factors

11, 13

That is, P and P + 2.

Dear Reader, if you understand the arithmetic, please continue....

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A logical proof begins with two sentences:

Statement 1
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.

Statement 2
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 prime factors.

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.) 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?

Statement 3
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!

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First published July 19, 2014 with subsequent revisions

To prove Statement 2, begin with the following elaboration of Statement 1:

If a natural number of the form X^2-1 is greater than (X-1)^2 / 2 by a prime and less than (X+1)^2 / 2 by a prime then X-1 and X+1 are twin primes.

What does this really mean?

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 even number.

All prime numbers can be expressed as half the difference between an odd N of the form X^2-1 and the previous perfect square.

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 prime".

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First published July 29 (added "A la Fermat" August 12) 2014

To prove Statement 3, begin with the following huge "cheats" for an algorithm:

1. You only need to look at a SINGLE pair of Ns for EACH quadratic interval.
2. This pair can only be in ONE place relative to the square of the even number between them.

From these premises we can demonstrate two distinctive methods: "Trivial division" or "A la Fermat".

This algorithm's efficiency exploits the easy factoring of X^2-X*2. A fast primality test can be used when the first factor is not trivial.

This algorithm has near-linear scalability - restricted only by the efficiency of a fast primality test, such as Miller-Rabin.

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The base case of the proof - the elaborated Statement 1 - says that the X-1 and X+1 factors can be one of the following pairs (and in no particular order):

X-1 is composite, X+1 is prime
X-1 is prime, X+1 is prime
X-1 is prime, X+1 is composite
X-1 is composite, X+1 is composite

The following four successive values of X^2-1 illustrate the four possible combinations of composite and prime, with red being prime numbers:




- (X-1)^2

- (X^2)-1































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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.

That the 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.

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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 p1p2, ..., 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 p1p2, ..., 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.

Suggested Proof

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.

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First published July 31, 2014

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

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To clarify Steps I, II, and V, take this example:

Consider any finite list of square numbers ("perfect squares").

4, 9, 16, 25, 36, 49

By the method of common differences we know that this growth is quadratic.

4, 9, 16, 25, 36, 49

      5, 7, 9, 11, 13

        2, 2, 2, 2

(To quote and credit Purplemath: Finding the Next Number in a Sequence: The Method of Common Differences: "Since these values, the "second differences", are all the same value, then I can stop. It isn't important what the second difference is (in this case, "2"); what is important is that the second differences are the same, because this tells me that the polynomial for this sequence of values is a quadratic.")

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.

99 = 3 * 3 * 11
121 = 11 * 11

143 = 11 * 13
169 = 13 * 13

195 = 3 * 5 * 13 = (13 * 15)
225 = 3 * 3 * 5 * 5 = (15 * 15)

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Still Don't Get It?

Let's try a picture...

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.

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Now what I've told you does not seem to be all that earth-shattering, 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 X-1 and X+1 factors are also prime numbers.

- We know that the perfect squares are growing quadratically.
- 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 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.

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 self-evidently 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 non-negative 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. 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".

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Recommended sites:

Number Spiral

Number Spirals

Divisor Plot

Interesting Graphing Applets

© 2007-2014 Michael M. Ross