Pull back the curtain on

exploring the undesigned
intelligence of the numberverse

lists/generators: Perfect Squares, Composites, Primes


About This Site
Natural Numbers is my personal exploration into number theory. It is autodidactic, and it is iconoclastic. It is factual, and it is speculative. It is not recommended for footnotes!
- Michael M. Ross

A thought experiment...
"Go forth and multiply"

It's out there...
The Twin-Prime Conjecture
"My personal proofs"

The numbers are here...

"...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....

First published October 20, 2014

"Legendre's by Lego": It's Elementary!

There is a simple geometric principle that demonstrates at least one prime exists between every perfect square interval. It rests upon the most elementary geometry of rectangular shapes. Imagine you're making squares with counters or Lego pieces or whatever. Begin with a 2-by-2 square. You're allowed to put down only one piece at a time by adding each one to the existing shape. So you're first task is to get from 2-by-2 to 3-by-3. Can you do this while keeping the shape a rectangle? You can change the shape with each new piece you put down but only to another rectangular shape. (Don't get creative: You must keep the distance between each adjacent piece the same.) Let's just try it:

It becomes clear right away that with certain odd numbers you cannot keep the shape rectangular. These are prime numbers. The rule is simple: You cannot add counters to make one perfect square into the next bigger perfect square - 2*2 to 3*3 or 10*10 to 11*11 - without breaking the rectangular symmetry at least once (actually, twice or more.)

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First published October 12, 2014

A Formula for Counting Primes in Quadratic Intervals

Yes. Why not?

1. Take the difference of two consecutive perfect squares.
2. Divide the difference by 2.
3. Divide this result by 3 and subtract the quotient from the result.
4. Repeat Step 3 for every prime less than or equal to the difference.
5. Compare the final result with the actual prime count in the interval.



Take the square interval 81 - 100 = 19.

Apply division and subtraction as follows:

19 / 2 = 9.5
9.5 / 3 = 3.17
6.33 / 5 = 1.27
5.06 / 7 = 0.72
4.34 / 11 = 0.39
3.95 / 13 = 0.3
3.65 / 17 = 0.21
3.44 / 19 = 0.18

There are three primes between 81 and 100: 83, 89, 97.

Why does this work?


a. The frequency of every prime factor can be considered a fraction.
b. The frequency of 2 is 1/2, the frequency 3 is 1/3, the frequency of 5 is 1/5, and so on.
c. Repeat subtraction of these fractional amounts eliminates all the composites.
d. The remainder must represent the integers that are prime.

I've created a small program (for Windows) to demonstrate the algorithm. Download (9KB)


It's only a minor mental step from this formula to proving Legendre's Conjecture.

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First published September 10, 2014

Are the "Landau problems" really one problem?

Four green bottles hanging on the wall,
Four green bottles hanging on the wall,
And if one green bottle should accidentally fall,
There'd be three green bottles hanging on the wall.

The Landau problems are four conjectures of number theory that were stated by Edmund Landau to be "unattackable at the present state of science" 102 years ago.

One thing can be said for certain about Landau's problems: What a sad state of affairs! What other scientific discipline would tolerate the humiliation of such a situation for 102 years? The working hypotheses must be that all four are true based upon all the available evidence. Yet for number theorists, a hypothesis is no more than a hunch. These four conjectures are in the deadbeat category of unproved, and they could remain there for infinity employing many able mathematicians.

Why? Because academic mathematicians are concerned with infinity, and a proof must be deterministically true to infinity. Such proofs are extremely elusive, if not impossible, for any conjecture that depends on prime number distribution that is more strictly defined than asymptotically. Because academic mathematicians do not have deterministic knowledge about prime numbers, the Prime Number Theorem, also called the "asymptotic law of the distribution of prime numbers" is the best they can do: a trend that's proven but still probabilistic - and, honestly, not that interesting. (Even the holy grail of deterministically proving the Riemann Hypothesis will only refine this trend. Despite the enormous amount of attention the RH gets, it's essentially geeky. It will not provide a deterministic solution to the prediction of prime numbers. It will simply refine the PNT.)

Yet we know, from very first counting principles, that every prime number is accountable for merely by the rules of arithmetic. It's only that mathematicians will never be granted the computational power to apply those rules of arithmetic to infinity - or even close! Do you remember that one about the permutations of a standard deck of playing cards? The factorial of 52. If we consider each playing card to be a prime number, those primes would account for 52! composites. That is 80658175170943878571660636856403766975289505440883277824000000000000 different composites. Imagine counting them. (By comparison, the age of the universe in seconds is about 432043200000000000.) Well you get the idea.

However, what really makes the Landau problems problematic is not the limitations of computational power, but the limitations of number theory. Number theory, or higher arithmetic, is the study of integers. It's a collection of "magic tricks" that enable mathematicians to manipulate and understand big numbers more easily. However, there appears to be one thing that's missing from that box of tricks: A means to relate the quadratic growth of square numbers with the asymptotic growth of prime numbers.

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My thumbnail thesis is that starting about 150 years ago, academic mathematics began to lose touch with the bedrock principles of the Infinitude of Primes and the Fundamental Theorem of Arithmetic out of lack of things to do. About the same time, some overlooked classical problems also reemerged, such as the twin-prime observation. Because of their late arrival, the Landau problems were caught between Euler's ordinal universe and Cantor's cardinal universe. Then add to that the parallel universes of complex numbers and planes, combinatorics and topologies, probability and statistics, computer logic and algebra - the whole panoply of the mathematical sciences.

I suggest that the Landau problems are really one problem. The inability to call them proven with the current standard of academic rigor - despite their obvious, overwhelming, likelihood of being true - is because of the following:

1. They've stood for over a century, so it's assumed they must be intractable.
2. They are not particularly attractive subjects for academic mathematicians (particularly if sticking one's neck out can result in ridicule).
3. They are not accessible to the standard alchemy of number theory.

I add to these three two other reasons of the cosmic joke variety:

4. One of these is quite possibly provable by arithmetic or algebra without needing any advanced concepts or techniques. *
5. One of these does not need to be proved true. (In fact, it needs to proved false.) *

This has created a logjam. Things that are easily observable - and for which a possibility of error approaches the infinitesimally tiny - are ignored. And yet, if you step back it's quite silly to say that you cannot admit something as true until it is proved that not a single exception to it will be found until infinity. That's an absurdist point of view if you think about it. Human beings are heuristic beings, and even mathematicians are not permitted firsthand knowledge of infinity. We have to imagine it.

It seems to me that some, not all, mathematicians are not pattern-seekers but think atomistically and linearly. That's fine - if there is authenticity (and even originality)! For me, there are more interesting ways to understand the natural numbers than the classical model in which they march, like little tin soldiers, to infinity on a line. My mental-model is cyclicality around a geometric backbone.

I subscribe to the philosophy of holism (from the Greek holos). My only faith is that natural systems taken as a whole have properties that cannot be understood only in terms of their component parts. Thus, "The whole exists prior to and is more elementary than its parts."

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* I refer to one and the same problem: Legendre's conjecture. It is clearly and demonstrably true - and has been since the time Legendre and Euler lived.

(Wikipedia states, matter-of-factly without irony and without attribution: "a counterexample near 10^18 would require a prime gap fifty million times the size of the average gap."

Such a counterexample will never be found. If it were found, the Prime Number Theorem would, ipso facto, be proved false.

In reality, the number of primes between perfect squares increases indefinitely and absolutely with minor perturbations that never exceed a known error limit because the quadratic growth of square numbers "more than negates" the decline in the density of primes. And I will explain why the PNT may be admitted as proof of this.

Because the three other Landau problems are closely interconnected with this one, if this one falls - that is, is stamped with a credible seal of a proof - the other, harder ones, become less intimidating and more vulnerable to attack (academic math lingo for being provable).

A common response a mathematician will make when presented with the facts of Legendre's conjecture is something like this: The PNT cannot be used to prove Legendre's conjecture because the PNT is an asymptotic law that's a discrete limit applied to all the numbers less than particular given value of N and not to any specific N2 and (N+1)2.

I'm not buying this, and I think it's fairly easy to show that this reflexive notion is false. However, regardless, I will not stay up at night worrying about the theoretical possibility that the vast sea of the asymptotic law of prime number distribution will be divided, in some miraculous fashion, to permit a square gap of a vast magnitude to be prime-free. ("It just ain't gonna happen.")

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Consider the possibility - unlikely as it is - that permitting a proof without perfect precision - might actually matter to some other branch of science?

This is, most likely, the weakest argument you will find on this whole page because we all know that science will use math that cannot be proved, when it needs to, with a working assumption that it is true. In fact, the natural sciences and even applied mathematics thrive and prosper on the back of uncertainty. Consider quantum physics! A much stronger argument is that no mathematical proof can be a philosophical proof because of at least three reasons: the theory of supersets, the theory of incompleteness, and the theory of complexity. Since philosophy is not the subject of this article, I will not explain why I choose these three in particular, but I'm not being very original.

I recognize the joy of an intellectual pursuit. However, the arcaneness - even alchemy - of the science that is being employed to solve the Landau problems - in particular, the daisy-chaining of obscure dependent theorems, theories, and other conjectures - can never produce an elegant proof that can be appreciated by the rest of us. And that's a sad state affairs because even the most esoteric physics is explicable conceptually to the averagely intelligent and curious. (I'm including general and special relativity, quantum mechanics, the big bang and inflation - but maybe not the Higgs boson :( - in this category.) Even that's not to say that obscurity is necessarily an impediment to popularity: Consider the proof of Fermat's Last Theorem.

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Legendre's Conjecture

"There is a prime number between n2 and (n+1)2 for every positive integer n."

I will double up on this conjecture with my own:

"Of the list of composite numbers between any interval n2 and (n+1)2 there is at least one semiprime 2y and one semiprime 3z?".

(I could go further and say the ratio of the real prime count to the quadratic interval is nearly linear, but I'm getting ahead of myself.)

Let's step back...

The Prime Number Theorem, without any of its elaborations, states: "The number of primes not exceeding x is asymptotic to x/log x." This answers the question: How many primes are there less than the number x? The answer is: Π(x) ~ x/log x, approximately. There is a reason that it's called the prime-counting function - that's how it was originally conceived (see the historical note).

What we care about with respect to Legendre's conjecture is not this overall trend, but the prime count, "projected" and real, between every two consecutive perfect squares - and, all-importantly, the proportional deviation between the projected and real counts as the number line progresses.

This question is not secondary to the PNT: It is actually, to my mind, the more important question. This is where I can happily state - without having the faintest knowledge of the existence of the conjecture we are discussing - that I asked and answered this question to my own satisfaction six or seven years ago, writing about it here in complete ignorance of academic number theory - like many other topics you will find on these pages...! A program and results are here: Counting primes by quadratic interval

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Let's break the monotony of this monologue with some light refreshment:

Contrast the real prime count with the "PNT-predicted" count *
This is not the graph you usually see about the PNT, which shows a step-like ascending pattern. These are plots of the PNT value for each interval (which creates a trend line) and the real count (which produces a not-unexpected "wave" pattern) due to noise in the error term. The real prime count exceeds the PNT limit in "most" intervals, and where it does not it falls short by a "marginal amount". This is the PNT error for each interval.

Left: The real prime count per quadratic interval versus the "asymptotic line".
Right: The deviation between the real and log-derived count for each interval.

* I recognize that the PNT does not actually predict a count for discrete intervals because it only defines a limit (a log-derived rate of growth). Nevertheless there is a formula that applies it to quadratic intervals: ((n+1)2/log((n+1)2)).
Furthermore, it is graph-able as a line that passes through those intervals, and, empirically, it gets within shooting distance of the discrete real data..

The PNT is a blunt instrument - and I don't like it very much - but it has its uses, whether intended for this purpose or not.

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I pause to note here the historical and ironical fact that Legendre, himself, was the first person to conjecture the size of the prime-counting function, in 1798 to be exact. (There was a revolution on, but he found time.) To quote from the Prime Pages: "The constant 1.08366 was based on his limited table for values of Π(x) (which only went to x = 400,000). In the long run 1 is a better choice..." ( This quote is suspiciously inauthentic though.)

(FYI I know that you know that the "pi symbol" in the prime-counting function, Π(x), has nothing to do with the other pi.)

This is, in fact, the conjecture that Legendre should be remembered for. And the limit he came up with increases my respect even more. Could the same individual who figured out something that precociously and with such accuracy actually have stated the trite conjecture that became one of Landau's problems?

It's not necessary to embark on the journey to infinity, because the arithmetic does not change with magnitude. The growth of the prime count continues to increase to infinity if the PNT continues to infinity. If one is true, the other is also. The delta between the unrefined prime-counting function and the actual prime-counting data is so tiny that it negates the possibility that its deviation could ever come close to challenging Legendre's conjecture. In fact, even entertaining the theoretical possibility of this occurring requires the repudiation of not one but two bedrock principles of mathematical science:

- The evidence: that is the known and observable data.
- The theory: that is the prime number theorem.

Logically, therefore, one could contend that Legendre's Conjecture does not require its own proof.

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However, to be as clear as possible, there is the reality of an informal "statistical proof" and the high-probability of an arithmetic proof.

Legendre's Conjecture is statistically true

First, scientists do not use statistical proof as a means to attain certainty, but to falsify claims and explain theory. Second, it conforms to the hypothetic-deductive model or method for defining the scientific method. Third, let me note nothing I'm saying here is in the least bit original (or well-said for that matter).. "It's all out there. Connect the dots."

Consider the gap, or interval, between two square numbers, X2 and (X+1)2 strictly as excluding the end-points (that is, the square numbers themselves), its proven empirically (for "small" numbers) and statistically (for "large" numbers) that there can be no cases where the unique prime count is less than two.

The prime number theorem states that X / log(X) is a good approximation of the prime-counting function. This can be refined by various methods. However, even in its most unrefined state, for numbers greater than 107 the error terms are tiny in comparison to the ratio of primes to all integers.

Given the truth of the prime number theorem, we know that the following formula shows that the prime count, within an error term, increases over successive quadratic intervals to infinity if PNT is correct.

((n+1)2/log((n+1)2)) - (n2/log(n2))

This is not a matter of dispute. The absolute prime count growth will continue if you plug in any number for which it is computationally possible to use the natural logarithm.

It is not necessary to invoke the Riemann Hypothesis to refine the error term. because the growth of the prime number count dwarfs it. For "small" numbers the empirical computed data is a piece of cake (my generated data < 108). For "large" numbers, we can use data that uses raw computational horsepower up to 1025 at present. Beyond this, we have rigorous proofs that use analytical continuation without the RH.

The real error term is far smaller than any deviation that could negate the effect of quadratic growth. For any computed magnitude greater than, say, 1010 the largest negative deviations are a fraction of one percentage point of the asymptotic line.

That's it: A statistical proof that "There is a prime number - in fact, at least two - between n2 and (n+1)2 for every positive integer n" without a single statistic.

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Two more pictures...

In the beginning, the real prime count expansion establishes a near-linear-ratio growth pattern with the square interval count (as projected by the PNT-derived interval count formula).

Left: A tiny plot of the early prime expansion.
Right: The linear growth pattern for my generated data < 108.

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Spinning up the data

"A Resplendent Rule"

"The Rule of 2 and 3": Two demonstrations

This one shows all the primes and all the 2*y and 3*z prime factors for each square interval:
- Column 1 and 2: The pair of squares
- Column 3: The primes in the pair
- Column 4: The 2* prime factors
- Column 5: The 3* prime factors

This particular Landau conjecture is true until proven false (which it simply cannot be).

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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 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 numbers.)

Download Excel worksheet: Distinct Factor Analysis (macro-enabled) (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: Squared-1000 (30KB) and Cubed-1000 (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: Squared-10000 (351KB) and Cubed-10000 (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 proximate-prime polynomial is ALWAYS A MULTIPLE OF 6.

For example:

Prox. Prime Poly.
1st Term
4th Term
7th Term
10th Term
13th Term
 n^2 + n + 10157 
 n^2 - n + 10331 
 2n^2 - 2n + 10627 
 n^2 - n + 11777 
 n^2 - n + 12107 
 2n^2 - 2n + 12277 
 2n^2 - 2n + 12409 
 3n^2 - 3n + 12653 
 n^2 + 5n + 12785 
 n^2 + n + 12887 

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 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.

Analysis for N<107 Download (11MB)

Previous Articles
Instantly factor a semiprime!
Reveal the DNA of semiprimes
The Q square of factoring


Tools for Testing Natural Numbers


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The Philosophy of Natural Numbers

If there is nothing there exists the possibility of something. With 0 and 1 there exists the constructs with which the number line and all the abstractions of mathematics are possible. Numbers are a perfect expression of the Platonic ideal. It would seem that any imagined universe would necessarily receive this preexistent knowledge just by virtue of its existence.

Undesigned Intelligence 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.





Recommended sites:

Number Spiral

Number Spirals

Divisor Plot

Interesting Graphing Applets

© 2007-2014 Michael M. Ross