Prime Spirals

Prime Spirals - Numberphile

Prime numbers, Ulam Spirals and other cool numbery stuff with Dr James Grime.
More links & stuff in full description below ↓↓↓

James Clewett on spirals at:

And more to come soon...

* subscribing to numberphile does not really change your physical appearance!

And golden line in this context was made up by Brady!

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41 and more Ulam's Spiral - Numberphile

More on prime numbers and Ulam's Spiral - this time focusing on 41 and Arthur C. Clarke.
More links & stuff in full description below ↓↓↓

This video features Dr James Clewett. More Clewett videos at:

See our other Ulam Spiral video at:
And more to come soon...

The book discussed is The Garden of Rama.

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Ulam Spiral Visualization - The Pattern of Prime Numbers

The Ulam spiral is a way of visualizing the distribution of prime numbers (in blue). This pattern is one the great unsolved mysteries in mathematics and has important consequences in Cryptography.

Prime Spirals

Inspired by the Prime Spiral video with Dr James Grime on Numberphile. I created this in Lightwave 3D and compiled it in After Effects. Explanation of what is going on is in the video itself. It basically shows how Prime numbers are distributed when varying an Archimedean spiral... I hope. I'm not really a mathematician.

Prime numbers on an archimedes spiral.

Here is the MATLAB script I used to create this:

Awesome Prime Number Constant - Numberphile

Have you ever heard of Mills' Constant? Video supported by (& free book):
More links & stuff in full description below ↓↓↓

Several people have pointed out the n=4 prime is 2521008887 (we missed an 8)

More on prime numbers:

This video features Dr James Grime -

The Mills Proof is at:

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Prime numbers: A PATTERN!

The graph at the end of this video shows an impressive truth about prime numbers (calculated out to the first 10 million primes, roughly). This pattern is solid, unchanging, and beautiful truth about prime numbers that may help us break the code of prime numbers and the fundamental properties of our mathematical system

Cool stuff on Ulam Spiral:

PRIME NUMBERS - Amazing pattern in three dimensions

Numbers can be represented in many different ways. By using three dimensions and nine colors, positive integers are transformed into physical objects. A pattern for all primes is found within this model. To know more, please visit

A Pattern in Prime Numbers ?

An interesting phenomenon I stumbled upon recently.

Hope you find it as fascinating as me.

If there are any mathematicians in the audience who can explain this,
(in a simple way) feel free to comment.

(Oh, and yes, I returned the voice to Professor Hawking, but he left me a copy ;)
...ok, I made it with speakonia :) )

Also I made a large rendering of the pattern available for download here:

(It is the smallest download button on the page...,
and please let me know when the link expired.)
Could be a nice poster :) as suggested by user bobbooty.

Music:

e-world by zero-project

is licensed under a Creative Commons license:

The Ulam Prime Spiral



The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily.

Inspired by the square spiral of Stanislaw Ulam, the prime numbers are plotted on a spiral number line. Each prime on the wound line is marked with a dot. Adjusting the tightness of the winding and the density allows different patterns to emerge.

Contributed by: Christopher Skiscim
Based on a program by: Greg Keogh

Primes and Twin Primes: An Awesome Journey Pt.1 of 4

Part 1 of 4. These videos convey the thought process in discovering several methods to study Prime Numbers. Great visualizations will guide you through the beauty of the primes, while compelling insights will lay a foundation for the Twin Prime Conjecture. Recommended to watch in HD mode. Go to for more information and visualizations.

Golden Ratio And Prime Number Spirals - An Amazing Pattern

I showed you how to make the seed arrangement using the golden ratio here:


If you remove all of the seeds whose index numbers are NOT prime numbers, the pattern that remains is really friggin' cool.

If you have any more information on this than what I mention in the video please let me know.

Prime Spiral Beta

There are some prime numbers and some other things going on here... still in development / sorry for the quality.

Prime Spiral Animation in Tableau

Based on a viz by Sarah Battersby

Gaussian Prime Spirals



The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily.

Start a loop with a point having integer coordinates in the complex plane (called a Gaussian integer) and trace a path as follows: ? Move right until a Gaussian prime p is encountered, then turn left 90. ? Continue, always moving straight in the curren...

Contributed by: Joseph O'Rourke and Stan Wagon

Audio created with WolframTones:

Ulam Spiral

Illustration of the Ulam spiral, with tink sounds for the composite numbers and sosumi sounds for 1 and the primes.

Prime Spiral: Sieve of Eratosthenes

Video shows the Sieve of Eratosthenes in action.

All numbers are first assumed to be prime - drawn in black.
Then all those with factors are systematically eliminated.

In the end...
Prime numbers, and one, are shown black.
Numbers with one prime factor are shown white.
Numbers with two prime factor are shown blue.
Numbers with three prime factor are shown dark red.
Numbers with four prime factor are shown greeny yellow.

Doodling in Math: Sick Number Games

I don't even know if this makes sense. Boo cold.
Try playing with this:



My personal website, which you might like:

Extending Ulam Beyond Primes

Picture with number lists:

Numberphile video on prime spiral:

What is ULAM SPIRAL? What does ULAM SPIRAL mean? ULAM SPIRAL meaning, definition & explanation

What is ULAM SPIRAL? What does ULAM SPIRAL mean? ULAM SPIRAL meaning - ULAM SPIRAL definition - ULAM SPIRAL explanation.

Source: Wikipedia.org article, adapted under license.

The Ulam spiral or prime spiral (in other languages also called the Ulam cloth) is a graphical depiction of the set of prime numbers, devised by mathematician Stanislaw Ulam in 1963 and popularized in Martin Gardner's Mathematical Games column in Scientific American a short time later. It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers.

Ulam and Gardner emphasized the striking appearance in the spiral of prominent diagonal, horizontal, and vertical lines containing large numbers of primes. Both Ulam and Gardner noted that the existence of such prominent lines is not unexpected, as lines in the spiral correspond to quadratic polynomials, and certain such polynomials, such as Euler's prime-generating polynomial x2?-?x?+?41, are believed to produce a high density of prime numbers. Nevertheless, the Ulam spiral is connected with major unsolved problems in number theory such as Landau's problems. In particular, no quadratic polynomial has ever been proved to generate infinitely many primes, much less to have a high asymptotic density of them, although there is a well-supported conjecture as to what that asymptotic density should be.

In 1932, more than thirty years prior to Ulam's discovery, the herpetologist Laurence M. Klauber constructed a triangular, non-spiral array containing vertical and diagonal lines exhibiting a similar concentration of prime numbers. Like Ulam, Klauber noted the connection with prime-generating polynomials, such as Euler's.

According to Gardner, Ulam discovered the spiral in 1963 while doodling during the presentation of a long and very boring paper at a scientific meeting. These hand calculations amounted to a few hundred points. Shortly afterwards, Ulam, with collaborators Myron Stein and Mark Wells, used MANIAC II at Los Alamos Scientific Laboratory to extend the calculation to about 100,000 points. The group also computed the density of primes among numbers up to 10,000,000 along some of the prime-rich lines as well as along some of the prime-poor lines. Images of the spiral up to 65,000 points were displayed on a scope attached to the machine and then photographed. The Ulam spiral was described in Martin Gardner's March 1964 Mathematical Games column in Scientific American and featured on the front cover of that issue. Some of the photographs of Stein, Ulam, and Wells were reproduced in the column.

In an addendum to the Scientific American column, Gardner mentioned the earlier paper of Klauber. Klauber describes his construction as follows, The integers are arranged in triangular order with 1 at the apex, the second line containing numbers 2 to 4, the third 5 to 9, and so forth. When the primes have been indicated, it is found that there are concentrations in certain vertical and diagonal lines, and amongst these the so-called Euler sequences with high concentrations of primes are discovered.

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