The Simpsons

The longest running animation TV show of all time and one of the best cartoons ever. Known for covering humor on every subject. Here' are a couple on Fermat's Last Theorem, Euler's Formula and the Complexity classes P and NP.

Interplanetary Transport Network

The Interplanetary Transport Network (ITN) is a collection of gravitationally determined pathways through the solar system that require very little energy for an object to follow. The ITN makes particular use of Lagrange points as locations where trajectories through space can be redirected using little or no energy. These points have the peculiar property of allowing objects to orbit around them even though there is no material object at them.

Factorization Using the Elliptic Curve Method

Click here for a Java implementation of integer factorization using the elliptic curve method (ECM). Source code included.

Prime Number Spiral

The Prime Number Spiral software is a tool for exploring the Ulam spiral.

Gnuplot

Gnuplot is a portable command-line driven interactive data and function plotting utility for UNIX, IBM OS/2, MS Windows, DOS, Macintosh, VMS, Atari and many other platforms. The software is copyrighted but freely distributed (i.e., you don't have to pay for it). It was originally intended as to allow scientists and students to visualize mathematical functions and data. It does this job pretty well, but has grown to support many non-interactive uses, including web scripting and integration as a plotting engine for third-party applications like Octave. Gnuplot has been supported and under development since 1986.

Project Euler

Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.

The motivation for starting Project Euler, and its continuation, is to provide a platform for the inquiring mind to delve into unfamiliar areas and learn new concepts in a fun and recreational context.

uCalc

UCALC Windows Graphing Calculator comes with the following features: Expression Evaluator, Unit Converter, User Solution Modules, Graphing, Equation Solver, User Functions & Variables, Summation Tables, Integrator, and General Ledger.

UCalc Fast Math Parser allows programs to evaluate math expressions that are defined at run time. Ease of implementation, flexibility, sturdiness and speed are at the core of the product's design. It includes direct support for Visual Basic, C++ (Microsoft and Borland), PowerBASIC (PB/DLL and PB/CC), and Delphi.

Euler Angles

A good read on three-dimensional space and how to rotate the orthogonal coordinate system.

Command Line Quaternions

Quaternions are numbers with 4 parts: one for time, three for space. This project hopes to create command line functions that generate thousands of points of quaternions, to be fed into animation software. Think: analytical animation!

Wolfram Web Resources

In addition to Mathworld, Wolfram Web Resources hosts an extensive list of other sites, including: Demonstrations, Functions, Gallery, Integrals, and Tones,

Mathematica 6

Mathematica 6 has been released and you can see some nice demonstration over at the Wolfram Demonstration Project.

Calculus Videos

The Department of Mathematics at University of Houston has created this library of calculus videos viewable using Apple's QuickTime. They also have a list of shorter videos for some calculus concepts.

Gnofract 4D

Gnofract 4D is a free, open source program which allows anyone to create beautiful images called fractals. The images are automatically created by the computer based on mathematical principles. These include the Mandelbrot and Julia sets and many more. You don't need to do any math: you can explore a universe of images just using a mouse. It runs on Unix-based systems such as Linux and FreeBSD.

SpeQ Mathematics

SpeQ is a small, extensive mathematics program with a simple, intuitive interface. All calculations are entered in a sheet. In there you can freely add, edit and execute all calculations. You can define variables and functions, and plot graphs of your functions. You can save your calculations for later re-use.

Maths Films

The aim of this website is to provide a source of animated mathematical films, images and text to support the teaching of mathematics in ways that build on the intuitions and powers we have all possessed all our lives.

What is the 4th Dimension?

Four-dimensional geometry is Euclidean geometry extended into one additional dimension. The prefix "hyper-" is usually used to refer to the four- (and higher-) dimensional analogs of three-dimensional objects, e.g. hypercube, hyperplane, hypersphere. n-dimensional polyhedra are called polytopes.

Here are some interesting notes, images and movies for understanding what the fourth dimension would look like.

Mesecina

Mesecina is a software to visualize and study the medial axis and related computational geometry structures in the plane.

Cornell University Library Historical Mathematics Monographs

The Cornell University Library Historical Mathematics Monographs is a collection of selected monographs with expired copyrights chosen from the mathematics field. These were monographs that were brittle and decaying and in need of rescue. These monographs were digitally scanned and facsimile editions on acid free paper were created.

FreeTechBooks.com

FreeTechBooks.com has a large number of free mathematics books.

yEd - Java Graph Editor

yEd is a very powerful graph editor that is written entirely in the Java programming language. It can be used to quickly and effectively generate drawings and to apply automatic layouts to a range of different diagrams and networks.

What Do Numbers Sound Like?

This website contains a list of math songs. The numbers assigned are actually the major scale degrees that correlate. If you don't know basic music theory, think "Do = 1, Re = 2, Mi = 3, Fa = 4, So = 5, La = 6, Ti = 7, Do + Octave = 8, Re + Octave = 9, and 0 = Quarter Rest." Listen to $\pi$ up to 1,000 digits on the piano alone or on the piano, bass & flute together.

Open (to interpretation) question: Which number sounds the best? Which transcendental constant sounds the best?

BBP-Type Formulas

The BBP (named after Bailey-Borwein-Plouffe) is a formula for calculating `\pi` discovered by Simon Plouffe in 1995,
`\pi = \sum_(n=0)^\infty(\frac{4}{8n+1}-\frac{2}{8n+4}-\frac{1}{8n+5}-\frac{1}{8n+6})(1/(16))^n`.

Amazingly, this formula is a digit-extraction algorithm for `\pi` in base 16.

Following the discovery of this and related formulas, similar formulas in other bases were investigated. This class of formulas are now known as BBP-type formulas.

Functional Programming Library for XSLT (FXSL)

Dimitre Novatchev, the author of FXSL, was kind enough to give a more thorough explanation of what FXSL can do. Here's the current list of all the features.

FXSL has quite a bit of support for Math operations:

• Exponential and logarithmic functions.


• Trigonometic and hiperbolic-trigonometric functions.


• Reverse trigonometric and reverse hyperbolic functions.


• Finding the root of a function of one real argument using Newton-Raphston and the binary-search algorithms.


• Generation of sequences of random numbers with values in a given interval and with certain distributions.


• Numerical differentiation.


• Numerical integration.


• Testing primality of natural numbers.


• Generation of Fibonacci numbers.

An XSL Calculator: The Math Modules of FXSL

Here's a helpful website for doing math problems using XSLT. Section #2 leads to an XSL calculator using the FXSL library.

mimeTex Tutorial

This website offers a LaTex math tutorial for mimeTex. It also has an online editor for you to try out the examples as you learn about LaTex

Mathematics Toolbar

The Mathematics Toolbar consists in a Microsoft Word template document (files with extension « .dot ») displaying a special command bar to the user. With the various buttons offered, the user can then create mathematical formula.

Burr Tools

For people too lazy to solve puzzles themselves, you can use Burr Tools to help solve them for you. This tool is also useful for looking at three-dimensional orthogonal projections.

Infinite Fractal Loop

The Infinite Fractal Loop is a web ring dedicated to fractal art. I found it while browsing Keith's Fractal Art.

VisualComplexity

VisualComplexity.com is a unified resource space for anyone interested in the visualization of complex networks. They present a large number of network data similar to ManyEyes. An interesting example of visualization can be seen at ZipDecode.

Liquid Journey

An interesting website that has some nice mathematical, chaotic animations using flash.

LaTex Equation Editor

This online LaTex equation editor belongs to the Hamline University Physics Department.

Number Systems of the World

Here's a nice collection in different languages of number systems of the world. Granted, this list is not exhaustive and only includes the first hundred integers and zero.

Natural Numbers

Here's an interesting website called NaturalNumbers.org. They have a couple of free programs that deal with primality testing and zeta functions. One of their more interesting pages has an analysis of the first 100 proximate-prime quadratic polynomials for n = 0 to 1000.

Apfloat for C++

Apfloat is a high performance arbitrary precision package. That means you can do calculations involving millions of digits with it. It uses Number Theoretic Transforms. It's simple to use. It's fast. It's freeware.

Number Spiral

NumberSpiral.com has an interesting report on number spirals which are similar to Ulam's Spiral. The image shown was created using their Vortex application.

More On Ulam's Spiral

Another Java application that can generate Ulam's Spiral.

LaTex Editor (LEd)

LEd, short for LaTex Editor, is a free environment for rapid TeX and LaTex document development.

Primality Testing Using RegEx

Here are a couple of interesting methods for doing primality testing using RegEx in Perl and Python.

Method #1 (Perl):
perl -wle 'print "Prime" if (1 x shift) !~ /^1?$|^(11+?)\1+$/'

Method #2 (Perl):
perl -wle 'print "Prime" if (1 x shift) !~ /^1?$|^(11+?)\1+$/' [number]

Method #3 (Python):
import re
def is_prime(num):
return not re.match(r"^1?$|^(11+?)\1+$", "1" * num)

Prime Numbers in PostScript

Here's an interesting website that shows you how to use a printer to generate prime numbers. It also has a PostScript file that generates ULAM's Spiral.

Ulam's Spiral

Ulam's Spiral, also known as the Prime Spiral, is a plot in which the positive integers are arranged in a spiral (left figure), with primes indicated in some way along the spiral. In the right plot above, primes are indicated in red and composites are indicated in yellow.

Le Village Premier has an interesting stand-alone Java application that can generate the spiral.

arXiv.org

The Cornell University Library hosts the arXiv.org website. arXiv.org has open access to hundreds of thousands of e-prints in Physics, Mathematics, Computer Science and Quantitative Biology.

Graph Paper

Here are three sites that allow you to create graph paper: WebEE, Incompetech and PDF Pad.

Use some of the grids to test Ulam's Spiral.

Leonhard Euler

Leonhard Euler (pronounced Oiler) was born on April 15, 1707 and died on September 7, 1783. He was a Swiss mathematician who was tutored by Johann Bernoulli. He worked at the Petersburg Academy and Berlin Academy of Science. He had a phenomenal memory, and once did a calculation in his head to settle an argument between students whose computations differed in the fiftieth decimal place. Euler lost sight in his right eye in 1735, and in his left eye in 1766. Nevertheless, aided by his phenomenal memory (and having practiced writing on a large slate when his sight was failing him), he continued to publish his results by dictating them. Euler was the most prolific mathematical writer of all times finding time (even with his 13 children) to publish over 800 papers in his lifetime. He won the Paris Academy Prize 12 times. When asked for an explanation why his memoirs flowed so easily in such huge quantities, Euler is reported to have replied that his pencil seemed to surpass him in intelligence. François Arago said of him "He calculated just as men breathe, as eagles sustain themselves in the air" (Beckmann 1971, p. 143; Boyer 1968, p. 482).

One of the most famous and appealing equations, `e^{i\pi} + 1 = 0` (where `e` is Euler's number, the base of the natural logarithm, `i` is the imaginary unit, one of the two complex numbers whose square is negative one (the other is `-i`), and `\pi` is pi, the ratio of the circumference of a circle to its diameter) was derived by him.

Mathematical Association of America (MAA)

The Mathematical Association of America is the largest professional society that focuses on mathematics accessible at the undergraduate level. Our members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists; statisticians; and many others in academia, government, business, and industry. We welcome all who are interested in the mathematical sciences.

International Mathematical Olympiad

The International Mathematical Olympiad (IMO) is the World Championship Mathematics Competition for High School students and is held annually in a different country. The first IMO was held in 1959 in Romania, with 7 countries participating. It has gradually expanded to over 90 countries from 5 continents. The IMO Advisory Board ensures that the competition takes place each year and that each host country observes the regulations and traditions of the IMO.

William Lowell Putnam Mathematics Competition

The William Lowell Putnam Mathematics Competition is a North American math contest for college students. Each year on the first Saturday in December, over 2000 students spend 6 hours (in two sittings) trying to solve 12 problems. Individual and team winners (and their schools, in the latter case) get some money and a few minutes of fame.

The Putnam exam problems and solutions can be found here.

GeoGebra

GeoGebra is a free and multi-platform dynamic mathematics software for schools that joins geometry, algebra and calculus. It received several international awards including the European and German educational software awards.

Create a Graph

The National Center for Education Statistics (NCES), located within the U.S. Department of Education and the Institute of Education Sciences, is the primary federal entity for collecting and analyzing data related to education. A couple of interesting links that they have are a couple of sites to create graphs with flash and without.

Cartoon

Console Calculator

Console Calculator is a fast and powerful command-line calculator. This calculator performs advanced mathematical functions, remembers recent entries, allows creation of user defined variables and custom functions. This calculator is an especially handy tool for software developers or engineering.

muParser

muParser is an extensible high performance math parser library. It is based on transforming an expression into a bytecode and precalculating constant parts of it.

MathsNet

MathsNet is an interactive website for learning mathematics. They also have a wide variety of mathematical games and puzzles.

WisWeb

WisWeb is the website of the Freudenthal Institute for Science and Mathematics for secondary education (students of 12 to 18 years old). The main focus of the site are applets; small computer programs that run over the internet.

TiddlyMath

TiddlyMath is a web notebook for mathematics. It allows you to create math formulas, diagrams and graphs to display on the web.

Many Eyes

You can explore data using your eyes! This site is set up to allow the entire internet to upload data, visualize it, and talk about their discoveries with other people.

Goldbach Weave

Check out Andy Wardley's website for an explanation of the Goldbach Weave.

Emprise JavaScript Charts

EJSChart is a 100% pure JavaScript charting solution.

Jmath

Jmath is a Computer Algebra System (CAS) targeted at high school students. This one uses Java, but is reasonably fast.

QPlot

QPlot is a nifty little application that allows you to visualize scatter data by entering or digitizing them.

The Perfect Circle

Rumor has it that they hold competitions for who can draw the most perfect circle by hand. No proof of this yet, but watch this video to be impressed.

Math Forum

My Math Forum: for a better understanding of mathematics.

Magic Number Machine

Magic Number Machine is a free, full-featured, graphically laid out, high-precision, scientific calculator for Mac OS X 10.4 and greater. Full source-code is included with the distribution.

Ideal if you need to enter large expressions or have accurate precision. "Data" drawers allow an easy way to generate statistical data, linear regression and Gaussian elimination. The extensive support of complex numbers and hexadecimal numbers is also a significant benefit for anyone who has to work with this type of data.

Requires Mac OS X 10.4 or greater. Native Intel Mac support (universal binary). Source code requires XCode 2.2 or greater.

TeXclip

TeXclip is the web application for people who prefer TeX than PowerPoint's equation editor.

Wikisky

Okay, this is not a math post per se, but who doesn't like astronomy? Wikisky can show you the stars above a given location at a certain time. Pan, zoom, browse and explore the night sky using a Google-Earth-like interface.

Mathematics in Movies

This site offers some clips of the mathematical scenes from different movies and TV shows. I hghly recommending watching Sneakers (1992), A Beautiful Mind (2001) and Proof (2005).

Goldbach Conjecture: Proved?

The Goldbach Conjecture states that all positive, even integers ≥ 4 can be expressed as the sum of two primes.

Several papers have recently surfaced at arXiv.org claiming to have proved this. However, none of these methods have been verified yet.

• A Proof of Goldbach Conjecture


• From the Goldbach Conjecture to the Theorem

Fibonacci Sequence

In a book completed in the year 1202, mathematician Leonardo of Pisa (also known as Fibonacci) posed the following problem: How many pairs of rabbits will be produced in a year, beginning with a single pair, if every month each pair bears a new pair that becomes productive from the second month on?

The total number of pairs, month by month, forms the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on. Each new term is the sum of the previous two terms. This set of numbers is now called the Fibonacci sequence.

Fibonacci numbers come up surprisingly often in nature, from the number of petals in various flowers to the number of scales along a spiral row in a pine cone. They also arise in computer science, especially in sorting or organizing data.

Amazingly, the ratios of successive terms of the Fibonacci sequence get closer and closer to a specific number, often called the golden ratio. It can be calculated as $\frac(1 + \sqrt5)(2)$, or 1.6180339887…. For instance, the ratio 55/34 is 1.617647…, and the next ratio, 89/55, is 1.6181818….


Fibonacci's immortal rabbit problem. A red rectangle designates a newborn pair, which doesn't produce offspring until the second month.

Proof Without Words

1³ + 2³ + … + n³ = (1 + 2 … + n)².
Mathematically, $\sum_{i=1}^{n} i^3 = (\sum_{i=1}^{n} i)^2$.

Proof Without Words

Area(Parallelogram) = ad - bc.

Proof Without Words

Law of Cosines: $\a^2 +b^2 - 2ab\cos\theta = c^2$.

Proof Without Words

1 + 4 + 9 + … + n² = n(n + 1)(2n + 1)/6.
Mathematically, $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$.

Scaling and Counting Concepts

Secret Worlds: The Universe Within (Powers of 10) shows you the scale of the universe and the microscopic world.

The MegaPenny Project helps you to visualize large numbers.

A Dot for Every Second in the Day

Very interesting website that shows you how things are scaled. Similar to Starship Dimensions.

Famous Unsolved Codes and Ciphers

This site contains an unofficial list of well-known unsolved codes and ciphers. A couple of the better-known unsolved ancient historical scripts are also thrown in, since they tend to come up during any discussion of unsolved codes.

Proof Without Words

$AB + AX = AD$.

XML MAIDEN Project

More Cascading Style Sheets (CSS) to help you create better looking math sites.

Indian American Mathematician is Awarded the Abel Prize

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2007 to Srinivasa S.R. Varadhan, Courant Institute of Mathematical Sciences, New York. He receives the prize “for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviation”.

Click here for more information.

MIT OpenCourseWare

MIT recently started offering its entire learning curriculum for free to anyone. Click here for their math portal.

Proof Without Words

Pythagorean Theorem: For a right triangle with legs a and b and hypotenuse c,
$a^2 + b^2 = c^2$.

Proof Without Words

The area of a regular dodecagon is 3R².

Proof Without Words

1/4 + (1/4)² + (1/4)³ + … = 1/3.
Mathematically, $\sum_{i=1}^{\infty} \frac{1}{4^i} = \frac{1}{3}$.

Proof Without Words

1 + 2 + 3 + … n = n(n + 1)/2.
Mathematically, $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$.

Islamic Quasicrystalline Architecture

Medieval Muslims Made Stunning Math Breakthrough
Magnificently sophisticated geometric patterns in medieval Islamic architecture indicate their designers achieved a mathematical breakthrough 500 years earlier than Western scholars, scientists said on Thursday.

By the 15th century, decorative tile patterns on these masterpieces of Islamic architecture reached such complexity that a small number boasted what seem to be "quasicrystalline" designs, Harvard University's Peter Lu and Princeton University's Paul Steinhardt wrote in the journal Science.

Click here for more information.

Lie Groups - E8

Mathematicians Map E₈
Mathematicians have mapped the inner workings of one of the most complicated structures ever studied: the object known as the exceptional Lie group E₈. This achievement is significant both as an advance in basic knowledge and because of the many connections between E₈ and other areas, including string theory and geometry.

Click here for more information.

Proof Without Words

The sum of the first n odd numbers is n².
Mathematically, $\sum_{i=1}^{n} (2i - 1) = n^2$.

Starship Dimensions

Okay, this isn't a mathematical post per se, but it does make use of dimensions and exponents.

Note: The images on that site are all exactly to scale, 10 pixels to a meter. Internet Explorer users may Click and Drag the starships to compare them as you like. At the bottom of the image are some contemporary vehicles and buildings for reference. Have fun!

GNU MP Bignum Library

The GNU Multiple Precision (MP) Bignum Library is a free library for arbitrary precision arithmetic, operating on signed integers, rational numbers, and floating point numbers. There is no practical limit to the precision except the ones implied by the available memory in the machine GMP runs on. GMP has a rich set of functions, and the functions have a regular interface.

Compare it to MIRACL for functionality and speed.

MathIWYG

MathIWYG allows you to enter math directly into your website using a Flash-based WYSIWYG (what-you-see-is-what-you-get) editor.

One: A Poem: A Raven

Here's an interesting poem by Mike Keith that counts the digits of π as well.

Folding Paper in Half 12 Times

Britney Gallivan has solved the Paper Folding Problem. This well known challenge was to fold paper in half more than seven or eight times, using paper of any size or shape. Click here for the rest of the article.

Tupper's Self-Referential Formula

J. Tupper concocted the amazing formula
$1/2 < \lfloor mod(\lfloor y/(17) \rfloor 2^(-17 \lfloor x \rfloor -mod(\lfloor y \rfloor,17)),2) \rfloor$,

where $\lfloor x \rfloor$ is the floor function and mod(b, m) is the mod function, which, when graphed over 0 ≤ x ≤ 105 and n ≤ y ≤ n + 16 with
n = $960,939,379,918,958,884,971,672,962,127,$
$852,754,715,004,339,660,129,306,651,505,519,$
$271,702,802,395,266,424,689,642,842,174,350,$
$718,121,267,153,782,770,623,355,993,237,280,$
$874,144,307,891,325,963,941,337,723,487,857,$
$735,749,823,926,629,715,517,173,716,995,165,$
$232,890,538,221,612,403,238,855,866,184,013,$
$235,585,136,048,828,693,337,902,491,454,229,$
$288,667,081,096,184,496,091,705,183,454,067,$
$827,731,551,705,405,381,627,380,967,602,565,$
$625,016,981,482,083,418,783,163,849,115,590,$
$225,610,003,652,351,370,343,874,461,848,378,$
$737,238,198,224,849,863,465,033,159,410,054,$
$974,700,593,138,339,226,497,249,461,751,545,$
$728,366,702,369,745,461,014,655,997,933,798,$
$537,483,143,786,841,806,593,422,227,898,388,$
$722,980,000,748,404,719$,

gives the self-referential "plot" illustrated above.

Latest, Greatest Twins in Their Prime

The Twin Internet Prime Search and PrimeGrid distributed computation projects have recently discovered the largest known twin primes, which is a pair of prime numbers separated by two. The pair discovered on January 15^th are $2,003,663,613 \times 2^{195,000} ± 1$. The two primes are 58,711 digits long. The discoverer was Eric Vautier from France.

LEGO Difference Machine

Check out the LEGO Difference Machine here.

Math + JavaScript = Cool

Check out ASCIIMathML and ASCIIsvg to learn how to display math equations and graphs in your website using nothing but JavaScript. You've probably seen ASCIIMathML in action in some of my other posts.

Golomb Ruler

In mathematics, the term "Golomb Ruler" refers to a set of non-negative integers such that no two distinct pairs of numbers from the set have the same difference. Conceptually, this is similar to a ruler constructed in such a way that no two pairs of marks measure the same distance. An Optimal Golomb Ruler (OGR) is the shortest Golomb Ruler possible for a given number of marks. However, finding (and proving) OGR's becomes exponentially more difficult as the number of marks increases, and it is for this reason that we have turned to the web for help in finding the OGR's with 24 and more marks.

For more information, check out the distributed.net: Project OGR page and more info at MathWorld and Wikipedia.

Antikythera Mechanism

The Antikythera mechanism is an ancient mechanical analog computer (as opposed to digital computer) designed to calculate astronomical positions. It was discovered in the Antikythera wreck off the Greek island of Antikythera, between Kythera and Crete, and has been dated to about 150-100 BC. Click here for more information.

InstaCalc Online Calculator

InstaCalc is a vision for a fast, simple and powerful calculator. You can embed it in your website or blog.

The Curta Calculator

Here are some links about the Curta calculator. Nice little machine.

• VCalc.net


• Curta.org

44th Known Mersenne Prime Found!

Less than a year after their last discovery, the Central Missouri State University (CMSU) team, led by professors Curtis Cooper and Steven Boone, has broken their own record for the largest known prime number: 2^32,582,657-1.

Great Mathematical Books

Journey Through Genius, by William Dunham © 1990.

The Mathematical Universe, by William Dunham © 1994.

Proofs Without Words II, by Roger B. Nelsen © 2000.

Proofs Without Words, by Roger B. Nelsen © 1993.

Fermat's Last Theorem, by Simon Singh © 1997.

The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes, by David Darling © 2004.

The Mathematics of Ciphers: Number Theory and RSA Cryptography, by S. C. Coutinho © 1998.

Axiomatic Set Theory, by Patrick Suppes © 1972.

Abel's Proof, by Peter Pesic © 2003.

Hilbert's Tenth Problem, by Yuri V. Matiyasevich © 1996.

Prime Numbers, by David Wells © 2005.