Busy Beaver Number 5 is Alive!

In theoretical computer science, the objective of the busy beaver is to find a terminating program of a given size that (depending on definition) either produces the most output possible, denoted by BB(n), or runs for the longest number of steps.

The latest value for n = 5 was recently discovered.

The known values are BB(1) = 1, BB(2) = 6, BB(3) = 21, BB(4) = 107, BB(5) = 47,176,870.

Click here for more information.

An Amazing Approximation to e

An amazing pandigital approximation to e that is correct to 18,457,734,525,360,901,453,873,570 decimal places is given by:

`e\approx(1+9^(−4^(6⋅7)))^(3^(2^85))`

It was discovered by Richard Sabey in 2004.

Proof:

`(1+9^(−4^(6⋅7)))^(3^(2^85))=(1+9^(−4^42))^(3^(2^85))`

`=(1+9^(−4^42))^(3^(2*2^84))`

`=(1+9^(−4^42))^(3^(2*2^84))`

`=(1+9^(−4^42))^(9^(2^(84)))`

`=(1+9^(−4^42))^(9^(4^42))`

`=(1+\frac{1}{9^(4^42)})^(9^(4^42))`

`=(1+\frac{1}{n})^n`.

Permutation Rotations

In this paper, we discuss certain properties of permutation rotations on each other.

Click here to read my paper.

The Golden Ratio's Siblings

On real numbers whose fractional part remain constant after squaring.

Click here to read my paper.

52nd Known Mersenne Prime Found!

The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 2^136,279,841-1, having 41,024,320 decimal digits. Luke Durant, from San Jose, California, found the prime on October 12th.

Click here for more information.

A Chiral Aperiodic Monotile

An update to the aperiodic tile. This update shows that a shape that tiles the plane aperiodically without reflections is possible. The original one did not.

Click here for the article.

Mathematicians have finally discovered an elusive ‘einstein’ tile

A 13-sided shape known as “the hat” has mathematicians tipping their caps.

It’s the first true example of an “einstein,” a single shape that forms a special tiling of a plane: Like bathroom floor tile, it can cover an entire surface with no gaps or overlaps but only with a pattern that never repeats.

Click here for more information.

Value-Counting Up to N

Some interesting properties arise when value-counting the integers sequentially up to N using N digits or fingers and comparing the number of values to the prime-exact equation; with a simple method for testing primes and prime powers (particularly Mersenne and Fermat primes).

Click here to read my paper.

Primality Testing and Factoring Using Pascal's Triangle

An interesting if not impractical way of primality testing and factoring a number using Pascal’s Triangle.

Click here to read my paper.

Collatz Conjecture

I've previously linked to Jason Davies website for another article. He has another JavaScript program for the Collatz Conjecture.

To pretty it up, remove the circle fill in collatz.css and modify the circle append (line 83) in collatz.js as follows:

      nodeEnter.append("circle")
          .attr("fill", function(d) {
            var cc;
            var i = parseInt(d.data);
            if ((i && (i & (i - 1)) === 0) && (i <= 16)) {
              cc = "#0000ff";
            }
            else if ((i % 3) === 0) {
              cc = "#c8c8c8";
            }
            else if ((i % 6) === 1) {
              cc = "#ffff00";
            }
            else if (((i % 2) === 1) || (((i % 3) === 2) && (((i / 2) % 2) === 0))) {
              cc = "#ffa500";
            }
            else {
              cc = "#000000";
            }
            return cc;
          })
         .attr("r", 5);

I was only interested in the initial node for those in orange and yellow.

Minimal Set for Powers of 2

The minimal set for powers of 2 is currently nondeterministic and can be shown to be more complex than previously proposed.

Click here for my analysis on it.

Mathematicians Discover the Perfect Way to Multiply

Four thousand years ago, the Babylonians invented multiplication. Last month, mathematicians perfected it.

On March 18, two researchers described the fastest method ever discovered for multiplying two very large numbers. The paper marks the culmination of a long-running search to find the most efficient procedure for performing one of the most basic operations in math.

“Everybody thinks basically that the method you learn in school is the best one, but in fact it’s an active area of research,” said Joris van der Hoeven, a mathematician at the French National Center for Scientific Research and one of the co-authors.

Click here and here for more information.

A 53-Year-Old Network Coloring Conjecture Is Disproved

A paper posted online last month has disproved a 53-year-old conjecture about the best way to assign colors to the nodes of a network. The paper shows, in a mere three pages, that there are better ways to color certain networks than many mathematicians had supposed possible.

Click here for more information.

Andrew Booker, a Mathematics Professor at the University of Bristol, Just Solved a Deceptively Simple Puzzle That Has Boggled Minds for 64 Years

A mathematician in England has cracked a math puzzle that's stumped computers and humans alike for 64 years: How can the number 33 be expressed as the sum of three cubed numbers?

While it might seem simple on its face, this question is part of an enduring number-theory conundrum that goes back to at least 1955 and may have been mulled over by Greek thinkers as early as the third century. The underlying equation to solve looks like this:

`x^3 + y^3 + z^3 = k`

That answer is:

`(8,866,128,975,287,528)^3 + (–8,778,405,442,862,239)^3 + (–2,736,111,468,807,040)^3 = 33`.

Click here for more information.

Karen Uhlenbeck is first woman to win prestigious maths Abel prize

Mathematician Karen Uhlenbeck has become the first woman to win the Abel prize, sometimes called the Nobel prize of mathematics. She has been awarded the 6 million Norwegian kroner ($700,000) prize for her work in the fields of gauge theory and geometric analysis, which have been credited with far-reaching impact in both mathematics and physics.

Click here for more information.

Mathematician Sir Michael Atiyah dies aged 89

One of the world's foremost mathematicians, Prof Sir Michael Atiyah, has died at the age of 89.

Sir Michael, who worked at Cambridge University before he retired, made outstanding contributions to geometry and topology.

Sir Michael was a recipient of the highest honour in mathematics, a Fields Medal. He died on Friday.

Click here for more information.

51st Known Mersenne Prime Found!

The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 2^82,589,933-1, having 24,862,048 digits. A computer volunteered by Patrick Laroche from Ocala, Florida made the find on December 7, 2018. The new prime number, also known as M82589933, is calculated by multiplying together 82,589,933 twos and then subtracting one. It is more than one and a half million digits larger than the previous record prime number.

Click here for more information.

Riemann Hypothesis Solved By Sir Michael Atiyah After 160 Years, He Says

One of the world's most renowned mathematicians showed how he solved the 160-year-old Riemann hypothesis at a lecture on Monday — and he will be awarded $1 million if his solution is confirmed.

Sir Michael Atiyah, who has won the two biggest prizes in mathematics — the Fields Medal and Abel Prize — took the stage at the Heidelberg Laureate Forum in Germany on Monday to present his work.

To solve the hypothesis you need to find a way to predict the occurrence of every prime number, even though primes have historically been regarded as randomly distributed.

Aityah's solution will need to be checked by other mathematicians and then published before it is fully accepted and he can claim the prize from the Clay Mathematics Institute of Cambridge.

Click here for more information.

Every Positive Integer Is A Sum Of Three Palindromes

For integer g ≥ 5, we prove that any positive integer can be written as a sum of three palindromes in base g.

Click here for more information.

"Nobel of Mathematics" Stolen Minutes After Awarded

One of the winners of the award known as the Nobel Prize for mathematics had his gold medal stolen minutes after it was given to him. Caucher Birkar, a Kurdish refugee turned Cambridge University math professor, was among four winners of the prestigious Fields Medal on Wednesday in Rio de Janeiro.

It was an embarrassing debut for crime-ridden Rio, the first Latin American city ever to host the Fields ceremony, which takes place every four years. Less than an hour had passed since Birkar, a 40-year-old specialist in algebraic geometry, had been handed his 14-karat gold medal when his briefcase went missing.

Click here for more information.

Creator of 'Grand Unified Theory of Mathematics' Wins Prestigious Math Prize

A mathematician who developed what some consider the "grand unified theory of mathematics" has won one of the most prestigious prizes in mathematics.

Robert Langlands, an emeritus professor at the Institute for Advanced Study at Princeton University, has won the Abel Prize, a prestigious mathematics prize that honors a lifetime of groundbreaking work, organizers of the prize announced yesterday (March 20).

Click here for more information.

The Science of Magic Angle Sculptures

John V. Muntean was inspired to create the Magic Angle Sculptures through his work with magic angle sample spinning, a scientific technique that mechanically simulates a molecule tumbling through space. The effect is to rapidly interchange the three axes of the Cartesian coordinates (x, y, and z). A complex observable phenomenon in three-dimensional space (such as the nuclear magnetic moments of a static molecule) can be represented by 3 x 3 tensors or sets of nine numbers; spinning at the magic angle simplifies that quantity to single isotropic values.

Click here for his videos.

50th Known Mersenne Prime Found!

Persistence pays off. Jonathan Pace, a GIMPS volunteer for over 14 years, discovered the 50th known Mersenne prime, 2^77,232,917-1 on December 26, 2017. The prime number is calculated by multiplying together 77,232,917 twos, and then subtracting one. It weighs in at 23,249,425 digits, becoming the largest prime number known to mankind. It bests the previous record prime, also discovered by GIMPS, by 910,807 digits.

Click here for more information.

Mathematicians Awarded $3 Million for Cracking Century-Old Problem

Christopher Hacon, a mathematician at the University of Utah, and James McKernan, a physicist at the University of California at San Diego, won this year's Breakthrough Prize in Mathematics for proving a long-standing conjecture about how many types of solutions a polynomial equation can have. Polynomial equations are mainstays of high-school algebra — expressions like `x^2+5x+6 = 1` — in which variables are raised to the whole number exponents and added, subtracted and multiplied. The mathematicians showed that even very complicated polynomials have just a finite number of solutions.

Click here for more information.

Mathematical Secrets of Ancient Tablet Unlocked After Nearly a Century of Study

Dating from 1,000 years before Pythagoras’s theorem, the Babylonian clay tablet is a trigonometric table more accurate than any today, say researchers.



At least 1,000 years before the Greek mathematician Pythagoras looked at a right angled triangle and worked out that the square of the longest side is always equal to the sum of the squares of the other two, an unknown Babylonian genius took a clay tablet and a reed pen and marked out not just the same theorem, but a series of trigonometry tables which scientists claim are more accurate than any available today.

The 3,700-year-old broken clay tablet survives in the collections of Columbia University, and scientists now believe they have cracked its secrets.

Click here for more information.

The Formula That Plots (Almost) Everything

Hold onto your logic hats! In this article we're going to explore one of the most amazing formulas in maths: Tupper's self-referential formula.

The protagonist of our story is the following inequality:

`1/2<\floor{mod(\floor{\frac{y}{17}}2^(-17\floor{x}-mod(\floor{y},17)),2))`

The plot works by either coloring a square or not coloring it: a square with coordinates (x, y) is colored if the inequality is true for x and y. If not the square is left blank.

If you plot the plot for many values of and , the outcome is the following:



I'll let that sink in a moment. No, your eyes are not deceiving you, the formula plots a bitmap picture of itself! Hence the name Tupper's self-referential formula (though Tupper never called this function that himself in his 2001 paper).

There is one missing detail, however. I haven’t told you the value of the number N on the y-axis.

Click here to read more information and see where Euler's equation appears.

Math 'Genius' Maryam Mirzakhani Dies At Age 40

Maryam Mirzakhani, an Iranian-born mathematician who was the first woman to win the coveted Fields Medal, died Saturday in a US hospital after a battle with cancer. She was 40.

Click here for more information.

Mathematicians Deliver Formal Proof Of Kepler Conjecture

A team led by mathematician Thomas Hales has delivered a formal proof of the Kepler Conjecture, which is the definitive resolution of a problem that had gone unsolved for more than 300 years. The paper is now available online through Forum of Mathematics, Pi, an open access journal published by Cambridge University Press. This paper not only settles a centuries-old mathematical problem, but is also a major advance in computer verification of complex mathematical proofs.

The Kepler Conjecture was a famous problem in discrete geometry, which asked for the most efficient way to cram spheres into a given space. The answer, while not difficult to guess (it's exactly how oranges are stacked in a supermarket), had been remarkably difficult to prove. Hales and Ferguson originally announced a proof in 1998, but the solution was so long and complicated that a team of a dozen referees spent years working on checking it before giving up..

Click here for more information.

When Mathematicians Get Married

Eccentric French maths genius's 'scribblings' go online

Nearly 18,000 pages of notes by eccentric French maths genius Alexandre Grothendieck were posted online Wednesday by his alma mater, Montpellier University in southern France.

Grothendieck, who died aged 86 in 2014, "revolutionised an entire area of mathematics, algebraic geometry," said Jean-Michel Marin, head of an institute that bears the mathematician's name at the university.

Click here for more information.

Math Champion Wins With Answer About Pecking Chicks

A 13-year-old boy from Texas won a national math competition on Monday with an answer rooted in probabilities — and a dash of farming.

The boy, Luke Robitaille, took less than a second to buzz in at the Raytheon Mathcounts National Competition with the correct answer.

The question: In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of unpecked chicks?

Click here for more information.

New Twist on Sofa Problem that Stumped Mathematicians and Furniture Movers

The Moving Sofa problem asks, what is the largest shape that can move around a right-angled turn? UC Davis mathematician Dan Romik has extended this problem to a hallway with two turns, and shows that a 'bikini top' shaped sofa is the largest so far found that can move down such a hallway.

Click here for more information.

French Mathematician Yves Meyer Wins Top Prize for 'Wavelet Theory'

A French mathematician known for his pioneering work on a theory used for applications ranging from image compression to the detection of gravitational waves from the merging of black holes has earned one of the world's top prizes in mathematics.

Yves Meyer, a professor emeritus in mathematics at the École normale supérieure Paris-Saclay in France, will receive the Abel Prize, the Norwegian Academy of Sciences and Letters (which awards the prize) announced today (March 21) in Oslo. The prize, which comes with a cash award of 6 million Norwegian krone ($710,000), will be bestowed by King Harald V of Norway on May 23.

Meyer was honored largely "for his pivotal role in the development of the mathematical theory of wavelets," the academy said. His work on wavelets began in the mid-1980s.

Click here for more information.

If the Moon Were Only 1 Pixel

A tediously accurate map of the solar system.

Calca

I found this little gem of a tool called Calca. It is a text editor symbolic calculator.

Two-hundred-terabyte maths proof is largest ever

A computer cracks the Boolean Pythagorean triples problem — but is it really maths?

Three computer scientists have announced the largest-ever mathematics proof: a file that comes in at a whopping 200 terabytes, roughly equivalent to all the digitized text held by the US Library of Congress. The researchers have created a 68-gigabyte compressed version of their solution — which would allow anyone with about 30,000 hours of spare processor time to download, reconstruct and verify it — but a human could never hope to read through it.

Click here for more information.

Fermat's Last Theorem Prize Approved

It was a problem that had baffled mathematicians for centuries -- until British professor Andrew Wiles set his mind to it.

"There are no whole number solutions to the equation xⁿ + yⁿ = zⁿ when n is greater than 2."

Otherwise known as "Fermat's Last Theorem," this equation was first posed by French mathematician Pierre de Fermat in 1637, and had stumped the world's brightest minds for more than 300 years.

In the 1990s, Oxford professor Andrew Wiles finally solved the problem, and this week was awarded the hugely prestigious 2016 Abel Prize -- including a $700,000 windfall.

Click here for more information.

Mathematicians Discover Prime Conspiracy

Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers — those numbers that are divisible only by 1 and themselves. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them.

Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9.

Click here for more information.

49th Known Mersenne Prime Found!

On January 7th, GIMPS celebrated its 20th anniversary with the discovery of the largest known prime number, 2^74,207,281-1. Curtis Cooper, one of many thousands of GIMPS volunteers, used one of his university's computers to make the find.

Click here for more information.

With Discovery, 3 Scientists Chip Away At An Unsolvable Math Problem

There are now 15 known convex pentagons, or nonregular pentagons with the angles pointing outward, that can "tile the plane."

Click here for more information.

2014 Fields Medal and Nevanlinna Prize Winners Announced

Maryam Mirzakhani became the first woman to be awarded the Fields Medal for her work connecting topology, geometry and dynamical systems.

Click here for more information.

2048

If you haven't tried it, check out the fun new math game 2048 on the web and any smartphone or device.

The Breasts Equation

This equation is making the rounds this week. However, it doesn't work on IE as of yet.

Google: `e^{-\frac{((x-4)^2+(y-4)^2)^2}{999}}+e^{-\frac{((x+4)^2+(y+4)^2)^2}{999}}+0.1\timese^{-((x+4)^2+(y+4)^2)^2}+0.1\timese^{-((x-4)^2+(y-4)^2)^2}`

Belgian Mathematician Wins Abel Prize for Shaping Algebraic Geometry

Pierre Deligne netted the prize, one of the most prestigious in mathematics and worth about $1 million, for proving a deep conjecture about algebraic geometry which has helped to transform number theory and related fields. Click here for more information.

Kenneth I. Appel, Mathematician Who Harnessed Computer Power, Dies at 80

Kenneth I. Appel, who helped usher the venerable mathematical proof into the computer age, solving a longstanding problem concerning colors on a map with the help of an I.B.M. computer making billions of decisions, died on April 19 in Dover, N.H. He was 80.

Click here for more information.

48th Known Mersenne Prime Found!

On January 25th, the largest known prime number, 2^57,885,161-1, was discovered on Great Internet Mersenne Prime Search (GIMPS) volunteer Curtis Cooper's computer. The new prime number, 2 multiplied by itself 57,885,161 times, less one, has 17,425,170 digits.

Click here for more information.

Animated Factorization Diagrams

A very cool visual representation of numbers and their factors.

Prime Number Patterns (II)

Another interesting article using circles on the distribution of primes and twin primes.

Proof Claimed For Deep Connection Between Primes

The usually quiet world of mathematics is abuzz with a claim that one of the most important problems in number theory has been solved.

Mathematician Shinichi Mochizuki of Kyoto University in Japan has released a 500-page proof of the abc conjecture, which proposes a relationship between whole numbers — a 'Diophantine' problem.

Click here to continue reading.

Prime Number Patterns

Prime Number Patterns is a visual representation of determining which natural numbers are prime, deficient, perfect, or abundant.

NASA MathTrax

NASA MathTrax is a graphing tool for middle school and high school students to graph equations, physics simulations or plot data files. The graphs have descriptions and sound so you can hear and read about the graph. Blind and low vision users can access visual math data and graph or experiment with equations and datasets.

'The Scale of the Universe,' by Two Teenage Brothers

Another Flash animation of the size of the universe.

Solar System Explained From the Inside Out

An excellent image showing the scale of our solar system.

Japanese man calculates pi to 10 trillion digits

The latest number-crunching champ isn't a supercomputer--it's a hacked-together PC.

Shigeru Kondo of Iida, Nagano Prefecture, worked with software designed by Northwestern University grad student Alexander Yee, and followed up their 2010 feat of reckoning pi to 5 trillion digits.

The result was achieved earlier this month after 371 days of computation and numerous hard drive failures.

Click here for more information.

Babbage Analytical Engine designs to be digitised

A project to construct one of the earliest mechanical computers based on sketches by its designer, Charles Babbage, has received a major boost.

The Science Museum in London has agreed to help by digitising the mathematician's original plans.

Eventually the images will be used to create a full working model of the Analytical Engine.

Click here for the full article.

Crazy Japanese Math

This is one way to keep kids (i.e., boys) interested in math. Too bad I can't read it.

Economy Statistics

Due to the ongoing bad economy, I decided to add a couple of statistical graphs to show that.

U.S. National Debt Clock : Real Time

US debt problem visualized: Debt stacked in 100 dollar bills

The Batman Equation

FRACTRAN

I came across this mathematical programming article on Wikipedia called FRACTRAN.

FRACTRAN is a Turing-complete esoteric programming language invented by the mathematician John Conway.

What an amazing methodology. I decided to buy the referenced book "Nonplussed!: Mathematical Proof of Implausible Ideas". It's easy to read, but the proof is still difficult to follow. I really need to read it again.

π = 4!

Extended Midy's Theorem

This is my proof, and an extension, of Midy's Theorem. Someone had referenced this in Wikipedia almost six years ago, but it was recently removed (still in the history) to make Wikipedia more official. Since this blog came about after my proof, I never went back and added it.

Microsoft Mathematics 4.01

Microsoft Mathematics provides a graphing calculator that plots in 2D and 3D, step-by-step equation solving, and useful tools to help students with math and science studies.

Be sure to download the Microsoft Mathematics Add-In for Word and OneNote as well.

Scale Of The Universe

Great flash website that shows the scale of the universe.

Vi Hart

An interesting article on Vi Hart.

Also check out her blog for more cool, math videos.

NetCalc

NetCalc.org is a prototype web-based graphic calculator aiming to provide advanced mathematical functionality. Functions can be viewed graphically, in table form and as a summary of area and mean values.

Benoît Mandelbrot, Novel Mathematician, Dies at 85

Benoît B. Mandelbrot, a maverick mathematician who developed an innovative theory of roughness and applied it to physics, biology, finance and many other fields, died on Thursday in Cambridge, Mass. He was 85.

Click here for more information.

Check out Skytopia's website for 3D Mandelbrot fractals.

Simpsons Math

I just came across this awesome website dedicated to all the math trivia found in The Simpsons TV show.

P vs. NP

According to Vinay Deolalikar a research scientist at HP Labs, P is not equal to NP.

His preliminary manuscript can be found here.

Sorting Algorithms

My last post on sorting algorithms had static displays. This site shows animations for all the different sorts.

Sortvis

Sortvis is a sorting algorithm visualization project.

Click here for more information and to download the code.

XKCD Cartoon - 3x9

Tell Me A Joke

The Wolfram|Alpha search engine returns a different mathematical joke for a "Tell me a joke" query.

Martin Gardner: 1914-2010

Martin Gardner, a prolific writer and popularizer of mathematics, and one of the most influential figures in skepticism passed away on Saturday, May 22, 2010.

Click here for more information.

NIST Releases Successor to Venerable Handbook of Math Functions

The National Institute of Standards and Technology (NIST) has released the Digital Library of Mathematical Functions (DLMF) and its printed companion, the NIST Handbook of Mathematical Functions, the much-anticipated successors to the agency's most widely cited publication of all time. These reference works contain a comprehensive set of tools useful for specialists who work with mathematical modeling and computation.

Click here for more information.

No 'Simple Theory of Everything'

The "exceptionally simple theory of everything," proposed by a surfing physicist in 2007, does not hold water, says Emory mathematician Skip Garibaldi.

Garibaldi, a rock climber in his spare time, did the math to disprove the theory, which involves a mysterious structure known as E8. The resulting paper, co-authored by physicist Jacques Distler of the University of Texas, will appear in an upcoming issue of Communications in Mathematical Physics.

Click here for more information.

Click here for original post.

Happy π Day!

Today is π Day. Google even changed their logo to promote that.

JSLab PlotTool

A graph plotting tool made entirely in JavaScript. It uses no graphics and the generated source code is W3C compliant.

Currently real valued functions of one variable and parameter functions for plane curves are supported. You can plot using a variety of options like line resolution and axis dimensions. The first beta also adds support for limited zoom functionality. As the graph is computed real time there is no limit for the zoom level.

MathJax

MathJax is an open source, Ajax-based math display solution designed with a goal of consolidating advances in many web technologies in a single definitive math-on-the-web platform supporting all major browsers.

Geeky Math Equation Creates Beautiful 3-D World

The quest by a group of math geeks to create a three-dimensional analogue for the mesmerizing Mandelbrot fractal has ended in success.

They call it the Mandelbulb. The 3-D renderings were generated by applying an iterative algorithm to a sphere. The same calculation is applied over and over to the sphere’s points in three dimensions. In spirit, that’s similar to how the original 2-D Mandelbrot set generates its infinite and self-repeating complexity.

Click here for more information.

Saturn’s Hexagon May Be Solar System’s Coolest Mystery

The Cassini spacecraft has returned the best images yet of the strange hexagonal jet stream that flows around the northern pole of Saturn.

First discovered by the Voyager spacecraft in the early 1980s, the hexagon remains a beautiful mystery to astronomers, and one they’ve been waiting for another shot to see for almost three decades.

Click here for more information.

Not So Fortunate Numbers

A Fortunate number, named after Reo Fortune, for a given positive integer n is the smallest integer m > 1 such that p_n# + m is a prime number, where the primorial p_n# is the product of the first n prime numbers.

For example, p₇# = 2×3×5×7×11×13 = 510,510. The smallest prime number after 510,511 is 510,529. Thus, 510,529 - 510,510 = 29 is a Fortunate number.

Fortune's conjecture states that no Fortunate number is composite. A Fortunate prime is a Fortunate number which is also a prime number. As of 2009, all the known Fortunate numbers are also Fortunate primes.

------------------------------------------------------------

Instead of only using primorial numbers in our sequence, what happens when we use all numbers such that for the largest prime dividing each number in our sequence, the primorial of that prime also divides that number (i.e., 630 = 2×3²×5×7 belongs to the sequence since p₄# = 210 divides 630)?

512 → 2⁵, 16,384 → 2⁹, and 524,288 → 2¹⁴ fail to generate Fortunate primes.

Do only perfect powers of two fail to generate Fortunate primes? Is there a pattern?

------------------------------------------------------------

For all numbers less than a billion in our sequence, the following fail to generate a Fortunate prime:

F#  Generator
9      512
9      8,388,608
15    4,194,304
15    67,108,864
21    524,288
25    147,456
25    373,248
25    393,216
25    1,062,882
25    1,259,712
25    4,251,528
25    4,718,592
25    5,308,416
25    5,971,968
25    10,077,696
25    17,915,904
25    21,233,664
25    35,831,808
25    42,467,328
25    172,186,884
27    16,384
35    1,119,744
35    1,492,992
35    33,554,432
35    47,775,744
35    150,994,944
35    362,797,056
49    22,118,400
49    31,104,000
49    32,805,000
49    42,187,500
49    56,623,104
49    90,000,000
49    286,654,464
49    364,500,000
49    720,000,000
49    859,963,392
55    679,477,248
65    10,616,832
77    159,252,480
77    188,956,800
85    322,486,272
119  314,572,800

J - An Amazing Programming Language

J is a modern, high-level, general-purpose, high-performance programming language. J is portable and runs on Windows, Unix, Mac, and PocketPC handhelds, both as a GUI and in a console.

Plouffe's Inverter

Simon Plouffe has a database of more than 215,000,000 mathematical constants like Pi, E, Catalan or Euler-Mascheroni constant with more than 2 billion digits.

Wolfram | Alpha

Check out the Wolfram|Alpha computational knowledge engine.

47th Known Mersenne Prime Found!

On April 12^th, the 47^th known Mersenne prime, 2^42,643,801-1, a 12,837,064 digit number was found by Odd Magnar Strindmo from Melhus, Norway! This prime is the second largest known prime number, a "mere" 141,125 digits smaller than the Mersenne prime found last August.

Click here for more information.

Best Visual Illusion of the Year Contest

The Best Visual illusion of the Year Contest is a celebration of the ingenuity and creativity of the world’s premier visual illusion research community. Contestants from all around the world have submitted novel visual illusions (unpublished, or published no earlier than 2008), and an international panel of judges has rated them and narrowed them to the TOP TEN.

Free Mathematics Books

Found this website with links for free mathematical e-books.

Quite Basic -- Sieve of Eratosthenes

A BASIC programming website that uses the Sieve of Eratosthenes as an example.

Gallery of Illustrations

An interesting website that has plenty of mathematical illustrations.

Number Puzzles

Found some great number puzzles on the Internet.

Bridges
Dominosa
Light-up
Loop
Nonograms
Nurikabe
Shikaku
Sudoku

Drunk Men Work Here

This is a very interesting website. Problems 170 and 129 are a couple of great logic games.

Security Codes

I was entering the four-digit security code to our house, and I realized that it doesn't end with an ENTER (E) key to accept it. The problem with that is you can keep pressing numbers until the right combination is found. Since most keypads use all ten digits (0-9), the total number of combinations should be $10^n$. However, it is the ENTER key that makes it so difficult.

Let D equal the number of digits for the security code. Let K equal the number of digits on the keypad.

For a binary keypad (0, 1) and a three-digit code, our set consists of (000E, 001E, 010E, 011E, 100E, 101E, 110E, 111E). Thus, we have at most $2^3 \times (3+1)$ possible keys to enter, where the plus one is for the ENTER key. Generally speaking, we'd have $K^D \times (D+1)$ possible keys to enter.

Without the ENTER key, we could keep pressing the keypad until all combinations have formed. The minimum number of keys required would be $K^D + D$.

So for a binary keypad, we'd have:

D	String	Keys (wo/ E)	Keys (w/ E)
1	01	2	4
2	00110	5	12
3	0001110100	10	32
4	0000111100101101000	19	80
5	000001111100010010101110110011010000	36	192

Thus, one only needs to know the string concatenations to be able to guess the right combination if no ENTER or code-stopper key is required.

Update: See De Bruijn Sequence and ProjectEuler #265 for a related problem.

Cinderella

Experience Geometry on your desktop and on the web. Easily create startling geometric constructions! Starting from simple triangle relations, continuing with trigonometric theorems up to fractals and transformation groups Cinderella lets you create and manipulate visualizations in an intuitive, yet powerful way.

Click here for more information.

Wolfram Mathematica Online Integrator

This online integrator from Wolfram is actually pretty cool.

Worldmapper

Worldmapper is a collection of world maps, where territories are re-sized on each map according to the subject of interest.

How online games are solving uncomputable problems

Online games that tap your brainpower without you noticing can crack problems that have defeated the most powerful computers, says Lewis Dartnell. Get involved with distributed computing with these online games and downloads.

Click here for more information.

The Eyeballing Game

The Eyeball website gives you some mathematical figures and asks you to eyeball them correctly. I scored a 5.92 on my first try.

Just scored 3.13 on my second try.

More games at the Games for the Brain website.

Titanic Primes Raced to Win $100,000 Research Award

Researchers have discovered the two largest known prime numbers, a whopping 12,978,189 and 11,185,272 digits long, as part of a 12 year old, world-wide volunteer computing project, the Great Internet Mersenne Prime Search ("GIMPS"). The primes can be written shorthand as 2^43,112,609-1 and 2^37,156,667-1. The larger number qualifies for a $100,000 research award, most of which GIMPS will donate to the University of California, Los Angeles (UCLA), and to charity.

Click here for more information.

F#

I'm learning F#, Microsoft's latest functional programming language for the .NET family. Very nice and interesting. I recommend buying these books on it: F# for Scientists, Foundations of F#, and Expert F#.

Proof of the Existence of God

Click here for the explanation.