## Thursday, December 17, 2009

### 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.

## Friday, December 11, 2009

### 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.

### 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.

## Thursday, November 05, 2009

### 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 pn# + m is a prime number, where the primorial pn# 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.

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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?

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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

## Tuesday, July 28, 2009

### 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.

## Monday, June 15, 2009

### 47th Known Mersenne Prime Found!

On April 12th, the 47th known Mersenne prime, 242,643,801-1, 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.

## Wednesday, June 03, 2009

### 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.

## Saturday, February 28, 2009

### 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.

## Tuesday, February 24, 2009

### Number Puzzles

Found some great number puzzles on the Internet.

Bridges
Dominosa
Light-up
Loop
Nonograms
Nurikabe
Shikaku
Sudoku

## Sunday, January 25, 2009

### Drunk Men Work Here

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

## Monday, January 05, 2009

### 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:

DStringKeys (wo/ E)Keys (w/ E)
10124
200110512
300011101001032
400001111001011010001980
500000111110001001010111011001101000036192

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.