## 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.

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.

## 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 p

For example, p

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 = 2×3

512 → 2

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:

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

_{n}# + m is a prime number, where the primorial p_{n}# is the product of the first n prime numbers.For example, p

_{7}# = 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 = 2×3

^{2}×5×7 belongs to the sequence since p_{4}# = 210 divides 630)?512 → 2

^{5}, 16,384 → 2^{9}, and 524,288 → 2^{14}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

## 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.

Labels:
calculator,
computer,
language,
programming

### 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.

Labels:
calculator,
computer,
math

## Monday, June 15, 2009

### 47th Known Mersenne Prime Found!

On April 12

Click here for more information.

^{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.

## 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.

## Sunday, March 29, 2009

## 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.

Labels:
illustrations,
images

## Tuesday, February 24, 2009

## Sunday, January 25, 2009

## 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:

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.

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.
Labels:
codes,
combination,
My Math,
Project Euler,
security

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