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 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.
------------------------------------------------------------
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
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
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.
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.
Click here for more information.
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.
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.
Update: See De Bruijn Sequence and ProjectEuler #265 for a related problem.
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.
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