Math Research, Tips and Tricks

Shor, I’ll do it

2011-11-02T10:59:00.000-07:00

I ran into this article by Scott Aaronson today and thought I would share it. To wet your taste, here's the first sentence of the second paragraph: Alright, so let’s say you want to break the RSA cryptosystem, in order to rob some banks, read your ex’s email, whatever. Overall, I found it to be a pretty good writeup for the layman. It's a little esoteric when it comes to explaining the QFT,

Stern–Brocot tree and Continuous Fractions

2010-03-13T01:28:00.000-08:00

This post continues the previous entries on Continuous Fractions. This time, we expand the concept to encompass the Stern-Brocot tree. The Stern-Brocot tree is a binary search tree where the two legs represent "less than" and "more than" (or Low,High or Left,Right, etc). As you can see from the image, it starts with 1 (or 1/1) at the top. The left side is less than 1 and the right side is more

Slashdot: 47th Mersenne Prime Confirmed

2009-06-16T01:51:00.000-07:00

The new prime, 2^42,643,801 - 1, is actually smaller than the one discovered previously.

Primality of 1

2009-05-18T00:18:00.000-07:00

When I was in school, I remember them teaching that a prime number was any number divisible by only 1 and itself. Thus, in school, they taught that 1 was prime. Since then, I have noticed that everyone insists that 1 is not prime. Since many of the patterns I find work better with 1 being prime, I decided to try to figure out why the discrepancy...From Wikipedia:Until the 19th century, most

Prime digits

2009-05-17T23:52:00.000-07:00

A coworker made a comment to me Friday about the distribution of digits in prime numbers. He was referring to a recent Slashdot article talking about the first digit in the prime number...but I have never liked thinking of the 1st digit as the far left... I'd prefer to think of the far right digit as digit 0. Can't help it - it's the whole positional notation thing (10^3 10^2 10^1 10^0 . 10^-1

Division by Zero

2009-04-27T22:27:00.000-07:00

I've been on this kick lately where I have started question some of the basic assumptions I learned in math... things like "you can't easily define pi" or the concept of an "irrational number". Today's philosophical debate is on the idea of division by zero.This came up because I was creating a ContinuedFraction interface for integers... The number 3, for instance, is 3 + 0/0 + 0/0 .... etc...

SeriesCF

2009-04-26T03:32:00.000-07:00

While working on my ContinuedFractions library today, it occured to me that our current representation of pi could be simplified by just specifying the numerators and denominators as series... I went ahead and changed the implementation to use a new SeriesCF class instead of the default ContinuedFraction class and I think the new approach is much cleaner.

PresenTeX

2009-04-25T14:25:00.000-07:00

While writing this package today, I came across the PresenTeX website. Very nice...from cf = d_0 + \cfrac{n_0}{d_1 + \cfrac{n_1}{d_2 + \cfrac{n_2}{d_3 + \cfrac{n_3}{\ddots\,}}}} I got:

Continued Fractions

2009-02-10T12:03:00.001-08:00

I was working on some continued fractions, and came across the fundamental recurrence formula above. It worked pretty well for some simple continued fractions (like √2) but seemed to be a bit unfriendly towards calculating π.I'm using this definition of π:Now, using that definition with the above equation, it was always completely wrong...So I tried changing the equation...So you'll notice a few

Intuitive Approach to Wilson's Theorem

2007-07-27T11:51:00.000-07:00

The other day I stumbled across something, only later to realize that it was Wilson's Theorem. Since the approach used to find it in the first place was more of a 'common-sense' approach than what you usually encounter for it, I thought I would go ahead and present it.Let's look at two examples.... For both, what we are looking for is:(n-1)! % n ≡ (n-1) mod n; iff n is primeExample 1: Composite

An Intuitive Guide To Exponential Functions & E

2007-07-20T10:03:00.001-07:00

Fairly simple explanation of what 'e' is and how it works.

Wikipedia: Mental Arithmetic

2007-07-15T20:34:00.000-07:00

I stumbled upon this today, and thought I would share. Some interesting stuff in there.We really should start teaching kids mental arithmetic at an early age rather than the "show your work or get an F"

Coprime, aka. Relatively Prime

2007-05-06T10:23:00.000-07:00

Two numbers are coprime (relatively prime) if they have no factors in common. I thought this image from Wikipedia was a good visual way of seeing it: Figure 1. The numbers 4 and 9 are coprime because the diagonal does not intersect any other lattice points

Subfactorial

2007-04-13T08:57:00.000-07:00

I thought this was a good way of explaining it:In practical terms, subfactorial is the number of ways of putting n letters into n envelopes (one in each envelope) with none in its correct envelope.

Pi

2007-04-05T17:13:00.000-07:00

This is probably the simplest explanation of Pi that I have ever seen:

Mock Theta Functions

2007-03-01T14:10:00.001-08:00

Sum of Series

2007-02-26T15:02:00.000-08:00

sum of odd integers < 2n = n2sum of all integers = n(n+1)/2

Pascals' Cototients

2007-02-16T12:19:00.000-08:00

Similar to yesterdays' post, this one is based off the totients of the values in Pascal's triangle.. but this time, we are displaying the Cototients (n - totient(n))...

Pascal's Totients

2007-02-15T15:33:00.000-08:00

I was trying to figure out new ways to look at Pascals' Triangle today, and decided to look not at the actual binomial coefficients, but at the Euler Totients of them. Take a gander.

Pascal and Binomial Coefficients

2007-02-12T07:39:00.000-08:00

I realized that I never really showed how Pascal's Triangle is related to the Binomial Coefficients... So here we go:(x+1)0 = 1(x+1)1 =1x + 1 =1 1(x+1)2 =(x+1)(x+1) = x*x + x*1 + 1*x + 1*1 = 1x2 + 2x + 1 = 1 2 1(x+1)3 =(x+1)(x+1)(x+1) = ((x+1)(x+1))(x+1) = (1x2 + 2x + 1)(x+1) = (1x2*x + 1x2)+(2x*x + 2x*1)+(1*x + 1*1)= 1x3 + 1x2 + 2x2 + 2x + x + 1 = 1x3 + 3x2 + 3x + 1 =1 3 3 1(x+1)4 =(x+1)(x+1)(x

A Modded Pacal Triangle

2007-02-07T12:02:00.000-08:00

As you generate very large Pascal Triangles, the numbers quickly get out of control. If you are planning on modding the values after obtaining them, why not mod them first?Here you see an example of doing a mod5 before writing the number down... So, for example, the middle element on Row#4 is 1 because (3+3)%5 = 6%5 = 1... Now, you can use the number 1 for further rows instead of the original

Binary Representation of Pascal's Triangle

2007-02-06T15:51:00.000-08:00

To look like this, one thing you might do is work out all the values of the previous lines and then mod them to determine whether they should be highlighted or not.Yet another way might be to just keep track of whether the values above are 1 or 0, and realize that the next one down is only 0 if both the parent hexes are 1.But, how about a way to do it without any regard to previous rows? This

Twin Primes

2007-01-26T16:42:00.000-08:00

So as I rewrite my Balanced Ternary (ī,0,1) library once again, something occurs to me...While it is not visually obvious which numbers are multiples of two, we do know that all multiples of 3 end in 0. Since no prime (except 3) can be a multiple of 3, that means that all primes are going to end in ī or 1. Well, since we know all primes are odd, that also means that the specific number with a 0

Prime Numbers, Assumptions and Thoughts

2007-01-18T11:18:00.000-08:00

Prime numbers have been a challenge to people for a very long time. If they are something that we still can't write an equation for, perhaps it is time to challenge our assumptions about these fascinating and extremely important numbers.First, it is commonly believed that there is no pattern to prime numbers. Research studied under this belief might very well be simply falling prey to

Slashdot | Largest Twin Prime Yet Discovered

2007-01-17T19:40:00.000-08:00

The Twin Internet Prime Search and PrimeGrid have recently discovered the largest known twin prime. A twin prime is a pair of prime numbers separated by the integer two. The pair discovered on January 15th was 2003663613 * 2195,000 ± 1. The two primes are 58,711 digits long. The discoverer was Eric Vautier, from France.