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September 7, 2008

Generating Pythagorean Triples

There is a simple formula that gives all the Pythagorean triples.

Suppose that m and n are two positive integers, with m < n. Then n2m2, 2mn, and n2 + m2 is a Pythagorean triple.

It’s easy to check algebraically that the sum of the squares of the first two is the same as the square of the last one.

Here are the first few triples for m and n between 1 and 10. You can also extend for numbers greater than 10.

   | m= 1          2           3           4            5           6              7            8            9
---+------------------------------------------------------------------------------------------------------------------
n= |
 2 | [3,4,5]
   |
 3 | [8,6,10]    [5,12,13]
   |
 4 | [15,8,17]   [12,16,20]  [7,24,25]
   |
 5 | [24,10,26]  [21,20,29]  [16,30,34]  [9,40,41]
   |
 6 | [35,12,37]  [32,24,40]  [27,36,45]  [20,48,52]  [11,60,61]
   |
 7 | [48,14,50]  [45,28,53]  [40,42,58]  [33,56,65]  [24,70,74]   [13,84,85]
   |
 8 | [63,16,65]  [60,32,68]  [55,48,73]  [48,64,80]  [39,80,89]   [28,96,100]  [15,112,113]
   |
 9 | [80,18,82]  [77,36,85]  [72,54,90]  [65,72,97]  [56,90,106]  [45,108,117] [32,126,130] [17,144,145]
   |
10 | [99,20,101] [96,40,104] [91,60,109] [84,80,116] [75,100,125] [64,120,136] [51,140,149] [36,160,164] [19,180,181]

August 30, 2008

Fibonacci Numbers and the Golden Ratio


Suppose you have a pair of cats given to you as babies. This pair of cats will begin to reproduce when they are two years old. They will produce another pair of cats every year. Each of these pairs of cats will also reproduce–one cat pair per year starting when they are two years old. How many cat pairs will you have after n years?

The answer to this question is a sequence of numbers known as the ‘Fibonacci sequence’:

1,1,2,3,5,8,13,…

These numbers can be described mathematically as follows:

F(n+2) = F(n+1) + F(n)

In other words, each Fibonacci number is the sum of the two preceding Fibonacci numbers. Another way of saying that is, If you know two successive Fibonacci numbers, you can find the next one by adding them together.

There are many reasons why Fibonacci numbers are interesting to mathemeticians. One of the reasons is that Fibonacci numbers are very closely related to the Golden Ratio, a number which the ancient GreeksĀ  find very interesting.

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