pythagorean triples

**lilmizz1993** · November 2nd 2008, 03:13 AM

It involves pythagorian triples, and Im a little bit confused. Are there any sites with the guideline on how to do this coursework? Or are there people who have done it who can give me hints and tips? Thanks! I get so frustrated with maths!

Please its urgent!!

**Soroban** · November 2nd 2008, 04:28 AM

Hello, lilmizz1993!

I can give some of the basics of Pythagorean Triples.
But a search will give you thousands of references.

A Pythagorean Triple (PT) is a set of natural numbers $(a,b,c)$ such that: . $a^2+b^2 \:=\:c^2$

PT's can be generated with: . $\begin{Bmatrix} a &=& m^2-n^2 \\ b &=& 2mn \\ c &=& m^2+n^2 \end{Bmatrix}$

. . where $m,n$ are natural numbers and $m > n.$

Examples: . $\begin{array}{c|c}(m,n) & (a,b,c) \\ \hline<br /> (2,1) & (3,4,5) \\ (3,2) & (5,12,13) \\ (4,3) & (7,24,25) \\ \vdots & \vdots \end{array}$

If $(m,n)$ have the same parity (both odd or both even),
. . we get multiples of "primitive" (earlier) PT's.

. . $\begin{array}{c|ccc} (m,n) & (a,b,c) \\ \hline<br /> (3,1) & (8,6,10) &=&2\cdot(4,3,5)\\ (4,2) & (12,16,20) &=& 4\cdot(3,4,5) \\ (5,1) & (24,10,26) &=& 2\cdot(12,5,13) \end{array}$

If $(m,n)$ are not relatively prime (share a common factor),
. . we get multiples of primitive PT's.

. . $\begin{array}{c|ccc} (m,n) & (a,b,c) \\ \hline<br /> (6,3) & (27,36,45) &=& 9\cdot(3,4,5) \\<br /> (10,5) & (75,100,125) &=& 25\cdot(3,4,5) \end{array}$

This may inspire you to do some exploring on your own.

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