ピタゴラス数: pythagorean triple

直角三角形の三辺の間には

x2 + y2 = z2

の関係が成立し、この方程式を満たす自然数の組をピタゴラス数 (pythagorean triple) と呼び、方程式をピタゴラス方程式と呼びます。

直角三角形を作る三つ子の数

3: 4: 5

み よ  こ  さん

5: 12:  13

ご 自由に いーさ

7:24: 25

なんと 似ている姉妹の 双子

8:15:17

やっぱり 囲碁は いーな

見つけ方

There is a simple formula that gives all the Pythagorean triples. Suppose that m and n are two positive integers, with m < n. Then n2 - m2, 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. Why is it that every triple can be generated in this manner?

Here are the first few triples for m and n between 1 and 10. Notice any patterns?

It is well known that if you choose arbitrary positive integers m>n, that these equations generate all Pythagorean Triples (x, y, z):

x=m²-n², y=2mn, z=m²+n²

m n x y z 
2 1 3 4 5 
3 1 8 6 10 
3 2 5 12 13 
4 1 15 8 17 
4 2 12 16 20 
4 3 7 24 25 
5 1 24 10 26 
5 2 21 20 29 
5 3 16 30 34 
5 4 9 40 41 
6 1 35 12 37 
6 2 32 24 40 
6 3 27 36 45 
6 4 20 48 52 
6 5 11 60 61 
7 1 48 14 50 
7 2 45 28 53 
7 3 40 42 58 
7 4 33 56 65 
7 5 24 70 74 
7 6 13 84 85

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Last-modified: 2010-05-22 (土) 08:46:00 (3462d)