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Pythagore Triple

Tool to generate Pythagorean triples. A Pythagorean triple is a set of three natural integer numbers (a,b,c), such that a^2+b^2=c^2

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Pythagore Triple -

Tag(s) : Arithmetics, Geometry

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# Pythagore Triple

## Generate Pythagorean Triples

### With two sides

Tool to generate Pythagorean triples. A Pythagorean triple is a set of three natural integer numbers (a,b,c), such that a^2+b^2=c^2

### What is a Pythagorean Triple? (Definition)

A Pythagorean triplet is a set of three natural numbers $$a$$, $$b$$ and $$c$$ such that $$a^2+b^2=c^2$$

Example: (3,4,5) is a triplet of Pythagoras because $$3^2+4^2=5^2$$

### How to find a Pythagorean Triple?

It exists heuristics to find a Pythagore Triple but the easiest method consists in testing iteratively all possibilities of a and b when s is given, the value of c is constrained by s=a+b+c.

The following equations can be deducted:

$$a^2 + b^2 = (s − a − b)^2 \\ a <= (s − 3)/3 \\ b < (s − a)/2$$

Example: If $$s = 12$$, then $$a <= 3$$ and $$b < 4.5$$, a quick test allows to find $$a = 3, b = 4$$ and get the triple $$\{3,4,5\}$$.

### What is the list of Pythagorean Triples?

The first Pythagorean triples (side inferior to 100)

 (3,4,5) (5,12,13) (6,8,10) (7,24,25) (8,15,17) (9,12,15) (9,40,41) (10,24,26) (11,60,61) (12,16,20) (12,35,37) (13,84,85) (14,48,50) (15,20,25) (15,36,39) (16,30,34) (16,63,65) (18,24,30) (18,80,82) (20,21,29) (20,48,52) (21,28,35) (21,72,75) (24,32,40) (24,45,51) (24,70,74) (25,60,65) (27,36,45) (28,45,53) (28,96,100) (30,40,50) (30,72,78) (32,60,68) (33,44,55) (33,56,65) (35,84,91) (36,48,60) (36,77,85) (39,52,65) (39,80,89) (40,42,58) (40,75,85) (42,56,70) (45,60,75) (48,55,73) (48,64,80) (51,68,85) (54,72,90) (57,76,95) (60,63,87) (60,80,100) (65,72,97)

### Is there a right-angled isosceles triangle with integer sides?

There is no Pythagorean triplet with 2 identical values. Indeed if 2 sides are $$a$$ (natural integer), the last side is $$a \sqrt2$$ which can not be an integer.

Example: $$a = 1$$ the triplet becomes $$(1, 1, \sqrt2)$$. By scaling, it is not possible to obtain an both isosceles and right triangle with integer sides.

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