Search for a tool
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

Results

Pythagore Triple -

Tag(s) : Arithmetics, Geometry

Share dCode and you

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our community Discord for help requests!

Thanks to your feedback and relevant comments, dCode has developped the best Pythagore Triple tool, so feel free to write! Thank you !

# 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.

## Source code

dCode retains ownership of the online 'Pythagore Triple' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Pythagore Triple download for offline use on PC, tablet, iPhone or Android !

## Need Help ?

Please, check our community Discord for help requests!