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
Pythagore Triple - dCode
Tag(s) : Arithmetics, Geometry
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!
Pythagore Triple
Generate Pythagorean Triples
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
Answers to Questions
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 source code of the script Pythagore Triple online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Pythagore Triple script for offline use on PC, iPhone or Android, ask for price quote on contact page !
Questions / Comments
Lire en Français
or 
