Pythagorean triple named after Pythagorean theorem
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.
The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula 
  
    
      
        
          a
          
            2
          
        
        +
        
          b
          
            2
          
        
        =
        
          c
          
            2
          
        
      
    
    {\displaystyle a^{2}+b^{2}=c^{2}}
  ; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides 
  
    
      
        a
        =
        b
        =
        1
      
    
    {\displaystyle a=b=1}
   and 
  
    
      
        c
        =
        
          
            2
          
        
      
    
    {\displaystyle c={\sqrt {2}}}
   is a right triangle, but 
  
    
      
        (
        1
        ,
        1
        ,
        
          
            2
          
        
        )
      
    
    {\displaystyle (1,1,{\sqrt {2}})}
   is not a Pythagorean triple because 
  
    
      
        
          
            2
          
        
      
    
    {\displaystyle {\sqrt {2}}}
   is not an integer. Moreover, 
  
    
      
        1
      
    
    {\displaystyle 1}
   and 
  
    
      
        
          
            2
          
        
      
    
    {\displaystyle {\sqrt {2}}}
   do not have an integer common multiple because 
  
    
      
        
          
            2
          
        
      
    
    {\displaystyle {\sqrt {2}}}
   is irrational.
Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system. It was discovered by Edgar James Banks shortly after 1900, and sold to George Arthur Plimpton in 1922, for $10.When searching for integer solutions, the equation a2 + b2 = c2 is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation.
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