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==Corollary: No set of three prime numbers can form a right triangle==

==Corollary: No set of three lines with prime numbers as lengths can form a right triangle==

Conceptualization:

Indirect proof: Let's describe a situation where lines with the lengths of three prime numbers ''do'' form a right triangle and see what follows.

Let's call the two prime legs ''l''₁ and ''l''₂ and ''h'' the prime hypoteneuse.

Let's call the two prime legs ''l''₁ and ''l''₂ and ''h'' the prime hypoteneuse.

One property, by definition, of a prime number ''p''₁ is that it has no integer factor ''f'' greater than one and less than itself.

One definition of a prime number ''p'' is that it is a natural number with no positive integral factor ''f'' greater than one and less than itself.

===Conceptualization of two prime factors===

===Second hypothesis===

 

Conditional sub-proof of indirect proof: Let's introduce a second hypothesis and see what follows from it.

 

:Simple assumption: One or more sets of ''two'' natural numbers, a pair ''n'' and ''o'', for example, exist that have a positive integral factor ''f'' apart from one and themselves and common to both of them.

 

:Corollary to assumption: Through the definition of prime numbers such as described in the first "conceptualization" section, if any pair of positive integers ''n'' and ''o'' have an common integral factor ''f'' apart from one and themselves, they are not prime because they ''both'' have a non-prime type of positive integral factor; in fact it's the same one.

 

 

[[Category:Plane Geometry]]

[[Category:Plane Geometry]]

[[Category:Mathematics]]

[[Category:Mathematics]]

[[Category:Geometric Theorems]]

[[Category:Geometric Theorems]]