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<center>[[image:Pythagoras.gif]]</center>
<center>[[image:Pythagoras.gif]]</center>
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==Corollary: No three lines with prime numbers as lengths can form a right triangle==
==Corollary: No three lines with odd prime numbers as lengths can form a right triangle==
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.
Indirect proof: Let's describe a situation where lines with the lengths of three odd prime numbers ''do'' form a right triangle and see what follows.
===Conceptualization===
===Conceptualization===
Let's call the two prime legs ''l''<sub>1</sub> and ''l''<sub>2</sub> and ''h'' the prime hypoteneuse.
Let's call the two odd prime legs ''l''<sub>1</sub> and ''l''<sub>2</sub> and ''h'' the odd prime hypoteneuse.
One definition of odd prime number ''op'' is that it is an odd natural number with no positive integral factor ''f'' other than one and itself.
===Hypothesis===
:Simple assumption: The hypoteneuse is always larger than either of its lengths.
:Simple assumption: One or more sets of ''two'' natural numbers, a pair ''n''<sub>1</sub> and ''n''<sub>2</sub>, for example, exist that have a positive integral factor ''f'' apart from one and themselves and common to both of them.
That is, it is to be proven that the first proposition (that is, the proposition that is the simple assumption, the ''assumption for a conditional proof'') leads to the second.
That is, it is to be proven that the first proposition (that is, the proposition that is the simple assumption, the ''assumption for a conditional proof'') leads to the second.
===Conceptualization (cont'd)===
===Conceptualization (cont'd)===
The conceptualization of the Pythagorean theorem in the case of the three prime numbers, that is using the ''assumption for an indirect proof'' is therefore: ''l''<sub>1</sub><sup>2</sup> + ''l''<sub>2</sub><sup>2</sup> = ''h''<sup>2</sup>.
The conceptualization of the Pythagorean theorem in the case of the three odd prime numbers, that is using the ''assumption for an indirect proof'' is therefore: ''l''<sub>1</sub><sup>2</sup> + ''l''<sub>2</sub><sup>2</sup> = ''h''<sup>2</sup>.
====Reasoning====
====Reasoning====
1. Let's divide one of the two prime number "''h''" factors contained in the "''h''<sup>2</sup>" by both sides of the equation:
1. Let's divide one of the two odd prime number "''h''" factors contained in the "''h''<sup>2</sup>" by both sides of the equation:
:(''l''<sub>1</sub><sup>2</sup> / ''h'') + (''l''<sub>2</sub><sup>2</sup> / ''h'') = ''h''.
:(''l''<sub>1</sub><sup>2</sup> / ''h'') + (''l''<sub>2</sub><sup>2</sup> / ''h'') = ''h''.
2. And since ''l''<sub>1</sub> and ''l''<sub>2</sub> are whole numbers, that which results when they are multiplied by themselves, namely ''l''<sub>1</sub><sup>2</sup> and ''l''<sub>2</sub><sup>2</sup>, certainly are as well.
2. And since ''l''<sub>1</sub> and ''l''<sub>2</sub> are whole numbers, that which results when they are multiplied by themselves, namely ''l''<sub>1</sub><sup>2</sup> and ''l''<sub>2</sub><sup>2</sup>, certainly are as well.
[[Category:Plane Geometry]]
[[Category:Plane Geometry]]
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Geometric Theorems]]
[[Category:Geometric Theorems]]