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→Corollary: No three lines with prime numbers as lengths can form a right triangle
<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 a prime number ''p'' is that it is a natural number with no positive integral factor ''f'' other than one and itself.
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===
===Hypothesis===
:Simple assumption: The hypoteneuse is always larger than either of its lengths.
:Simple assumption: The hypoteneuse is always larger than either of its lengths.
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 propositions (that is, the propositions that are the simple assumption, the ''assumptions for a conditional proof'') lead 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.
3. If two primes can have no common factor, squaring the primes add no new factor, so their squares don't either.
3. Because of the simple assumption, the square of the odd prime legs ''l''<sub>1</sub><sup>2</sup> and ''l''<sub>2</sub><sup>2</sup> only have an prime factor squared larger than odd prime factor ''h'' and therefore the leg lengths squared are not divisible by the odd prime hypoteneuse by the given definition of an odd prime number (otherwise it would give it a factor both greater than one and itself).
4. Because of the simple assumption, the square of the prime legs ''l''<sub>1</sub><sup>2</sup> and ''l''<sub>2</sub><sup>2</sup> only have a prime factor doubled which must be less than prime factor ''h'' and therefore the leg lengths squared are not divisible by the prime hypoteneuse by the given definition of a prime number (otherwise it would give it a factor both greater than one and itself).
4. If the sum of square of the lengths happen to be a whole number, it must be divided by an odd hypoteneuse , which can't be larger than the sum of the two sides. Odd numbers take the form of 2n + 1, which when squared gives 4n<sup>2</sup> + 4n + 1, which also must be an odd number. Two odd numbers summmed always gives an even number, which would result in not being divisible by the odd hypoteneuse.
5. Therefore the sum of each of the squared legs divided by the hypoteneuse can never be a whole number like the ''h'' that remains when ''h'' is divided through, contradicting the rephrased original premise of the indirect proof and demonstrating the opposite of the assumption to be true, namely the three lengths in a pythagorean triangle can never be all prime.
5. Therefore the sum of each of the squared legs divided by an odd prime hypoteneuse can never be a odd number like the ''h'' that remains when ''h'' is divided through, contradicting the rephrased original premise of the indirect proof and demonstrating the opposite of the assumption to be true, namely the three lengths in a pythagorean triangle can never be all prime.
[[Category:Plane Geometry]]
[[Category:Plane Geometry]]
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Geometric Theorems]]
[[Category:Geometric Theorems]]