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Line 18: − + − + − + + + − 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.+ − ===Second hypothesis===+ − − Conditional sub-proof of indirect proof: Let's introduce a second hypothesis and see what follows from it. − − − − :Corollary to assumption: Through the definition of prime numbers such as described in the first "conceptualization" section, if any pair of positive integers ''n''<sub>1</sub> and ''n''<sub>2</sub> 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. − − Conditional sub-proof to be proven: ''Assuming no pair of prime numbers can possess the qualities of the pair set with the described factor, no three lines having prime numbers as lengths can form a right triangle.'' Line 42:
Line 36: − + − + − − 3. 2nd simple assumption: The hypoteneuse is always larger than either of its lengths − − 4. Because of this second 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 prime hypoteneuse is not divisible by the squared legs. − − 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.
→Reasoning: hid till I can look at it again
<center>[[image:Pythagoras.gif]]</center>
<center>[[image:Pythagoras.gif]]</center>
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__NOTOC__
==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]]