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Pythagorean theorem (view source)

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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==
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==Corollary: No three lines with odd prime numbers as lengths can form a right triangle==
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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.
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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===
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Let's call the two prime legs ''l''<sub>1</sub> and ''l''<sub>2</sub> and ''h'' the prime hypoteneuse.
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Let's call the two odd prime legs ''l''<sub>1</sub> and ''l''<sub>2</sub> and ''h'' the odd prime hypoteneuse.
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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.
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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.
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===Hypothesis===
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===Second hypothesis===
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:Simple assumption:  The hypoteneuse is always larger than either of its lengths.
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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 proposition (that is, the proposition that is the simple assumption, the ''assumption for a conditional proof'') leads to the second.
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===Third hypothesis===
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:Second 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.
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: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.
      
===Conceptualization (cont'd)===
 
===Conceptualization (cont'd)===
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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>.
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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====
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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:
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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.
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3. 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).
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4. 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.
      
[[Category:Plane Geometry]]
 
[[Category:Plane Geometry]]
 
[[Category:Mathematics]]
 
[[Category:Mathematics]]
 
[[Category:Geometric Theorems]]
 
[[Category:Geometric Theorems]]
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