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Undo revision 1828081 by AbelBanquo (talk)
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 the sum of square of the lengths happened to be a whole number divisible by some hypoteneuse, it must be divided by an odd prime hypoteneuse. 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 summed always gives an even number, which would result in the sum of square of the leg lengths not being fully divisible by any odd prime hypoteneuse.
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 a single odd prime factor ''h'' available with which to divide 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. Therefore the sum of each of the squared legs divided by an odd prime hypoteneuse can never be a whole number and thus an odd prime 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 odd primes.
4. If the sum of square of the lengths happened to be a whole number divisible by some hypoteneuse, it must be divided by an odd prime hypoteneuse. 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 summed always gives an even number, which would result in the sum of square of the leg lengths not being fully divisible by any odd prime hypoteneuse.
5. Therefore the sum of each of the squared legs divided by an odd prime hypoteneuse can never be a whole number and thus an odd prime 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 odd primes.
[[Category:Plane Geometry]]
[[Category:Plane Geometry]]
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Geometric Theorems]]
[[Category:Geometric Theorems]]