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→Corollary: No three lines with odd prime numbers as lengths can form a right triangle: ditto
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 summmed always gives an even number, which would result in the sum of square of the lengths not being fully divisible by any odd prime hypoteneuse.
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 summmed always gives an even number, which would result in the sum of square of the 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 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 prime.
5. Therefore the sum of each of the squared legs divided by an odd prime hypoteneuse can never be a 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]]