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→Corollary: No three lines with prime numbers as lengths can form a right triangle
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 a prime number ''p'' is that it is a natural number with no positive integral factor ''f'' other than one and itself.
===Second 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.
: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.
: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.
: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.
: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.
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)===