For a right angle triangle, the length of the perpendicular sides are a, b. The length of the hypotenuse is c. a and b are positive integers and c is a prime number. Prove that 2 (a+b+1) is a...

For a right angle triangle, the length of the perpendicular sides are a, b. The length of the hypotenuse is c. a and b are positive integers and c is a prime number. Prove that 2 (a+b+1) is a perfect square number. 

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Prove that if with a,b,c positive integers and c a prime number then 2(a+b+1) is a perfect square:

This hypothesis is false. Consider a 8-15-17 right triangle. (`8^2+15^2=64+225=289=17^2` )

17 is a prime number but 2(8+15+1)=2(24)=48 is not a perfect square.

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The hypothesis is true for:

3-4-5   2(a+b+1)=16
5-12-13 2(5+12+1)=36
9-40-41 2(9+40+1)=100

etc... but the hypothesis is not true since there exists a counterexample.

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