SOLUTION: let m and n be two consicutive even integers and 1/m+1/n=p/q (in the lowest form) prove that (p, q, q+1) is a pythagorous triplet.

Algebra ->  Numeric Fractions Calculators, Lesson and Practice -> SOLUTION: let m and n be two consicutive even integers and 1/m+1/n=p/q (in the lowest form) prove that (p, q, q+1) is a pythagorous triplet.      Log On


   

Question 475364: let m and n be two consicutive even integers and 1/m+1/n=p/q (in the lowest form) prove that (p, q, q+1) is a pythagorous triplet.
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Instead of writing n we can write m+2, in which



And we want to prove that

Begin by letting m = 2k, so our equation becomes



Simplify the LHS by dividing both top and bottom by 2:



The denominator is equal to 2k(k+1), in which k+1, 2k, 2k+1 are all pairwise relatively prime. Hence, the LHS is irreducible, so we can say that





And that , so we are done.