Determine all triples (m,n,p) of positive rational numbers such that the numbers m + 1 n p , n + 1 p m , p + 1 m n are integers.

Question

Determine all triples (m,n,p) of positive rational numbers such that the numbers   m  +      1          n      p        ,  n  +      1          p      m        ,  p  +      1          m      n       are integers.

Callum Richmond · Accepted Answer

(      m      n      p      +      1              )        3                    (      m      n      p              )        2             is an integer. Consider   m  n  p  =      X    Y  , where   gcd  (  X  ,  Y  )  =  1. Then X=Y=1.   m  n  p  =  1   m  =      w    x  ,   n  =      y    z  ,   p  =            x      z              w      y      we can see using the three statements with divisibility solving that the only solutions are (1,1,1) ,   (  2  ,  1  ,      1    2    ) and   (  4  ,      1    2    ,      1    2    ), and all its symmetric permutations.So x∣2w, z∣2y and wy∣2xz.

Determine all triples (m,n,p) of positive rational numbers such that the numbers m+1/np,n+1/pm,p+1/mn are integers.

Answered question

Answer & Explanation

New Questions in Other