Determine all possible positive real triplets (a, b, c) satisfying a^b =c; b^c = a and c^a = b

If a^b=c, then a^ab= c^a= b, and then a^abc= b^c= a, so abc=1.

Writing c=1/ab, then a^b=a^-1.b^-1, and then a^bc= a^-c.b^-c= a^-(c+1). If a≠1, this implies bc=-(c+1) which is impossible if b and c are positive, so a=1.

By symmetry, you can also prove b=1 and c=1, so (1,1,1) is the only answer.

Posted by Old Original Oskar! on 2007-06-08 09:31:56