Prove that there does not exist any triplet (A, B, C) of positive integers, that satisfy this equation:

                                (36A + B)(36B + A) = 2^C

Gamer's excellent proof also can also be extended to prove there does not exist any quintuplet (A,B,C,D,E) of positive integers such that

(EDA + B)(EDB + A) = E^C

For any given D and E, there is no minimum (A,B,C) which satisfies the equation, because A and B must be divisible by E, so (A/E, B/E, C-2) also must satisfy the equation.  Because there is no minimum solution for any given values of D and E,  there is no solution at all.

Posted by Steve Herman on 2008-07-12 23:32:06