Determine all possible pair(s) (P, Q) of nonnegative integers such that 1 + 2^P*3^Q is a perfect square.

Up to the point where the sum of p and q reaches 5391, there are only 5 sets of (p,q).

list
   10   for Sum=1 to 10000000
   20     for P=0 to Sum
   30        Q=Sum-P
   40        V=1+2^P*3^Q
   50        Sr=int(sqrt(V)+0.5)
   60        if Sr*Sr=V then print P,Q,V,Sr
   70     next
   80   next
OK

produces      

 p       q   1+2^P*3^Q  sqrt(1+2^P*3^Q)


 0       1         4           2
 3       0         9           3
 3       1        25           5
 4       1        49           7
 5       2       289          17
 

before overflowing at a sum of p and q equal to 5391.

Posted by Charlie on 2010-03-19 12:58:28