Determine all possible pair(s) of nonnegative integers (P, Q) that satisfy this equation.

                             P⁴ + (P+1)⁴ = Q² + (Q+1)²

Prove that these are the only pair(s) that exist.

list
   10   for Tot=0 to 999999
   20       for P=0 to Tot
   30         Q=Tot-P
   40         if P^4+(P+1)^4=Q^2+(Q+1)^2 then
   50             :print P;Q,P^4+(P+1)^4
   60       next
   70   next
OK

finds only

run
 0  0    1

by the time I stopped the program while the total of P and Q had reached 230,987:

Break in 50
?tot
 230987
OK

so it looks as if (0,0) is the only pair.

Posted by Charlie on 2010-01-09 16:58:37