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Quaint Quartic-Quadratic Sum (Posted on 2010-01-09) Difficulty: 4 of 5
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.

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Some Thoughts computer exploration (spoiler) | Comment 1 of 3

   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

finds only

 0  0    1

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

Break in 50

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

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