Consider three positive integers x< y< z in arithmetic sequence.

Determine analytically all possible solutions of each of the following equations:

(I) x³ + z³ = y³ + 10yz

(II) x³ + y³ = z³ - 2, with y< 116.

I didn't get the meaning of "arithmetic sequence" at first.  Thanks for the additional input.

I was unable to find a solution that fit both equations.  For the first equation, (1,4,7) and (3,6,9) fit.  For the second equation, (5,6,7) works.

For a solution that fits both equations, I had first tried by combining the equations to get [x³=5yz-1].  From this, I determined that x must end in either 4 or 9 since 5yz ends in either 0 or 5.  Subtracting the two equations yields [z³-y³=5yz+1].  Using a for the arithmatic sequence step leads to [3ax²+9a²x+7a³=5x²+3ax+2a²].  However, I could not find an integer solution that satisfied both equations.

Posted by hoodat on 2007-02-26 18:20:24