What are the two smallest positive whole numbers the difference of whose squares is a cube and the difference of whose cubes is a square ?

(In reply to Nontrivial Implied by Richard)

The program in two versions: Quick Basic and UBASIC:
DEFDBL A-Z
sum = 3
DO
  FOR a = 1 TO (sum + 1) / 2 - 1
    b = sum - a
    diffSq = b * b - a * a
    diffCu = b * b * b - a * a * a
    cuRt = INT(diffSq ^ (1 / 3) + .5)
    sqRt = INT(SQR(diffCu) + .5)
    IF sqRt * sqRt = diffCu AND cuRt * cuRt * cuRt = diffSq THEN
      PRINT a, b
      ct = ct + 1: IF ct > 40 THEN END
    END IF
  NEXT
  sum = sum + 1
LOOP

   10   for Sum=3 to 30000000


   15     Max=int((Sum+1)/2-1)


   20    for A=1 to Max


   30       B=Sum-A


   40       DiffSq=B*B-A*A


   50       DiffCu=B*B*B-A*A*A


   60       CuRt=int(DiffSq^(1/3)+0.5)


   70       SqRt=int(sqrt(DiffCu)+0.5)


   80       if SqRt*SqRt=DiffCu and CuRt*CuRt*CuRt=DiffSq then


   90       :print A,B


  100    next


  110   next

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UBASIC has additional precision available. It was stopped when sum was in the 5000+ range, and found only the two answers I posted. At these high numbers the cubing would get to the murky boundaries of the precision available to QuickBasic.

Posted by Charlie on 2003-12-06 13:52:06