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Squares and Cubes (Posted on 2003-12-06) Difficulty: 2 of 5
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 ?

See The Solution Submitted by Ravi Raja    
Rating: 3.7500 (4 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Nontrivial Implied | Comment 9 of 21 |
(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

--------
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
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