Find all pairs (A, B) of distinct nonzero integers with A ≠ -1 and B ≠ -1 such that (1 + 1/A) is a nonzero integer multiple of (1 + 1/B).

Prove that there are no others.

Note: (1 + 1/A) is can be a negative as well as positive integer multiple of (1 + 1/B). So, equations like: (1 + 1/A) = -2(1 + 1/B) or, (1 + 1/A) = (1 + 1/B) are permissible. Remember, A and B must be distinct.

DEFDBL A-Z
CLS
FOR tot = 2 TO 999999
 FOR absa = 1 TO tot - 2
  absb = tot - absa
  parta = 1 + 1 / absa: partb = 1 - 1 / absb
  rat = parta / partb
  ratrnd = INT(rat + .5)
  IF ABS(rat - ratrnd) < .0000001 THEN PRINT absa; -absb, parta; partb, parta / partb
 NEXT
NEXT

finds only

 a  b          1 + 1/a              1 + 1/b                ratio
 1 -2          2                    .5                       4
 1 -3          2                    .6666666666666666        3
 2 -2          1.5                  .5                       3
 2 -4          1.5                  .75                      2
 3 -3          1.333333333333333    .6666666666666666        2

 

Posted by Charlie on 2013-10-15 12:41:22