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Complex Conclusion (Posted on 2012-08-29) Difficulty: 3 of 5
Each of a and c is a positive integer and each of b and d is an integer with b ≥ d.

Determine all possible pairs (a + bi, c + di) of complex numbers whose sum equals their product.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts computer exploration | Comment 1 of 6

DEFDBL A-Z
FOR actot = 2 TO 2000
  FOR a = 1 TO actot - 1
    c = actot - a
    ac = a * c
    bd = ac - a - c
    absbd = ABS(bd)
    SELECT CASE bd
       CASE 0
         IF a = 1 THEN PRINT a; 0; c; "any neg value"
         IF c = 1 THEN PRINT a; "any pos value"; c; 0
       CASE IS > 0
         FOR d = 1 TO SQR(bd)
           b = bd / d
           IF b = INT(b) THEN
             IF b + d = b * c + a * d THEN
               PRINT a; b; c; d
             END IF
           END IF
         NEXT
       CASE IS < 0
         FOR f1 = 1 TO SQR(absbd)
           f2 = absbd / f1
           IF f2 = INT(f2) THEN
             b = f1: d = -f2
             IF b + d = b * c + a * d THEN
               PRINT a; b; c; d
             END IF
             b = f2: d = -f1
             IF b + d = b * c + a * d THEN
               PRINT a; b; c; d
             END IF
           END IF
         NEXT f1
    END SELECT
  NEXT
NEXT actot

finds only

1  1  1 -1

as a, b, c and d respectively, where the sum is 2 and the product is 2. (It finds it twice, actually, as in the latter part f1 and f2 are equal.)

All possibilities were searched where a+c totals up to 2000.

Edited on August 29, 2012, 12:56 pm
  Posted by Charlie on 2012-08-29 12:54:45

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