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Egyptian Number (Posted on 2013-02-09) Difficulty: 3 of 5
An Egyptian number is a positive integer that can be expressed as a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is 1. For example, 32 = 2 + 3 + 9 + 18 is Egyptian because 1/2+1/3+1/9+1/18=1 . Prove that all integers greater than 23 are Egyptian.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 2.3333 (3 votes)

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Solution re: Lucky numbers? Comment 5 of 5 |
(In reply to Lucky numbers? by xdog)

To complete the proof offered there (with 2a+2 and 2a+9 both being Egyptian when a is Egyptian):

The program produces the necessary Egyptian breakdown of nos. 24 - 55, which I have sorted manually below the program:

DECLARE SUB addOn ()
CLEAR , , 25000
DEFDBL A-Z
DIM SHARED done(23 TO 55), highneeded, memb, tot, rectot, maxmemb
maxmemb = 12
DIM SHARED lst(maxmemb)
highneeded = 55
memb = 0

addOn

SUB addOn
  IF memb = 0 THEN strt = 1:  ELSE strt = lst(memb)
  FOR newOne = strt TO highneeded
    IF newOne + tot <= highneeded THEN
     tot = tot + newOne
     saverectot = rectot
     rectot = rectot + 1 / newOne
     memb = memb + 1
     lst(memb) = newOne
     IF tot >= 24 THEN
       IF done(tot) = 0 THEN
         IF ABS(rectot - 1) < .00000001# THEN
           done(tot) = 1
           PRINT tot,
           FOR i = 1 TO memb
             PRINT lst(i);
           NEXT: PRINT
           WHILE done(highneeded - 1) = 1 AND done(highneeded) = 1
             highneeded = highneeded - 1
             IF highneeded = 23 THEN END
           WEND
         END IF
       END IF
     END IF
     IF rectot < 1 AND memb < maxmemb THEN
      addOn
     END IF
     tot = tot - newOne
     rectot = saverectot
     memb = memb - 1

    ELSE
     EXIT FOR
    END IF
  NEXT newOne
END SUB

24            2  4  6  12
25            5  5  5  5  5
26            4  4  6  6  6
27            3  6  6  6  6
28            4  4  4  8  8
29            2  3  12  12
30            2  3  10  15
31            2  4  5  20
32            2  3  9  18
33            3  3  9  9  9
34            2  8  8  8  8
35            2  6  9  9  9
36            2  6  8  8  12
37            2  3  8  24
38            2  6  6  12  12
39            2  6  6  10  15
40            4  4  8  8  8  8
41            2  6  6  9  18
42            2  4  12  12  12
43            2  4  10  12  15
44            3  3  6  8  24
45            2  4  9  12  18
46            2  4  8  16  16
47            3  4  8  8  12  12
48            3  4  8  8  10  15
49            3  4  6  12  12  12
50            2  4  8  12  24
51            3  3  5  10  30
52            2  5  5  20  20
53            2  5  6  10  30
54            2  3  7  42
55            2  4  7  14  28


  Posted by Charlie on 2013-02-11 11:09:19
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