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In how many different ways can 2/15 be represented as 1/a + 1/b + 1/c, if a and b and c are positive integers with a >= b >= c.

 Nov 15, 2021
 #1
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At the website   http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html

 

they found 72 ways to express 2/15 into Egyptian fractions with 3 fractions.

 Nov 15, 2021
 #2
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Well, I found at least 80 "modern fractions" ! I only checked numbers up to 10,000.

 

1  = (1560, 130, 8)
2  = (1320, 132, 8)
3  = (1080, 135, 8)
4  = (1020, 136, 8)
5  = (920, 138, 8)
6  = (840, 140, 8)
7  = (720, 144, 8)
8  = (696, 145, 8)
9  = (600, 150, 8)
10  = (570, 152, 8)
11  = (520, 156, 8)
12  = (480, 160, 8)
13  = (440, 165, 8)
14  = (420, 168, 8)
15  = (408, 170, 8)
16  = (360, 180, 8)
17  = (345, 184, 8)
18  = (320, 192, 8)
19  = (312, 195, 8)
20  = (300, 200, 8)
21  = (280, 210, 8)
22  = (270, 216, 8)
23  = (264, 220, 8)
24  = (240, 240, 8)
25  = (720, 48, 9)
26  = (450, 50, 9)
27  = (270, 54, 9)
28  = (180, 60, 9)
29  = (126, 70, 9)
30  = (120, 72, 9)
31  = (90, 90, 9)
32  = (930, 31, 10)
33  = (480, 32, 10)
34  = (330, 33, 10)
35  = (255, 34, 10)
36  = (210, 35, 10)
37  = (180, 36, 10)
38  = (130, 39, 10)
39  = (120, 40, 10)
40  = (105, 42, 10)
41  = (90, 45, 10)
42  = (80, 48, 10)
43  = (75, 50, 10)
44  = (66, 55, 10)
45  = (60, 60, 10)
46  = (1320, 24, 11)
47  = (110, 30, 11)
48  = (420, 21, 12)
49  = (220, 22, 12)
50  = (120, 24, 12)
51  = (100, 25, 12)
52  = (70, 28, 12)
53  = (60, 30, 12)
54  = (45, 36, 12)
55  = (40, 40, 12)
56  = (1170, 18, 13)
57  = (156, 20, 13)
58  = (84, 20, 14)
59  = (70, 21, 14)
60  = (35, 30, 14)
61  = (240, 16, 15)
62  = (90, 18, 15)
63  = (60, 20, 15)
64  = (40, 24, 15)
65  = (30, 30, 15)
66  = (120, 16, 16)
67  = (48, 20, 16)
68  = (45, 18, 18)
69  = (36, 20, 18)
70  = (30, 20, 20)
71  = (28, 21, 20)
72  = (24, 24, 20)
73  = (1920, 128, 8)
74  = (1720, 129, 8)
75  = (2520, 126, 8)
76  = (2070, 46, 9)
77  = (3720, 124, 8)
78  = (3000, 125, 8)
79  = (4920, 123, 8)
80  = (7320, 122, 8)

 Nov 15, 2021

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