There are 72 positive integer solutions thato exist.
The equation 2/x + 3/y = 1/2006 can be rewritten as 2*2006/x + 3*2006/y = 1.
Since the equation 1/(n+1) + (n-1)/(n+1) = 1, using the factor value of 1; the prime factors of 2006: 2, 17, 59; and the primes 2 and 3 for each separate fraction of the original equation; and the composite numbers of the primes, one can find 27 of the positive integer solutions.
Initially missed (as pointed out by K Sengupta in the following post), a 28th solution is available as the following equation: 2*n/(2+3)*n + 3*n/(2+3)*n also equals 1, where n<>0.
An additional 44 can be found using a little brute force where the x value is > 2006*2 and the y value < 2006*3.
The following are x, y values that will provide solutions where both x and y are positive.
4013 24150234
4014 12078126
4015 8054090
4016 6042072
4018 4030054
4020 3024045
4024 2018036
4029 1426266
4036 1012027
4046 716142
4063 479434
4071 415242
4080 361080
4114 242726
4130 210630
4148 183549
4189 142426
4216 124372
4248 108324
4301 89562
4366 74222
4420 65195
4484 57171
4590 47790
4720 40120
4879 33866
5015 30090
5168 26904
5428 23069
5746 19942
6018 18054
6324 16461
7021 14042
7480 12980
7493 12954
8024 12036
10030 10030
10948 9499
10974 9486
12036 9027
14455 8330
16048 8024
17936 7752
21063 7434
24898 7174
28084 7021
31860 6885
38114 6726
45784 6596
55165 6490
63189 6426
72216 6372
87556 6307
106318 6254
122366 6222
140420 6195
181543 6154
208624 6136
240720 6120
359074 6086
413236 6077
477428 6069
714136 6052
1010021 6042
1424260 6035
2016030 6030
3022039 6026
4028048 6024
6040066 6022
8052084 6021
12076120 6020
24148228 6019
Edited on October 1, 2006, 9:22 am
Edited on October 2, 2006, 8:43 pm
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Posted by Dej Mar
on 2006-09-30 07:53:54 |
Solution
