The following fractions are written on the board 1/n, 2/(n-1), 3/(n-2), ... , n/1 where n is a natural number. Alice calculated the differences of the neighboring fractions in this row and found among them 10000 fractions of type 1/k (with natural k). Prove that he can find even 5000 more of such these differences.
(In reply to
Just some guesses by Jer)
In the light of Math Man's finding, 2310, rather than 2730 is the first appearance of 30 unit fraction differences, as it also has 5 distinct prime factors (in fact the lowest such):
unreduced reduced difference
fractions fractions
210/2100 209/2101 1/10 19/191 1/1910
231/2079 230/2080 1/9 23/208 1/1872
330/1980 329/1981 1/6 47/283 1/1698
385/1925 384/1926 1/5 64/321 1/1605
441/1869 440/1870 21/89 4/17 1/1513
540/1770 539/1771 18/59 7/23 1/1357
561/1749 560/1750 17/53 8/25 1/1325
595/1715 594/1716 17/49 9/26 1/1274
616/1694 615/1695 4/11 41/113 1/1243
715/1595 714/1596 13/29 17/38 1/1102
771/1539 770/1540 257/513 1/2 1/1026
826/1484 825/1485 59/106 5/9 1/954
925/1385 924/1386 185/277 2/3 1/831
946/1364 945/1365 43/62 9/13 1/806
1155/1155 1154/1156 1 577/578 1/578
1156/1154 1155/1155 578/577 1 1/577
1365/945 1364/946 13/9 62/43 1/387
1386/924 1385/925 3/2 277/185 1/370
1485/825 1484/826 9/5 106/59 1/295
1540/770 1539/771 2 513/257 1/257
1596/714 1595/715 38/17 29/13 1/221
1695/615 1694/616 113/41 11/4 1/164
1716/594 1715/595 26/9 49/17 1/153
1750/560 1749/561 25/8 53/17 1/136
1771/539 1770/540 23/7 59/18 1/126
1870/440 1869/441 17/4 89/21 1/84
1926/384 1925/385 321/64 5 1/64
1981/329 1980/330 283/47 6 1/47
2080/230 2079/231 208/23 9 1/23
2101/209 2100/210 191/19 10 1/19
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Posted by Charlie
on 2021-02-21 14:06:13 |
re: Just some guesses
