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)
The 14 at n=210 correspond to
15/195 14/196 1/13 1/14 1/182
21/189 20/190 1/9 2/19 1/171
36/174 35/175 6/29 1/5 1/145
70/140 69/141 1/2 23/47 1/94
85/125 84/126 17/25 2/3 1/75
91/119 90/120 13/17 3/4 1/68
105/105 104/106 1 52/53 1/53
106/104 105/105 53/52 1 1/52
120/90 119/91 4/3 17/13 1/39
126/84 125/85 3/2 25/17 1/34
141/69 140/70 47/23 2 1/23
175/35 174/36 5 29/6 1/6
190/20 189/21 19/2 9 1/2
196/14 195/15 14 13 1
Where, in each row are: the two unreduced adjacent fractions, the fractions reduced and the difference.
syms num den prev curr diff n d
prev=sym(0);
for num=1:209
den=sym(210-num);
curr=num/den;
diff=curr-prev;
[n,d]=numden(diff);
if n==1
%disp([num den num-1 den+1 curr prev diff])
fprintf('%5d/%-5d %5d/%-5d %9s %9s %9s
',num,den,num-1,den+1,curr,prev,diff)
end
prev=curr;
end
For 2730:
91/2639 90/2640 1/29 3/88 1/2552
105/2625 104/2626 1/25 4/101 1/2525
196/2534 195/2535 14/181 1/13 1/2353
351/2379 350/2380 9/61 5/34 1/2074
456/2274 455/2275 76/379 1/5 1/1895
546/2184 545/2185 1/4 109/437 1/1748
651/2079 650/2080 31/99 5/16 1/1584
715/2015 714/2016 11/31 17/48 1/1488
820/1910 819/1911 82/191 3/7 1/1337
910/1820 909/1821 1/2 303/607 1/1214
1015/1715 1014/1716 29/49 13/22 1/1078
1170/1560 1169/1561 3/4 167/223 1/892
1261/1469 1260/1470 97/113 6/7 1/791
1275/1455 1274/1456 85/97 7/8 1/776
1365/1365 1364/1366 1 682/683 1/683
1366/1364 1365/1365 683/682 1 1/682
1456/1274 1455/1275 8/7 97/85 1/595
1470/1260 1469/1261 7/6 113/97 1/582
1561/1169 1560/1170 223/167 4/3 1/501
1716/1014 1715/1015 22/13 49/29 1/377
1821/909 1820/910 607/303 2 1/303
1911/819 1910/820 7/3 191/82 1/246
2016/714 2015/715 48/17 31/11 1/187
2080/650 2079/651 16/5 99/31 1/155
2185/545 2184/546 437/109 4 1/109
2275/455 2274/456 5 379/76 1/76
2380/350 2379/351 34/5 61/9 1/45
2535/195 2534/196 13 181/14 1/14
2626/104 2625/105 101/4 25 1/4
2640/90 2639/91 88/3 29 1/3
For 39270:
561/38709 560/38710 1/69 8/553 1/38157
595/38675 594/38676 1/65 9/586 1/38090
715/38555 714/38556 13/701 1/54 1/37854
1156/38114 1155/38115 34/1121 1/33 1/36993
1870/37400 1869/37401 1/20 89/1781 1/35620
4081/35189 4080/35190 53/457 8/69 1/31533
5236/34034 5235/34035 2/13 349/2269 1/29497
5985/33285 5984/33286 57/317 16/89 1/28213
6546/32724 6545/32725 1091/5454 1/5 1/27270
7140/32130 7139/32131 2/9 649/2921 1/26289
7260/32010 7259/32011 22/97 61/269 1/26093
7701/31569 7700/31570 151/619 10/41 1/25379
7855/31415 7854/31416 1571/6283 1/4 1/25132
8415/30855 8414/30856 3/11 601/2204 1/24244
9010/30260 9009/30261 53/178 39/131 1/23318
10626/28644 10625/28645 23/62 125/337 1/20894
11221/28049 11220/28050 1603/4007 2/5 1/20035
11781/27489 11780/27490 3/7 1178/2749 1/19243
11935/27335 11934/27336 31/71 117/268 1/19028
12376/26894 12375/26895 52/113 75/163 1/18419
12496/26774 12495/26775 568/1217 7/15 1/18255
13090/26180 13089/26181 1/2 4363/8727 1/17454
13651/25619 13650/25620 73/137 65/122 1/16714
14400/24870 14399/24871 480/829 11/19 1/15751
15555/23715 15554/23716 61/93 101/154 1/14322
17766/21504 17765/21505 423/512 19/23 1/11776
18480/20790 18479/20791 8/9 1087/1223 1/11007
18921/20349 18920/20350 53/57 172/185 1/10545
19041/20229 19040/20230 577/613 16/17 1/10421
19075/20195 19074/20196 545/577 17/18 1/10386
19635/19635 19634/19636 1 9817/9818 1/9818
19636/19634 19635/19635 9818/9817 1 1/9817
20196/19074 20195/19075 18/17 577/545 1/9265
20230/19040 20229/19041 17/16 613/577 1/9232
20350/18920 20349/18921 185/172 57/53 1/9116
20791/18479 20790/18480 1223/1087 9/8 1/8696
21505/17765 21504/17766 23/19 512/423 1/8037
23716/15554 23715/15555 154/101 93/61 1/6161
24871/14399 24870/14400 19/11 829/480 1/5280
25620/13650 25619/13651 122/65 137/73 1/4745
26181/13089 26180/13090 8727/4363 2 1/4363
26775/12495 26774/12496 15/7 1217/568 1/3976
26895/12375 26894/12376 163/75 113/52 1/3900
27336/11934 27335/11935 268/117 71/31 1/3627
27490/11780 27489/11781 2749/1178 7/3 1/3534
28050/11220 28049/11221 5/2 4007/1603 1/3206
28645/10625 28644/10626 337/125 62/23 1/2875
30261/9009 30260/9010 131/39 178/53 1/2067
30856/8414 30855/8415 2204/601 11/3 1/1803
31416/7854 31415/7855 4 6283/1571 1/1571
31570/7700 31569/7701 41/10 619/151 1/1510
32011/7259 32010/7260 269/61 97/22 1/1342
32131/7139 32130/7140 2921/649 9/2 1/1298
32725/6545 32724/6546 5 5454/1091 1/1091
33286/5984 33285/5985 89/16 317/57 1/912
34035/5235 34034/5236 2269/349 13/2 1/698
35190/4080 35189/4081 69/8 457/53 1/424
37401/1869 37400/1870 1781/89 20 1/89
38115/1155 38114/1156 33 1121/34 1/34
38556/714 38555/715 54 701/13 1/13
38676/594 38675/595 586/9 65 1/9
38710/560 38709/561 553/8 69 1/8
but there are indeed 62 occurrences.
Fixed formatting via adding the 5's and -5's, and changing the 8's to 9's in
fprintf('%5d/%-5d %5d/%-5d %9s %9s %9s
',num,den,num-1,den+1,curr,prev,diff)
Edited on February 21, 2021, 9:51 am
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Posted by Charlie
on 2021-02-20 18:20:27 |
re: Just some guesses (showing the crunched numbers)
