Let a be a prime of the form (2n+1), and let b be a prime of the form (2a+1), such that 1/b has an even period of length 2a, i.e. 0 followed by 2a decimal digits.
The 'splitadd' function splits these 2a digits into equal halves and adds them.
To give an (imaginary) example, say 1/b was 0.0123456789, then 'splitadd' would produce 01234+56789, and add these for a value of 58023.
Show that the result of the 'splitadd' function is always (10^a-1), or find a counterexample.
(In reply to
re(3): possible computer solution(s) by Charlie)
Actually the problem was not in the setting of digits, but rather that symbolics can't be directly compared in Matlab, with ==, but must be compared with isequal( a, b).
A corrected list follows:
3 7
verified
5 11
verified
11 23
verified
23 47
verified
29 59
verified
41 83
smaller period than 2a
53 107
smaller period than 2a
83 167
verified
89 179
verified
113 227
smaller period than 2a
131 263
verified
173 347
smaller period than 2a
179 359
smaller period than 2a
191 383
verified
233 467
smaller period than 2a
239 479
smaller period than 2a
251 503
verified
281 563
smaller period than 2a
293 587
smaller period than 2a
359 719
smaller period than 2a
419 839
smaller period than 2a
431 863
verified
443 887
verified
491 983
verified
509 1019
verified
593 1187
smaller period than 2a
641 1283
smaller period than 2a
653 1307
smaller period than 2a
659 1319
smaller period than 2a
683 1367
verified
719 1439
smaller period than 2a
743 1487
verified
761 1523
smaller period than 2a
809 1619
verified
911 1823
verified
953 1907
smaller period than 2a
1013 2027
smaller period than 2a
1019 2039
smaller period than 2a
1031 2063
verified
1049 2099
verified
1103 2207
verified
1223 2447
verified
1229 2459
verified
1289 2579
verified
1409 2819
verified
1439 2879
smaller period than 2a
1451 2903
verified
1481 2963
smaller period than 2a
1499 2999
smaller period than 2a
1511 3023
verified
1559 3119
smaller period than 2a
1583 3167
verified
1601 3203
smaller period than 2a
1733 3467
smaller period than 2a
1811 3623
verified
1889 3779
verified
1901 3803
smaller period than 2a
1931 3863
verified
1973 3947
smaller period than 2a
2003 4007
verified
2039 4079
smaller period than 2a
2063 4127
verified
2069 4139
verified
2129 4259
verified
2141 4283
smaller period than 2a
2273 4547
smaller period than 2a
2339 4679
smaller period than 2a
2351 4703
verified
2393 4787
smaller period than 2a
2399 4799
smaller period than 2a
2459 4919
smaller period than 2a
2543 5087
verified
2549 5099
verified
2693 5387
smaller period than 2a
2699 5399
smaller period than 2a
2741 5483
smaller period than 2a
2753 5507
smaller period than 2a
2819 5639
smaller period than 2a
2903 5807
verified
2939 5879
smaller period than 2a
2963 5927
verified
2969 5939
verified
3023 6047
verified
3299 6599
smaller period than 2a
3329 6659
verified
3359 6719
smaller period than 2a
3389 6779
verified
3413 6827
smaller period than 2a
3449 6899
verified
3491 6983
verified
3539 7079
smaller period than 2a
...
smaller period than 2a
15173 30347
smaller period than 2a
15233 30467
smaller period than 2a
15269 30539
verified
15401 30803
smaller period than 2a
15569 31139
verified
15629 31259
verified
15773 31547
smaller period than 2a
15791 31583
verified
15803 31607
verified
15923 31847
verified
16001 32003
smaller period than 2a
16091 32183
verified
16253 32507
smaller period than 2a
16301 32603
smaller period than 2a
16421 32843
smaller period than 2a
16493 32987
smaller period than 2a
16553 33107
smaller period than 2a
16673 33347
smaller period than 2a
16811 33623
verified
16823 33647
verified
16883 33767
verified
16931 33863
verified
17159 34319
smaller period than 2a
17183 34367
verified
17291 34583
verified
17333 34667
smaller period than 2a
17351 34703
verified
The verified cases are indeed a larger set than I had indicated.
The cases where b is not prime have not been annotated.
There were no failures in all the prime values for a up through 17,351, the 2000th prime including those in the range of the ellipsis.
Edited on April 9, 2022, 12:49 pm
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Posted by Charlie
on 2022-04-09 12:30:58 |
re(4): possible computer solution(s)
