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
possible computer solution(s) by Charlie)
An partial annotated listing follows. The portion left out is rather boring; enough is left to get the idea:
3 7
verified
5 11
verified
7 15
b is not prime
11 23
verified
13 27
b is not prime
17 35
b is not prime
19 39
b is not prime
23 47
verified
29 59
verified
31 63
b is not prime
37 75
b is not prime
41 83
smaller period than 2a
43 87
b is not prime
47 95
b is not prime
53 107
smaller period than 2a
59 119
b is not prime
61 123
b is not prime
67 135
b is not prime
71 143
b is not prime
73 147
b is not prime
79 159
b is not prime
83 167
verified
89 179
verified
97 195
b is not prime
101 203
b is not prime
103 207
b is not prime
107 215
b is not prime
109 219
b is not prime
113 227
smaller period than 2a
127 255
b is not prime
131 263
verified
137 275
b is not prime
139 279
b is not prime
149 299
b is not prime
151 303
b is not prime
157 315
b is not prime
163 327
b is not prime
167 335
b is not prime
173 347
smaller period than 2a
179 359
smaller period than 2a
181 363
b is not prime
191 383
verified
193 387
b is not prime
197 395
b is not prime
199 399
b is not prime
211 423
b is not prime
223 447
b is not prime
227 455
b is not prime
229 459
b is not prime
233 467
smaller period than 2a
239 479
smaller period than 2a
241 483
b is not prime
251 503
verified
257 515
b is not prime
263 527
b is not prime
269 539
b is not prime
271 543
b is not prime
277 555
b is not prime
281 563
smaller period than 2a
283 567
b is not prime
293 587
smaller period than 2a
307 615
b is not prime
311 623
b is not prime
313 627
b is not prime
317 635
b is not prime
331 663
b is not prime
337 675
b is not prime
347 695
b is not prime
349 699
b is not prime
353 707
b is not prime
359 719
smaller period than 2a
367 735
b is not prime
373 747
b is not prime
379 759
b is not prime
383 767
b is not prime
389 779
b is not prime
397 795
b is not prime
401 803
b is not prime
409 819
b is not prime
419 839
smaller period than 2a
421 843
b is not prime
431 863
smaller period than 2a
433 867
b is not prime
439 879
b is not prime
443 887
smaller period than 2a
449 899
b is not prime
457 915
b is not prime
461 923
b is not prime
463 927
b is not prime
467 935
b is not prime
479 959
b is not prime
487 975
b is not prime
491 983
smaller period than 2a
499 999
b is not prime
503 1007
b is not prime
509 1019
smaller period than 2a
521 1043
b is not prime
523 1047
b is not prime
541 1083
b is not prime
547 1095
b is not prime
557 1115
b is not prime
563 1127
b is not prime
569 1139
b is not prime
571 1143
b is not prime
577 1155
b is not prime
587 1175
b is not prime
593 1187
smaller period than 2a
599 1199
b is not prime
601 1203
b is not prime
607 1215
b is not prime
613 1227
b is not prime
617 1235
b is not prime
619 1239
b is not prime
631 1263
b is not prime
641 1283
smaller period than 2a
643 1287
b is not prime
647 1295
b is not prime
653 1307
smaller period than 2a
659 1319
smaller period than 2a
661 1323
b is not prime
673 1347
b is not prime
677 1355
b is not prime
683 1367
smaller period than 2a
691 1383
b is not prime
701 1403
b is not prime
709 1419
b is not prime
719 1439
smaller period than 2a
727 1455
b is not prime
733 1467
b is not prime
739 1479
b is not prime
743 1487
smaller period than 2a
751 1503
b is not prime
757 1515
b is not prime
761 1523
smaller period than 2a
769 1539
b is not prime
773 1547
b is not prime
787 1575
b is not prime
797 1595
b is not prime
809 1619
smaller period than 2a
811 1623
b is not prime
821 1643
b is not prime
823 1647
b is not prime
827 1655
b is not prime
829 1659
b is not prime
839 1679
b is not prime
853 1707
b is not prime
857 1715
b is not prime
859 1719
b is not prime
...
7703 15407
b is not prime
7717 15435
b is not prime
7723 15447
b is not prime
7727 15455
b is not prime
7741 15483
b is not prime
7753 15507
b is not prime
7757 15515
b is not prime
7759 15519
b is not prime
7789 15579
b is not prime
7793 15587
b is not prime
7817 15635
b is not prime
7823 15647
smaller period than 2a
7829 15659
b is not prime
7841 15683
smaller period than 2a
7853 15707
b is not prime
7867 15735
b is not prime
7873 15747
b is not prime
7877 15755
b is not prime
7879 15759
b is not prime
7883 15767
smaller period than 2a
7901 15803
smaller period than 2a
7907 15815
b is not prime
7919 15839
b is not prime
For all the a values other than ten, up to the 10,000th prime, 104,729, either b was not prime or the period was smaller than 2a (typically just period a, but could be a/2^k, k being a non-negative integer).
It looks like the conditions hold for only these 10 values of a, and in each case the conjecture is verified, though this doesn't constitute proof as perhaps there's a huge value of a for which b is prime but the conjecture fails.
Toward the high end, most were of a smaller period than 2a:
...
103643 207287
smaller period than 2a
104033 208067
smaller period than 2a
104183 208367
smaller period than 2a
104231 208463
smaller period than 2a
104369 208739
smaller period than 2a
104393 208787
smaller period than 2a
104399 208799
smaller period than 2a
104561 209123
smaller period than 2a
104579 209159
smaller period than 2a
104729 209459
smaller period than 2a
Edited on April 7, 2022, 10:20 am
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Posted by Charlie
on 2022-04-07 10:16:27 |
re: possible computer solution(s)
