This refers to David James Sycamore's A306861, which asks — for each prime — how many digit 1s must be appended to both sides of that prime (even if it already starts or ends with one or more 1s) before we generate another prime. For example, for the prime 3 only one 1 is required: 3 → 131. For the prime 263, ten are required: 263 → 11111111112631111111111. For the prime 34337, it takes one hundred 1s on each side, and for the prime 36161, it takes one thousand. Here's a three-column list giving the number for all primes less than 10000:
2 * 2833 5 6311 6
3 1 2837 3 6317 1
5 1 2843 1 6323 10
7 3 2851 3 6329 19
11 * 2857 2 6337 38
13 3 2861 6 6343 935
17 1 2879 3 6353 3
19 21 2887 1368 6359 4
23 1 2897 1 6361 3
29 1 2903 18 6367 44
31 2 2909 3 6373 2915
37 * 2917 12 6379 5
41 3 2927 4 6389 33771
43 2 2939 7 6397 2
47 1 2953 2 6421 >40000
53 1 2957 18 6427 50
59 42 2963 1 6449 10
61 14 2969 7 6451 2
67 3 2971 11 6469 8
71 73 2999 1914 6473 >50000
73 3 3001 3 6481 78
79 2 3011 4 6491 1
83 1 3019 3 6521 1
89 4 3023 4 6529 2
97 3 3037 5 6547 2
101 * 3041 1 6551 1
103 2 3049 591 6553 2
107 1 3061 6 6563 18
109 3 3067 6 6569 10
113 1 3079 2 6571 3
127 3 3083 28 6577 18
131 1 3089 3 6581 1
137 3 3109 2 6599 3
139 3 3119 12 6607 8
149 1 3121 2 6619 2
151 6 3137 1 6637 6
157 2 3163 11 6653 58
163 3 3167 1 6659 27
167 192 3169 2 6661 2
173 1 3181 5 6673 27
179 4 3187 2 6679 2
181 3 3191 288 6689 3
191 3 3203 3 6691 2
193 8 3209 7 6701 >40000
197 1 3217 96 6703 81
199 9 3221 7 6709 18
211 36 3229 159 6719 1
223 5 3251 1 6733 6
227 12 3253 9 6737 7
229 5 3257 3 6761 1
233 18 3259 2 6763 12
239 1 3271 5 6779 54
241 26 3299 16 6781 2
251 1 3301 5 6791 1
257 16 3307 102 6793 593
263 10 3313 2 6803 9
269 15 3319 3 6823 2
271 2 3323 3 6827 52
277 72 3329 10 6829 3
281 22 3331 2 6833 1
283 3 3343 20 6841 2
293 4 3347 18 6857 9
307 2 3359 4 6863 1
311 4 3361 5 6869 4
313 5 3371 1 6871 2
317 1 3373 141 6883 2
331 12 3389 15 6899 1
337 5 3391 5 6907 26
347 13 3407 21 6911 1
349 3 3413 7 6917 3
353 9 3433 12 6947 1
359 1 3449 12 6949 20
367 6 3457 5 6959 1
373 60 3461 13 6961 6
379 2 3463 2 6967 11
383 1 3467 3 6971 39
389 58 3469 39 6977 1413
397 >75000 3491 36 6983 1
401 1 3499 18 6991 44
409 2 3511 8 6997 2
419 24 3517 >50000 7001 3
421 42 3527 1 7013 30
431 4 3529 5 7019 984
433 6 3533 3 7027 3
439 30 3539 1 7039 2
443 1 3541 32 7043 15
449 4 3547 5 7057 2
457 3 3557 1 7069 12
461 27 3559 5 7079 7
463 3 3571 2 7103 54
467 67 3581 15 7109 1
479 3 3583 2 7121 3
487 5 3593 3 7127 1
491 12 3607 6 7129 35
499 6 3613 42 7151 3
503 1 3617 6 7159 2
509 1 3623 13 7177 3
521 12 3631 3 7187 696
523 6 3637 16214 7193 6
541 6 3643 437 7207 6
547 11 3659 255 7211 6
557 145 3671 1 7213 582
563 >60000 3673 5 7219 2
569 3 3677 6 7229 3
571 2 3691 17 7237 6
577 2 3697 3 7243 5
587 9 3701 33 7247 78
593 187 3709 6 7253 6
599 1 3719 1 7283 10
601 2 3727 7211 7297 6
607 38 3733 5 7307 31
613 117 3739 3 7309 20
617 6 3761 3 7321 3
619 3 3767 21 7331 24
631 2 3769 5 7333 2
641 1 3779 1 7349 1
643 5 3793 8 7351 5
647 3 3797 42 7369 8
653 73 3803 3 7393 2
659 312 3821 12 7411 3
661 3 3823 150 7417 2
673 3746 3833 3 7433 1
677 25 3847 18 7451 4
683 1 3851 1 7457 1
691 5 3853 1103 7459 11
701 1 3863 7 7477 2
709 45 3877 204 7481 >40000
719 1 3881 13 7487 6
727 572 3889 3 7489 3
733 42 3907 3 7499 1
739 46026 3911 552 7507 24
743 1 3917 3 7517 3
751 3 3919 20 7523 132
757 2 3923 6 7529 1
761 42 3929 1 7537 255
769 2 3931 14 7541 1
773 6 3943 3 7547 7
787 2 3947 3 7549 5
797 1 3967 11 7559 12
809 16 3989 1 7561 3
811 5 4001 3 7573 3
821 1 4003 5 7577 16
823 3 4007 1 7583 6
827 10 4013 4 7589 1
829 12 4019 1 7591 5
839 15 4021 5 7603 3
853 144 4027 6 7607 9
857 2068 4049 48 7621 15
859 5 4051 6 7639 3
863 9 4057 29 7643 1
877 18 4073 1 7649 3
881 3 4079 27 7669 23
883 8 4091 6 7673 13
887 4 4093 336 7681 3
907 3 4099 2 7687 3
911 4 4111 9 7691 3
919 5 4127 90 7699 60
929 12 4129 2 7703 4
937 2 4133 4 7717 18
941 16 4139 9 7723 1293
947 1 4153 8 7727 4
953 1 4157 21 7741 5
967 6 4159 >40000 7753 2
971 4 4177 57 7757 7
977 166 4201 2 7759 5
983 3 4211 1 7789 2
991 24 4217 22 7793 3
997 2 4219 3 7817 3
1009 447 4229 16 7823 1
1013 138 4231 804 7829 6
1019 4 4241 12 7841 10
1021 23 4243 3 7853 1
1031 1 4253 7 7867 15
1033 2 4259 1 7873 14
1039 11 4261 42 7877 12
1049 1 4271 1 7879 12
1051 3 4273 5 7883 1
1061 4 4283 3 7901 9
1063 12 4289 72 7907 13
1069 3 4297 2 7919 21
1087 6 4327 3 7927 810
1091 6 4337 3 7933 2
1093 14 4339 3 7937 7
1097 141 4349 3 7949 3
1103 1 4357 125 7951 3
1109 1 4363 2 7963 2
1117 41 4373 6 7993 2
1123 11 4391 42 8009 3
1129 8 4397 1 8011 11
1151 133 4409 49 8017 3
1153 3 4421 16 8039 1
1163 804 4423 2 8053 6
1171 173 4441 53 8059 126
1181 6 4447 26 8069 3
1187 1 4451 1 8081 1
1193 4 4457 591 8087 1
1201 5 4463 10 8089 492
1213 5 4481 12 8093 6
1217 3 4483 317 8101 5
1223 9 4493 1 8111 10398
1229 1 4507 30 8117 9
1231 3 4513 2 8123 4
1237 9 4517 9 8147 13
1249 * 4519 2 8161 2
1259 27 4523 4 8167 9
1277 1 4547 1 8171 1
1279 3 4549 6 8179 90
1283 1 4561 2 8191 5
1289 3 4567 323 8209 3
1291 173 4583 303 8219 3
1297 3 4591 2 8221 6575
1301 1 4597 2 8231 24
1303 8 4603 6 8233 3
1307 780 4621 27 8237 7953
1319 432 4637 30 8243 1
1321 6 4639 6 8263 150
1327 15 4643 3 8269 5
1361 4 4649 33 8273 216
1367 18 4651 5 8287 29
1373 1 4657 710 8291 6
1381 12 4663 23 8293 11
1399 17 4673 3 8297 4
1409 4 4679 4 8311 2
1423 2 4691 45 8317 20
1427 4 4703 1 8329 3
1429 12 4721 1 8353 3
1433 15 4723 2 8363 24
1439 4 4729 3 8369 1
1447 8 4733 1 8377 188
1451 12 4751 3 8387 1
1453 6 4759 2 8389 5
1459 17 4783 17 8419 3
1471 3 4787 51 8423 1
1481 252 4789 29 8429 1
1483 3 4793 30 8431 6
1487 72 4799 7 8443 12
1489 5 4801 54 8447 3
1493 16 4813 2 8461 3
1499 121 4817 1 8467 18
1511 36 4831 20 8501 4
1523 3 4861 563 8513 1
1531 6 4871 1 8521 6
1543 5 4877 2988 8527 6
1549 5 4889 1 8537 1
1553 3 4903 23 8539 3
1559 6 4909 47 8543 6
1567 5 4919 3 8563 2
1571 48 4931 6 8573 9
1579 12 4933 2 8581 3
1583 1 4937 1 8597 1
1597 15 4943 12 8599 18
1601 3 4951 3 8609 18
1607 9 4957 12 8623 342
1609 3 4967 16 8627 1
1613 1 4969 3 8629 2
1619 1 4973 1 8641 54
1621 17 4987 17 8647 3
1627 54 4993 3 8663 7
1637 1 4999 2 8669 13
1657 5 5003 6 8677 2
1663 5 5009 1 8681 6
1667 25 5011 2 8689 2
1669 11 5021 1 8693 9
1693 2 5023 2 8699 1554
1697 46 5039 7 8707 5
1699 2 5051 3 8713 5
1709 3 5059 60 8719 3
1721 7 5077 120 8731 2
1723 2 5081 7 8737 5
1733 1 5087 537 8741 570
1741 6 5099 1 8747 1
1747 8 5101 11 8753 1
1753 113 5107 2 8761 48
1759 2 5113 48 8779 2
1777 2 5119 3 8783 3
1783 12 5147 1 8803 2
1787 36 5153 1 8807 18
1789 14 5167 2 8819 9
1801 2 5171 483 8821 3
1811 3078 5179 2 8831 1
1823 9565 5189 4 8837 72
1831 3 5197 42 8839 5
1847 1 5209 2 8849 1
1861 2 5227 2 8861 4
1867 2 5231 1 8863 5
1871 36 5233 5 8867 3
1873 6 5237 4 8887 3
1877 6 5261 18 8893 30
1879 6 5273 3 8923 8
1889 1 5279 1 8929 33
1901 3 5281 71 8933 3
1907 4 5297 15 8941 3
1913 1 5303 16 8951 6
1931 1 5309 10 8963 702
1933 3 5323 378 8969 1
1949 34 5333 18 8971 18
1951 8 5347 6 8999 7
1973 138 5351 1 9001 57
1979 6 5381 186 9007 2
1987 3 5387 1 9011 6
1993 6 5393 7 9013 11
1997 1 5399 1 9029 60
1999 479 5407 42 9041 30
2003 40 5413 2 9043 3
2011 2 5417 6 9049 2
2017 2 5419 50 9059 1
2027 10 5431 5 9067 261
2029 311 5437 3 9091 17
2039 1 5441 4 9103 5
2053 83 5443 47 9109 24
2063 228 5449 12 9127 3
2069 1 5471 3 9133 3
2081 1 5477 426 9137 3
2083 2 5479 14 9151 72
2087 1 5483 3 9157 2
2089 6 5501 42 9161 3
2099 3 5503 107 9173 226
2111 12 5507 18 9181 8
2113 8 5519 1 9187 6
2129 1 5521 3 9199 78
2131 5 5527 5 9203 42
2137 2 5531 639 9209 1
2141 18 5557 359 9221 4
2143 2 5563 30 9227 1
2153 1 5569 2 9239 1
2161 6 5573 1 9241 2
2179 2 5581 3 9257 1
2203 2 5591 42 9277 2
2207 63 5623 2 9281 1
2213 1 5639 7 9283 156
2221 3 5641 5 9293 1
2237 4 5647 833 9311 6
2239 5 5651 1 9319 5
2243 130 5653 6 9323 43
2251 5 5657 75 9337 20
2267 7 5659 6 9341 31
2269 2 5669 1 9343 2
2273 3 5683 3 9349 3
2281 3 5689 3 9371 36
2287 120 5693 6 9377 1
2293 3 5701 5 9391 2
2297 1 5711 6 9397 17
2309 1 5717 6 9403 2
2311 2 5737 3 9413 3
2333 6078 5741 1 9419 24
2339 13 5743 5 9421 3
2341 11 5749 3 9431 6
2347 2 5779 20 9433 8
2351 3 5783 1 9437 1
2357 3 5791 2 9439 6
2371 11 5801 24 9461 19
2377 3 5807 1 9463 18
2381 192 5813 364 9467 1
2383 6 5821 5 9473 441
2389 14 5827 5 9479 9
2393 1 5839 2 9491 1
2399 6 5843 3 9497 15
2411 198 5849 24 9511 9
2417 1 5851 8 9521 4
2423 1 5857 11 9533 3
2437 3 5861 1 9539 3
2441 3 5867 31 9547 6
2447 1 5869 47 9551 1
2459 4 5879 1 9587 3
2467 6 5881 6 9601 2
2473 5 5897 19 9613 3
2477 1 5903 55 9619 672
2503 8 5923 2 9623 15
2521 5 5927 >40000 9629 1
2531 1 5939 24 9631 2
2539 5 5953 35 9643 3
2543 4 5981 1 9649 11
2549 28 5987 1 9661 2
2551 18 6007 2 9677 1
2557 2 6011 9 9679 2
2579 1 6029 3 9689 24
2591 >50000 6037 17 9697 30
2593 1332 6043 5 9719 25
2609 3 6047 43 9721 2
2617 18 6053 21 9733 66
2621 1 6067 6 9739 2
2633 48 6073 12 9743 63
2647 3 6079 17 9749 36
2657 6 6089 42 9767 9
2659 2 6091 11 9769 2
2663 1 6101 4 9781 2
2671 11 6113 6 9787 5
2677 420 6121 5 9791 228
2683 9 6131 39 9803 1
2687 6 6133 5 9811 15
2689 8 6143 3 9817 5
2693 18 6151 3 9829 2
2699 3 6163 6 9833 3
2707 30 6173 1 9839 1
2711 93 6197 1 9851 3
2713 12 6199 12 9857 1
2719 12 6203 3 9859 24
2729 1 6211 74 9871 56
2731 2 6217 45 9883 18
2741 9 6221 15 9887 7
2749 5 6229 2 9901 59
2753 6 6247 5 9907 3
2767 3 6257 16 9923 4
2777 9 6263 6 9929 24
2789 16 6269 1 9931 102
2791 3 6271 3 9941 1
2797 8 6277 2687 9949 3
2801 21 6287 190 9967 2
2803 18 6299 28 9973 2
2819 6 6301 3 etc
The six asterisks indicate infinity. No finite number of 1s will ever make those primes prime again. The ten greater-than symbols show lower limits for unknowns. The currently largest known number is 46026 for the prime 739.
Friday, May 24
Saturday, May 11
Infinite knight tours
I wanted to share two pictures illustrating a knight starting on a central square of an infinite chessboard jumping to every other square of that board without ever landing on the same square twice.
The first is what you usually see when researching the subject. A central 5-by-5 square is tackled first. Then two-squares-wide rings are travelled sequentially. It takes four trips to complete each ring. The trips may be made clockwise or counterclockwise. In my drawing I've alternated the directions. The tours are not limited to these possibilities as there are other variants. Just the central 5-by-5 square can be drawn in 64 different ways, but it will always end up in one of the four corners (16 ways per corner).
My second picture duplicates the green starting and red terminal square of the finite 25-by-25 approximation of our infinite chessboard.
But it uses a radically different construction method. Our central 5-by-5 starting square is joined by another to its right; and that by another above. If one follows the connections one will understand why this also will travel the entire board to infinity. I've created the 5-by-5 patterns haphazardly, as (again) there are plenty of alternatives from which to choose.
I drew my pictures using Mathematica code from Dr. Colin Rose.
The first is what you usually see when researching the subject. A central 5-by-5 square is tackled first. Then two-squares-wide rings are travelled sequentially. It takes four trips to complete each ring. The trips may be made clockwise or counterclockwise. In my drawing I've alternated the directions. The tours are not limited to these possibilities as there are other variants. Just the central 5-by-5 square can be drawn in 64 different ways, but it will always end up in one of the four corners (16 ways per corner).
My second picture duplicates the green starting and red terminal square of the finite 25-by-25 approximation of our infinite chessboard.
But it uses a radically different construction method. Our central 5-by-5 starting square is joined by another to its right; and that by another above. If one follows the connections one will understand why this also will travel the entire board to infinity. I've created the 5-by-5 patterns haphazardly, as (again) there are plenty of alternatives from which to choose.
I drew my pictures using Mathematica code from Dr. Colin Rose.
Thursday, April 18
Antipodal New Zealand
Back in 2001 I used Mathematica to create a very rough image of an antipodal New Zealand atop the countries of Portugal, Spain, and Morocco:
Antipodal New Zealand here (in case you didn't recognize it) is in red. I'm pleased to see that the latest version (12) of Mathematica does this sort of thing using a built-in GeoAntipode function and the country graphics are much improved:
What a difference! Carl Lange calculated that about 60% of New Zealand has above-sea-level land at its antipode.
Antipodal New Zealand here (in case you didn't recognize it) is in red. I'm pleased to see that the latest version (12) of Mathematica does this sort of thing using a built-in GeoAntipode function and the country graphics are much improved:
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