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manual distribution worksheet for 59536 primality tests |
What have I discovered? Of Leyland numbers with at least one million decimal digits, but fewer than one million one hundred decimal digits, there is only one prime. That prime was discovered by Gabor Levai long before I got to it. I saved all of my primality-test output where the 59536 entries are listed smallest to largest. If you want to see the one prime, search for "PRP".
The execution times vary wildly (due to processor circumstances) with an average of 9.45 hours per test. That would work out to 64 years if I hadn't been able to multi-process. I know now how to keep the execution times to 6 hours or less per test but that means running fewer processes per machine. Still, I might be able to shave a couple of months off the total time required for the next run.
Éric Angelini presented a proposal for three "digit-spine" sequences on his blog here, as well as to the MathFun community. I decided to take on the first one:
s = 1, 10, 2, 0, 3, 26, 9, 119, 532, 4, 6, 896, 118, 34, 15, ...
p = 2, 11, 2, 2, 3, 23, 7, 113, 523, 3, 5, 887, 113, 31, 13, ...
d = 1, 1, 0, 2, 0, 3, 2, 6, 9, 1, 1, 9, 5, 3, 2, ...
On Friday, Joshua Searle posted to the Sequence Fanatics Discussion list a neat procedure: take the binary complement of an integer multiplied by 3. Iterate. For example, starting with 3 we get the binary of 9 (1001), the complement of which (0110) is 6. Continuing, from 6 we get the binary of 18 (10010), the complement of which (01101) is 13. Arriving at zero, we stop.
0 3
1 6
2 13
3 24
4 55
5 90
6 241
7 300
8 123
9 142
10 85
11 0
Eleven steps to get to zero. The largest integer reached is 300 at step 7. We can shorthand the sequence data for 3 with (11,7,300) [steps to reach zero, steps to reach a maximum, the maximum]. Here are the statistics for integer starts up to 28:
0 (0,0,0)
1 (1,0,1)
2 (2,0,2)
3 (11,7,300)
4 (12,8,300)
5 (1,0,5)
6 (10,6,300)
7 (3,1,10)
8 (4,2,10)
9 (13,9,300)
10 (2,0,10)
11 (19,15,300)
12 (80,28,328536)
13 (9,5,300)
14 (2,1,21)
15 (15,11,300)
16 (16,12,300)
17 (81,29,328536)
18 (14,10,300)
19 (11,7,300)
20 (12,8,300)
21 (1,0,21)
22 (6,2,72)
23 (83,31,328536)
24 (8,4,300)
25 (73,21,328536)
26 (22,5,661)
27 (79,27,328536)
28 (7572,2962,123130640068522377168864228132316865867184046004226894)
Éric Angelini did a "smallest multiplication" bit yesterday that I felt was worth extending.
0 0 = 0 * 1
1 10 = 2 * 5
2 12 = 3 * 4
3 132 = 6 * 22
4 84 = 7 * 12
5 152 = 8 * 19
6 126 = 9 * 14
7 170 = 10 * 17
8 198 = 11 * 18
9 195 = 13 * 15
10 1008 = 16 * 63
11 1100 = 20 * 55
12 1218 = 21 * 58
13 713 = 23 * 31
14 1416 = 24 * 59
15 1150 = 25 * 46
16 1612 = 26 * 62
17 1728 = 27 * 64
18 1820 = 28 * 65
19 1914 = 29 * 66
20 1020 = 30 * 34
21 1216 = 32 * 38
22 1221 = 33 * 37
23 2345 = 35 * 67
24 2448 = 36 * 68
25 2925 = 39 * 75
26 12600 = 40 * 315
27 1927 = 41 * 47
28 2898 = 42 * 69
...
The column of indices on the far left is shown embedded (in bold) in their adjacent products. The constraint on the multiplier and multiplicand is that they must be distinct nonnegative integers with the multiplier the smallest such not yet used and the multiplicand the smallest such that yields the embedded index. A chart extending the products is here.