click to enlarge |

*ocean*effect.

click to enlarge |

click to enlarge |

click to enlarge |

manual distribution worksheet for 59536 primality tests |

An hour ago I completed the primality check of 59536 Leyland-prime candidates that I started on 2 May 2022. My 27 May 2022 reality check explains why it took way longer (just short of nine months) than I had originally planned. Preceding the primality testing was another month (or so) sieving the original Leyland numbers file (so as to exclude divisibility by primes up to 2*10^11), so ten months altogether, now done.

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, ...

Our sought-after sequence is 's'. This is followed by 'p', a sequence of examples (for each, there might be a second such) of primes that are as close to the terms of 's' as possible. Finally 'd', the absolute differences between the respective 's' and 'p' values. Determining 's' is that it must be the lexicographically earliest sequence of distinct nonnegative integers such that its digit-spine (the sequential digits) as well as the digit-spine of 'd' are identical.

For example, the third 'd' being 0 is a consequence of the zero in 10, the second term of 's'. This forces the third term of 's' to be the smallest prime (as it had not already been used). And so on. There will always be integers in 'd' that inform our upcoming values in 's'. And the digits of those values get added to the growing end of our 'd' sequence.

So far, all of our 'd' values are single digits. But there will come a point when concatenating a digit with the next one in line allows for a smaller 's' term than what is available for the single digit. This first happens at term #1622 where a d = 1 dictates s = 1010. The next d digit is 0. Concatenating that 0 with the previous 1 gives us d = 10 at #1622 and this allows the smaller s = 897. This makes me want to know at which term the first three-digit d appears.

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)

Note the large number of steps and maximum in the last one. Somewhat more surprising are subsequent records. Do all integer starts eventually reach zero? Tom Duff has provided a list of the progressive record number of steps to get to zero:

1: 1

2: 2

3: 11

4: 12

9: 13

11: 19

12: 80

17: 81

23: 83

28: 7572

33: 7573

74: 7574

86: 7578

180: 7580

227: 664475

350: 664882

821: 3180929

3822: 3180930

4187: 3180931

5561: 3181981

6380: 3181988

6398: 3182002

22174: 3182226

22246: 120796790

26494: 556068798

34859: 556068799

49827: 556068871

70772: 556068872

103721: 572086553

104282: 572086610

204953: 1246707529

213884: 1246707552

225095: 1246707555

407354: 1246707602

425720: 87037147316

1671887: 128018188027

6264400: >383000000000

6524469: >327000000000

7011851: 225197172159

7404616: >370000000000

Éric Angelini did a "smallest multiplication" bit yesterday that I felt was worth extending.

0 **0** = 0 * 1

1 **1**0 = 2 * 5

2 1**2** = 3 * 4

3 1**3**2 = 6 * 22

4 8**4** = 7 * 12

5 1**5**2 = 8 * 19

6 12**6** = 9 * 14

7 1**7**0 = 10 * 17

8 19**8** = 11 * 18

9 1**9**5 = 13 * 15

10 **10**08 = 16 * 63

11 **11**00 = 20 * 55

12 **12**18 = 21 * 58

13 7**13** = 23 * 31

14 **14**16 = 24 * 59

15 1**15**0 = 25 * 46

16 **16**12 = 26 * 62

17 **17**28 = 27 * 64

18 **18**20 = 28 * 65

19 **19**14 = 29 * 66

20 10**20** = 30 * 34

21 1**21**6 = 32 * 38

22 1**22**1 = 33 * 37

23 **23**45 = 35 * 67

24 **24**48 = 36 * 68

25 29**25** = 39 * 75

26 1**26**00 = 40 * 315

27 19**27** = 41 * 47

28 **28**98 = 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.

Subscribe to:
Posts (Atom)