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D(12,12)

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Hugo van der Sanden edited this page Dec 19, 2022 · 47 revisions

D(12,12)

This is an element of the OEIS sequence A292580. $D(12,12) = \min(x: \tau(x + i) = 12: 0 \le i < 12)$, where $\tau(x)$ is the number of divisors of $x$.

This page describes progress on the (completed) distributed effort to find and prove that minimal value. That effort proved that $D(12,12) = 120402988681658048433948$, which gives the following sequence of factorizations:

  • $120402988681658048433948 = 2^2 \cdot 3 \cdot 10033582390138170702829$
  • $120402988681658048433949 = 7^2 \cdot 17 \cdot 144541402979181330653$
  • $120402988681658048433950 = 2 \cdot 5^2 \cdot 2408059773633160968679$
  • $120402988681658048433951 = 3 \cdot 13^2 \cdot 237481240003270312493$
  • $120402988681658048433952 = 2^5 \cdot 3762593396301814013561$
  • $120402988681658048433953 = 59^2 \cdot 239 \cdot 144722262373095367$
  • $120402988681658048433954 = 2 \cdot 3^2 \cdot 6689054926758780468553$
  • $120402988681658048433955 = 5 \cdot 41^2 \cdot 14325162246479244311$
  • $120402988681658048433956 = 2^2 \cdot 7 \cdot 4300106738630644586927$
  • $120402988681658048433957 = 3 \cdot 113^2 \cdot 3143106708477772951$
  • $120402988681658048433958 = 2 \cdot 11^2 \cdot 497533011081231605099$
  • $120402988681658048433959 = 23^2 \cdot 1070301707 \cdot 212654891653$

Credits

Thank you to:

  • Natalia Makarova, Eugene Zhilitsky and CorporalTermit for contributing their time and their computers to run the code;
  • DemIS for compiling the Windows builds;
  • Dmitry Petukhov for inventing new optimizations;
  • and especially Vladimir Letsko for setting up Pentadecathlon dreams, without which this would not have happened.

The code

The primary code was built from the C code in this repository, using the Makefile target 'pcoul'. See the README for information on how to build and run it.

The list of patterns

The work is split into 1486 patterns (or "batches"), each representing a different pattern of powers of the primes (2, 3, 5, 7, 11) within the chain of 12 numbers. The full list of patterns can be found by running:

  pcoul -x:120402988681658048433948 -f11 -a 12 12

Each pattern is listed with its batch number in an output line starting "203", like:

203 b1485: . 2.11^2 3 2^2.7 5 2.3 . 2^3 3^5 2.5^2 7^5 2^2.3 [sq=2]

If the pattern fixes one or more squares, it will show "[sq=n]", as in the example above. Patterns that fix one square take a few seconds to run, those that fix more than one square complete in a fraction of a second.

There are a total of 1486 patterns. 982 of the patterns fix one or more squares, the remaining 504 patterns do not fix a square, and may take from a few minutes to a few days to run.

Running batches

Each batch must be run with a separate logfile. The values of -x and -f are required. The best arguments to use vary according to the pattern; a reasonable starting point may be something like -p2e8 -W1e7 -g6.For example:

  pcoul -x:120402988681658048433948 -f11 -p2e8 -W1e7 -g6 -rb100.log -b100 12 12

The work that needed to be done

Assignments and completed work are below. I (Hugo) ran the patterns fixing a square myself, only the patterns that do not fix a square are listed below.

Here is a CSV file of the patterns: D12-12.csv. It shows the batch ID, the LCM of the allocated values, the number of unallocated values that must be prime, the number of unallocated values that must be square, and the pattern itself. These values are expected to determine how long each batch will take to complete.

Complete:

  • Natalia LCM=19488845930400: all 24
  • Natalia LCM=56818792800: all 44
  • Natalia LCM=14642258400: all 78
  • Hugo LCM=8116970400: all 24
  • Natalia LCM=1331114400, p=7: all 4
  • Eugene LCM=1331114400, p=6: all 20
  • Natalia LCM=1331114400, p=5: all 4
  • Hugo LCM=42688800, p=7: all 16
  • CorporalTermit LCM=42688800, p=6: all 54
  • Hugo LCM=42688800, p=5: all 42
  • Eugene LCM=42688800, p=4: all 4
  • Natalia LCM=6098400, p=7: all 8
  • CorporalTermit LCM=6098400, p=6: all 46
  • CorporalTermit LCM=6098400, p=5: all 36
  • Natalia LCM=6098400, p=4: all 4
  • Natalia LCM=3880800, p=7: all 2
  • CorporalTermit LCM=3880800, p=6: all 28
  • Hugo LCM=3880800, p=5: 55, 81, 83, 110, 115, 363, 366, 370, 374, 379, 394, 398, 404, 429, 431, 458
  • CorporalTermit LCM=3880800, p=5: 463, 802, 807, 844, 846, 881, 903, 906, 913, 918, 939, 942, 945, 1219, 1224
  • Eugene LCM=3880800, p=5: 1261, 1263, 1298
  • Natalia LCM=3880800, p=4: all 4
  • Natalia LCM=554400, p=6: all 4
  • Eugene LCM=554400, p=5: all 20, except 1256
  • Hugo LCM=554400, p=5: 1256
  • Eugene LCM=554400, p=4: all 4

Timings

Times for a single batch vary widely. They will be faster if a) there are more values in the pattern fixed to be a square; b) there are more values fixed to be a prime; c) there are more values that become fixed to be a prime after a square is assigned; d) the least common multiple (LCM) of the prime powers in the pattern is higher; and e) the computer is faster.

Time for worst cases was about 4 days.

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