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38s and Bunches of 1s (Posted on 2017-06-30) Difficulty: 3 of 5
Prove every number in the sequence 38, 381, 3811, 38111, 381111, ... is composite.

See The Solution Submitted by Brian Smith    
Rating: 4.5000 (2 votes)

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re(2): No proof but a pattern found | Comment 3 of 7 |
(In reply to re: No proof but a pattern found by chun)

The second pattern, 37*103, 37*103003, 37*103003003, ..., is easily enough analyzed by the fact that 37*3 is 111. Only the first such repetition has a carry, makint the 37*1 into a 38.


Which means that 381, and all those with one more than a multiple of 3 1's are shown to be composite by the first pattern. The second pattern shows that all those with one less than a multiple of 3 1's, such as 3811, are composite.

What remains are the numbers with a number of 1's that is a multiple of 3, presumably based on the third pattern found.

Edited on June 30, 2017, 10:21 am
  Posted by Charlie on 2017-06-30 10:14:56

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