Problems & Puzzles: Puzzles
Puzzle 197. Always composite numbers?
Sequences of odd numbers that remain composite without any easily evident reason may be as attractive as primes.
Less glamorous examples must be hidden here and there, for sure....
Very, recently while tackling the
puzzle 195 (*), I faced one probably interesting
example of that kind of sequences:
Some facts about this sequence are:
1. Can you demonstrate that the members of that sequence are always composite, or can you find the first (probable) prime of them?
2. Can you report other interesting/beauty apparently always composite odd sequences ? (we prefer sequences with more number of members composites than the reported one above; this implicitly means a not very high rate of growth)
The Questions 1 & 2 of this puzzle was solved Ken Wilke, Jean Claude Rosa, Jim Howell, Jens Kruse Andersen and Faride FiroozBakht, basically the same way.
But only Patrick De Geest noticed that this question was posed and solved previously by Chris Caldwell and Harvey Dubner, in the Journal of Recreational Mathematics, Volume 28, Number 1, 1996-1997, pp. 1-9
Every term in our sequence is a composite of two factors according to the following pattern:
Other similar sequences of odd composites similar to the ours, and studied by Caldwell & Dubner, are:
(1)k0(1)k, for k>1
(1)k2(1)k, for k>0
(3)k2(3)k, for k>0
In particular Ken Wilke wrote:
Faride FiroozBakht, wrote:
J. C. Rosa has gotten a better example than the one I posed in this puzzle: 7(1)k7, for k=>0, seems to be always composite and seems to be free of a factorizing pattern as before...
In order to save some time, I would like to anticipate that 7(1)k7 is composite in the following general cases:
a) k=even; in this case 11 is a divisor
In short the only "interesting" cases to search for possible primes are when k=6m+3, for m=>0.
Can you try again with this sequence detecting or a prime or the reason why this is a composite number for any k?
J. K. Andersen solved the follow-up posed by J.C. Rosa. Here is his email: