Problems & Puzzles: Puzzles

Puzzle 1191 A chain of primes Carlos Rivera propose explore the following type of chain of primes. Starting with a prime number obtain the next couple of integers as a product of the decimal integers of such a prime and add +1 or add -1. Continue this in every branch until a) a composite appears, or b) a repeated prime appears. Example: Black-Bold = Composite; Red-Bold = repeated prime. 67 -> (43, 41)   43 -> (13, 11)   41 -> (5, 3)     13 -> (4, 2)     11 -> (2, 0)       5 -> (6, 4)       3 -> (4, 2)       2 -> (3, 1) Sequence large = 8 : 67, 43, 41, 13, 11, 5, 3, 2. Q. Find larger sequences of primes.

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From Set 28 to Oct 4, contributions came from JM Rebert, Michael Branicky, Vicente F. Izquierdo,

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JM wrote:

Let n be the length of a sequence starting with a prime p

n p        sequences

2 2        [2, 3]

3 11       [11, 2, 3]

4 41       [41, 5, 3, 2]

5 43       [43, 13, 11, 2, 3]

8 67       [67, 43, 41, 13, 11, 5, 3, 2]

9 467      [467, 167, 43, 41, 13, 11, 5, 3, 2]

10 3389    [3389, 647, 167, 43, 41, 13, 11, 5, 3, 2]

11 4799    [4799, 2269, 2267, 167, 43, 41, 13, 11, 5, 3, 2]

13 6689    [6689, 2593, 2591, 271, 269, 89, 13, 109, 107, 73, 71, 2, 3]

14 1455757 [1455757, 24499, 2593, 2591, 271, 269, 89, 13, 109, 107, 73, 71, 2, 3]

17 4588889 [4588889, 737281, 737279, 2351, 18523, 18521, 31, 29, 241, 239, 79, 2, 19, 17, 7, 53, 3]

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Michael wrote:

9 [467, 167, 43, 41, 13, 11, 5, 3, 2]
10 [3389, 647, 167, 43, 41, 13, 11, 5, 3, 2]
11 [4799, 2269, 2267, 167, 43, 41, 13, 11, 5, 3, 2]
13 [6689, 2593, 2591, 271, 269, 89, 13, 109, 107, 73, 71, 2, 3]
14 [1455757, 24499, 2593, 2591, 271, 269, 89, 13, 109, 107, 73, 71, 2, 3]
17 [4588889, 737281, 737279, 2351, 18523, 18521, 31, 29, 241, 239, 79, 2, 19, 17, 7, 53, 3]

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Vicente wrote:

The longest chain found is
17: 2,3,7,17,19,29,31,53,79,239,241,2351,18521,18523,737279,737281,4588889
There is no longer chain for either prime numbers or composite numbers until 10^23-1.
 

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