Problems & Puzzles: Puzzles

Puzzle 947. Circulant Matrixes and a sequence of primes Paolo Lava sent the following nice puzzle: Chains of n different primes through the following process: Take a prime and calculate the determinant of the circulant matrix of its digits (*). If it is a different prime repeat the process. 1 2 2 43 -> 7 3 89899 -> 43 ->7 Det = 7 4 3 3 4 Det = 43 8 9 8 9 9 9 8 9 8 9 9 9 8 9 8 8 9 9 8 9 9 8 9 9 8 Q. What is the K-th terms for K>3?  _____ (*) About circulant matrix: http://mathworld.wolfram.com/CirculantMatrix.html  and http://en.wikipedia.org/wiki/Circulant_matrix

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Contribution came from Emmanuel Vantieghem

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Emmanuel wrote on March 26, 2019

The number  m  you find in the attachement consists of  9987  nines and two eigths : one at the position  1  and one at the position  4130. It is prime (according to Mathematica and to Dario Alpern's Alpetron) and the determinant of the circulant matrix equals  89899.

It is the smallest prime with that property.

If you want an extension of this result, you should find a prime whose sum of digits is  m.  The smallest such number could have two eights and (m+2)/9  nines. Definitely a very very big number ...

If you allow negative numbers (as  -2, -3, -5, ...) to be called 'prime' then there is a much smaller 'chain' :

88999999999999999998899999999999999999999999999999999999999999999
9999999999999999999999999999999999 -> 887 -> 23 -> -5.
 

But also here, the 'next' step would be an enormous number ...

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