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' :
9999999999999999999999999999999999 -> 887 -> 23 -> -5.
But also here, the 'next' step
would be an enormous number ...