# Diophantine equation for generating computably enumerable set

By Matiyasevich's theorem, each member of computably enumerable set can be obtain from a diophantine equation system. For prime numbers, this system of diophantine equation is found. My question is:

For special computably enumerable set as like as perfect numbers, how can we construct this diophantine equation system?

Is there Matiyasevich's type theorem for the matrix algebra?

Jones' paper in 1982 contains a general method to transform any polynomial Diophantine equation over $$\mathbb N=\{0,1,\ldots\}$$ to one with at most 9 natural number unknowns.
• Thanks for your answer. Is there such semi results for matrix algebra? For example, is there diophantine representation of all permutation matrices or determinant $\pm{1}$ matrices? – Shahrooz Janbaz Dec 2 '18 at 7:07