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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?

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For Diophantine representations of perfect numbers, you may visit https://link.springer.com/article/10.1007%2FBF01629440

For Diophantine representations of Mersenne primes and Fermat primes, see http://matwbn.icm.edu.pl/ksiazki/aa/aa35/aa3531.pdf

For universal diophantine equation, please consult Jones' paper [J. Symbolic Logic 47(1982), 549-571] available from https://projecteuclid.org/euclid.jsl/1183741086.

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.

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  • $\begingroup$ 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? $\endgroup$ – Shahrooz Janbaz Dec 2 '18 at 7:07

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