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?


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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