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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.

For Diophantine representations of perfect numbers, you may read V.Y. Kryauchyukas, Diophantine representation of perfect numbers.

For James Jones' 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 of that name in J. Symbolic Logic 47 (1982), 549–571, doi:10.2307/2273588, also available from https://www.jstor.org/stable/2273588.

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.

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 available from https://projecteuclid.org/euclid.jsl/1183741086.

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.

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 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 available from https://projecteuclid.org/euclid.jsl/1183741086.