For Diophantine representations of perfect numbers, you may read V.Y. Kryauchyukas, _[Diophantine representation of perfect numbers](https://doi.org/10.1007/BF01629440)_.

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](https://doi.org/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.