Matiyasevich theorem/Examples of Diophantine sets

Here are some simple examples of Diophantine sets.

• The set of all even non-negative integers is defined by the Diohpantine equation

\[a-2x=0 \]

• The set of all full squares is defined by the Diohpantine equation

\[a-x^2=0 \]

• The set of all non-negative integers that are not full squares is defined by the Pell's equation

\[(x+1)^2-a(y+1)^2=1 \] provided that the unknowns \(x\) and \(y\) range over non-negative integers.

• The set of all Fibonacci numbers is defined by the Diophantine equation

\[(x^2-ax-a^2)^2=1 \]