Definition:Hilbert 23/10/Historical Note

Historical Note on Algorithm to determine whether Polynomial Diophantine Equation has Integer Solution
Hilbert's $10$th problem was the question:
 * Given a Diophantine equation with any number of unknown quantities and with rational integer numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

The MRDP Theorem has shown that recursively enumerable sets are equivalent to Diophantine sets.

Matiyasevich's Theorem shows that solutions to Diophantine equations may grow exponentially.

proved this in $1970$ by using a method involving Fibonacci numbers, which grow exponentially.