# Definition:Hilbert 23/10

## Contents

## Hilbert $23$: Problem $10$

### Algorithm to determine whether Polynomial Diophantine Equation has Integer Solution

There is no algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.

## Historical Notes

### Historical Note on Hilbert's $10$th Problem

**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.

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

### Historical Note on Hilbert $23$

The **Hilbert 23** were delivered by David Hilbert in a famous address at Paris in $1900$.

He considered them to be the oustanding challenges to mathematicians in the future.

There was originally going to be a $24$th problem, on a criterion for simplicity and general methods in proof theory, but Hilbert decided not to include it, as it was (like numbers $4$, $6$, $16$ and $23$) too vague to ever be described as "solved".

## Sources

- 1902: David Hilbert:
*Mathematical Problems*(*Bull. Amer. Math. Soc.***Vol. 8**,*no. 10*: 437 – 479)

- (translated by Mary Winston Newson from "Mathematische Probleme")

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Diophantine equation** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Diophantine equation**