# Diophantine equation

Polynomial (possibly a multivariate polynomial) with integer coefficients.
Sometimes systems of Diophantine equations are considered.
Problems generally involve finding integer solutions to the equations, notably determining if any solution exists, and if infinitely solutions exist.
The general problem is known to be undecidable: Hilbert's tenth problem.
The Pythagorean triples, and its generalization Fermat's last theorem, are the quintessential examples.

## There are infinitely many Pythagorean triples

Direct consequence of Euclid's formula.

## Fermat's last theorem

A generalization of the Pythagorean triple infinity question.

## Hilbert's tenth problem (1970, Determine if one Diophantine equation has a solution)

Once you hear about the uncomputability of such problems, it makes you see that all Diophantine equation questions risk being undecidable, though in some simpler cases we manage to come up with answers. The feeling is similar to watching people trying to solve the Halting problem, e.g. in the effort to determine BB(5).

## Decidability of Hilbert's tenth problem in modular arithmetic

www.jstor.org/stable/1970438 says for prime modulo there is an algorithm.

## Hilbert's tenth problem over other rings

mathoverflow.net/questions/11540/what-are-the-most-attractive-turing-undecidable-problems-in-mathematics/11557#11557 contains a good overview of the decidability status of variants over rings other than the integers.

## Waring's problem (Every number is a sum of positive numbers to the power of )

And when it can't, attempt to classify which subset of the integers can be reached. E.g. Legendre's three-square theorem.

## Waring's problem for squares

4 squares are sufficient by Lagrange's four-square theorem.
The subsets reachable with 2 and 3 squares are fully characterized by Legendre's three-square theorem and

## Sum of three cubes (Every number has infinitely many representations as the sum of three cubes)

Compared to Waring's problem, this is potentially much harder, as we can go infinitely negative in our attempts, there isn't a bound on how many tries we can have for each number.
In other words, it is unlikely to have a Conjecture reduction to a halting problem.

## Waring-Goldbach problem

It is exactly what you'd expect from the name, Waring was watching Netflix with Goldbach, when they suddenly came up with this.