A "reduced fraction" is a fraction that has the smallest possible integer numerator and denominator for its value.
For example:is not a reduced fraction, because there is another fraction equal to it but with smaller numerator and denominator:
Hilbert's tenth problem is decidable for quadratic equations by 
Ciro Santilli 40 Created 2024-07-12 Updated 2025-07-16
Ciro Santilli 40 Created 2024-07-12 Updated 2025-07-16TODO is it or is not:
- mathoverflow.net/questions/207482/algorithmic-un-solvability-of-diophantine-equations-of-given-degree-with-given
- math.stackexchange.com/questions/181380/second-degree-diophantine-equations
- mathoverflow.net/questions/142938/is-there-an-algorithm-to-solve-quadratic-diophantine-equations
- math.stackexchange.com/questions/798609/is-there-any-solution-to-this-quadratic-diophantine-equation
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.
Decidability of Hilbert's tenth problem of a given degree and number of variables by Ciro Santilli 40 Created 2024-07-12 Updated 2025-07-16
Ciro Santilli 40 Created 2024-07-12 Updated 2025-07-16 Decidability of Hilbert's tenth problem in modular arithmetic by Ciro Santilli 40 Created 2024-07-12 Updated 2025-07-16
Ciro Santilli 40 Created 2024-07-12 Updated 2025-07-16E.g. International Mathematical Olympiad, International Physics Olympiad, competitive programming, etc.
The most triggering thing possible is when programming competitions don't release their benchmarks as open source software afterwards: at least like that they might help someone to solve their real world problems. Maybe.
Some irrelevant people highlight that knowledge Olympiads can have good effects, because they are "an opportunity to meet university teachers and their research organizations". Ciro's argument is just that there are much more efficient ways to achieve those goals.
As an alternative way to get into university, this is not 100% horrible however, e.g. the University of São Paulo accepted students from olympiads in 2019 and then again 2023: jornal.usp.br/institucional/usp-oferece-200-vagas-em-mais-de-100-cursos-de-graduacao-para-alunos-participantes-de-olimpiadas-do-conhecimento/?a
Resources to study for knowledge olympiads:
- savchenkosolutions.com contains collaboratively created solutions to the Savchenko Problems in Physics. It appears to be a frontend for: github.com/savchenko-physics/savchenko-solutions. Closed license, kind of sad.
Unlike over non-commutative rings, polynomials do look like proper polynomials over commutative ring.
In particular, Hilbert's tenth problem is about polynomials over the integers, which is a commutative ring, and therefore brings mindshare to this definition.
There are unlisted articles, also show them or only show them.