Waring's problem by Ciro Santilli 37 Updated 2025-07-16
And when it can't, attempt to classify which subset of the integers can be reached. E.g. Legendre's three-square theorem.
Sum of three cubes by Ciro Santilli 37 Updated 2025-07-16
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.
Video 1.
3 as the sum of the 3 cubes by Numberphile (2019)
Source.
It is exactly what you'd expect from the name, Waring was watching Netflix with Goldbach, when they suddenly came up with this.
A polynomial with multiple input arguments, e.g. with two inputs and :
as opposed to a polynomial with a single argument e.g. one with just :
By default, we think of polynomials over the real numbers or complex numbers.
However, a polynomial can be defined over any other field just as well, the most notable example being that of a polynomial over a finite field.
For example, given the finite field of order 9, and with elements , we can denote polynomials over that ring as
where is the variable name.
For example, one such polynomial could be:
and another one:
Note how all the coefficients are members of the finite field we chose.
Given this, we could evaluate the polynomial for any element of the field, e.g.:
and so on.
We can also add polynomials as usual over the field:
and multiplication works analogously.
Polynomial over a ring by Ciro Santilli 37 Updated 2025-07-16
However, there is nothing in the immediate definition that prevents us from having a ring instead, i.e. a field but without the commutative property and inverse elements.
The only thing is that then we would need to differentiate between different orderings of the terms of multivariate polynomial, e.g. the following would all be potentially different terms:
while for a field they would all go into a single term:
so when considering a polynomial over a ring we end up with a lot more more possible terms.

Pinned article: Introduction to the OurBigBook Project

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