Different ways to represent every natural numbers What kind representations exist $\forall n\in \mathbb N$?
For example, each natural number $n$ can be expressed as:
$1)$ A sum of four squares (Lagrange)
$2)$ A sum of three triangular numbers (Gauss)
$3)$ A sum of Fibonacci numbers (Zeckendorfs theorem)
$4)$ A product of primes (Fundamental theorem of Number Theory)
$5)$ Factoradic digits
$6)$ Sum of "digits" in the usual sense
Are there more representations which work for every natural number?
 A: Here's a similar, unanswered question.
Here's a few interesting ones

*

*Bézout's identity — Let $a$ and $b$ be integers with greatest common divisor $d$. Then there exist integers $x$ and $y$ such that $ax + by = d$. Moreover, the integers of the form $az + bt$ are exactly the multiples of $d$.


*A sum of Lucas numbers. Let $m$ be a positive integer. Then, $m$ or $m − 2$ has an odd expression in Lucas numbers. You can find this result in this paper.


*A sum of $9$ cubes.


*Every integer greater than 11 is a sum of two composite numbers.


*And last but not least, Goldbach's Conjecture.
A: *

*We call a positive integer $n$ "powerful" if $p^2$ divides $n$ for every prime $p$ dividing $n$. Equivalently, if $n$ can be written as $a^2b^3$ for some positive integers $a,b$.

Every positive integer can be written (in infinitely many ways) as a difference of two powerful numbers, according to Wayne McDaniel,  Representations of every integer as the difference of powerful numbers, Fibonacci Quarterly 20 (1982) 85–87.


*Javier Cilleruelo, Florian Luca, and Lewis Baxter prove every positive integer is a sum of three palindromes. In fact, they do more than this: "For integer $g\ge5$, we prove that any positive integer can be written as a sum of three palindromes in base $g$". See https://arxiv.org/abs/1602.06208


*Rosales, García-Sanchéz, and García-García have a paper, "Every positive integer is the Frobenius number of a numerical semigroup with three generators." For an explanation of the terms in the title, I refer you to the publication in Math. Scand. 94 (2004) 5-12 (or just see what Wikipedia and/or other websites have to say about numerical semigroups).


*Howe, E.W., Kedlaya, K.S., Every positive integer is the order of an ordinary abelian variety over ${\bf F}_2$, Res. number theory 7, 59 (2021). https://doi.org/10.1007/s40993-021-00274-w does what it says in the title. "order" just means "number of points". ${\bf F}_2$ is the field of two elements. "ordinary abelian variety" is beyond my pay grade.
