Different ways to represent every natural numbers

Question

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?

Answer 1

Here's a similar, unanswered question.

Here's a few interesting ones

1. 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 $d$.

2. 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.

3. A sum of $9$ cubes.

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

5. And last but not least, Goldbach's Conjecture.

Answer 2

1. 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.

2. 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

3. 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).

4. 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.