Thread: Unique representations : list them all!

I am interested in the various unique representations which integers have. I cannot think of many, but I am interested to know which others there are. If you know some unique representations for specific subsets of $\displaystyle \mathbb{N}$ please list them as well. Let's see how far we can get!

(1) Every integer is uniquely representable as a product of prime powers.
(2) Every integer is uniquely representable as a product of an nth power and an integer divisible by no nth power, for any n (a square and a squarefree, a cube and a cubefree, etc.)
(3) Every prime of the form $\displaystyle 4n+1$ is uniquely representable as a sum of two squares.
(4) (forgetting the obvious!) Every integer is uniquely representable in base b, for any b>0.

(5) Every positive integer is uniquely representable as a sum of distinct, nonconsecutive Fibonacci numbers. (Zeckendorf's theorem)

Goldbach's conjecture - Wikipedia, the free encyclopedia
Okay, that's not a theorem lol. But some think hard that it's true ^^

I have this from an old combinatorial approach to number theory book

$\displaystyle \forall n \in \mathbb{N} $ $\displaystyle \exists k \in \mathbb{N}$ such that n can be uniquely represented in the form

$\displaystyle n=\binom{a_{k}}{k}+\binom{a_{k-1}}{k-1}+...+\binom{a_{b}}{b} $

where $\displaystyle a_{k}>a_{k-1}>...>a_{b}\ge b\ge 1 $

Originally Posted by Moo

Goldbach's conjecture - Wikipedia, the free encyclopedia
Okay, that's not a theorem lol. But some think hard that it's true ^^

It is not unique however.