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.