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.