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 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 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

such that n can be uniquely represented in the form



where

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.