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Math Help - Unique representations : list them all!

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    MHF Contributor Bruno J.'s Avatar
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    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 \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 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.
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    Super Member flyingsquirrel's Avatar
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    (5) Every positive integer is uniquely representable as a sum of distinct, nonconsecutive Fibonacci numbers. (Zeckendorf's theorem)
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  3. #3
    Moo
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    Goldbach's conjecture - Wikipedia, the free encyclopedia
    Okay, that's not a theorem lol. But some think hard that it's true ^^
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    I have this from an old combinatorial approach to number theory book

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

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

    where  a_{k}>a_{k-1}>...>a_{b}\ge b\ge 1
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    Quote Originally Posted by Moo View Post
    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.
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