Is there a "canonical" representation of integers using numbers other than primes?Consider the (cumbersome) statement: "Every integer greater than 1 can be written as a unique product of integers belonging to a certain subset, S of integers.When S is the set of primes, this is the Fundamental Theorem of Arithmetic. My question is this: Are there any other types of numbers, for which this is true.EDIT: As the answers show, this obviously cannot be done. What if we relax the integer condition, i.e. can there be any other canonical representation of positive integers using complex numbers?

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Is there a &quot;canonical&quot; representation of integers using numbers other than primes?Consider the (cumbersome) statement: &quot;Every integer greater than 1 can be written as a unique product of integers belonging to a certain subset, S of integers.When S is the set of primes, this is the Fundamental Theorem of Arithmetic. My question is this: Are there any other types of numbers, for which this is true.EDIT: As the answers show, this obviously cannot be done. What if we relax the integer condition, i.e. can there be any other canonical representation of positive integers using complex numbers?

Zackary Hatfield · Accepted Answer

If you mean that every positive integer gets a unique multiplicative factorization, then no, there is no other canonical representation. Why? Because then every prime number p can be factorized, but the only way that&#039;s possible is if the components of the factorizations include the primes them-selves. Furthermore, you can&#039;t add any other number to the list because then the factorization of this number would be non-unique.Alternatively, there are non-multiplicative representations of integers. The p-adic representation is just writing n in &quot;base p&quot;:   n  =      a    0    +      a    1    p  +      a    2        p    2    +  ⋯  +      a    r        p    r  . Even though the golden ratio is not a rational number, we can write integers in base golden ratio.Algebraic number theory studies number fields and rings of integers beyond just Q and Z. Of note, there is not necessarily unique factorization of the elements. For example, in       Z    [      −    5    ], we have  6  =  2  ⋅  3  =  (  1  +      −    5    )  (  1  −      −    5    )  .This lead to some headaches (I assume anyway), until mathematicians figured out that even though the numbers don&#039;t factor uniquely, the ideals of the integers factor uniquely into products of prime ideals, which has lead to other algebraic constructions based off of them designed ultimately to study the structure of numbers. (If you don&#039;t understand this section of my answer, don&#039;t worry about it. It&#039;s for a later time then.)

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