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A friend of mine thought about this problem, and I got very interested in it, but I couldn't develop it. Could some help me, because I got stuck trying to verify it's injectivity and surjectivity in a rigorous way.

Thanks in advance for any help :)

Prove that the function $f:(Z_{+} \times Z_{+}) \rightarrow Z_{+}^{*}$ and $f(x;y)=2^{y}(2x+1)$ is bijective.

Fernando Santiago
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This follows immediately from the existence and the uniqueness of prime decompositions:

For any natural number $n$, there is a unique nonnegative $y$ and a unique odd natural number $m$ such that $n = 2^y m$ and for an odd number $m$ there is a unique nonnegative integer $x$ such that $m = 2x + 1$. All in all, there are unique nonnegative ingers $x, y$ with $n = f(x,y)$.

k.stm
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  • Using UFD is overkill: if $a>1$ then any $n>0$ can be written uniquely in the form $,a^{\large y} x,\ a\nmid x,,$ where $,a^{\large y}$ is the largest power of $,a,$ that divides $,n,, $ with obvious simple proof. – Bill Dubuque Jul 29 '19 at 17:31
  • @BillDubuque The same argument carried out iteratively gives unique prime decomposition, so it’s not overkilling by much. : ⟩ – k.stm Jul 29 '19 at 21:36
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    No, the above uses only the property that a nonzero element cannot be divisible by unbounded powers of $a$, which is a much weaker property than being a UFD. – Bill Dubuque Jul 29 '19 at 21:40