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For reference, let [itex]\mathbb N^{\infty}[/itex] denote the set of all [itex]\infty[/itex]-tuples of the form [itex](n_{1}, n_{2}, \dots)[/itex] with [itex]n_{i}\in \mathbb N ,\,\forall i[/itex], and let [itex]\mathbb N^{*}=\{1,2,3,\dots\}[/itex].

Let [itex]n\in\mathbb N^{*}[/itex]. Then by the Fundamental Theorem of Arithmetic, we can represent [itex]n[/itex] as a unique factorization of primes. That is [tex]n=p_{1}^{e_{1}} p_{2}^{e_{2}}\cdots=\prod_{k=1}^{\infty} p_{k}^{e_{k}}.[/tex] Define a function [itex]\phi :\mathbb N^{*} \to \mathbb N^{\infty}[/itex] such that [itex]\phi(n)=(e_{1},e_{2},\dots)=(e_{k})_{k=1}^{\infty}[/itex]. Further, define a binary operation [itex]\oplus:\mathbb N^{\infty}\times\mathbb N^{\infty}\to\mathbb N^{\infty}[/itex] such that [tex]\oplus((a_{1},a_{2},\dots),(b_{1},b_{2},\dots))=(a_{1}+b_{1},a_{2}+b_{2},\dots).[/tex]

I will now show that [itex](\mathbb N^{\infty},\oplus)[/itex] is a commutative monoid.

Thus, [itex](\mathbb N^{\infty},\oplus)[/itex] is a commutative monoid.

Now I will show that [itex]\phi[/itex] is a commutative monoid homomorphism.

An interesting result to is to show that [itex]\phi[/itex] is actually a commutative monoid isomorphism. I leave this up to the reader.[EDIT: THIS IS INCORRECT IF WE'RE WORKING WITH [itex]\mathbb N^{\infty}[/itex]]

I've been messing around with this, and you can get some pretty cool new operations on numbers. Now that they are "basically" vectors, you can start making new operations like dot product, tensor product, their 'length' with respect to different norms. My favourite so far has been showing a homomorphism from [itex]\mathbb N^{\infty}\to\mathbb P[x]^{\infty}[/itex], the set of all polynomials with one variable. Once this is done, you can create a new binary operation such that you take two numbers from [itex]\mathbb N^{*}[/itex], convert both separately by [itex]\phi[/itex] to [itex]\mathbb N^{\infty}[/itex], then to [itex]\mathbb P[x]^{\infty}[/itex], and do normal polynomial multiplication, then convert that polynomial all the way back to [itex]\mathbb N^{*}[/itex].

I also feel (severely speculative), if you can figure out some way with all these new operations to define a homomorphic addition within [itex]\mathbb N^{\infty}[/itex], we could find a simpler solution to Fermat's Last Theorem.

You can also play around with it more finitely, which I can explain a bit later, but my fingers are tired from typing.

If you have input, or come up with your own results, or have criticisms, please contribute! I would like to see if these definitions bring out anything cool.

Let [itex]n\in\mathbb N^{*}[/itex]. Then by the Fundamental Theorem of Arithmetic, we can represent [itex]n[/itex] as a unique factorization of primes. That is [tex]n=p_{1}^{e_{1}} p_{2}^{e_{2}}\cdots=\prod_{k=1}^{\infty} p_{k}^{e_{k}}.[/tex] Define a function [itex]\phi :\mathbb N^{*} \to \mathbb N^{\infty}[/itex] such that [itex]\phi(n)=(e_{1},e_{2},\dots)=(e_{k})_{k=1}^{\infty}[/itex]. Further, define a binary operation [itex]\oplus:\mathbb N^{\infty}\times\mathbb N^{\infty}\to\mathbb N^{\infty}[/itex] such that [tex]\oplus((a_{1},a_{2},\dots),(b_{1},b_{2},\dots))=(a_{1}+b_{1},a_{2}+b_{2},\dots).[/tex]

I will now show that [itex](\mathbb N^{\infty},\oplus)[/itex] is a commutative monoid.

- Let [itex]u,v\in\mathbb N^{\infty}[/itex] with [itex]u=(a_{k})_{k=1}^{\infty}[/itex] and [itex]v=(b_{k})_{k=1}^{\infty}[/itex]. Then [itex]u\oplus v=(a_{k}+b_{k})_{k=1}^{\infty}=(b_{k}+a_{k})_{k=1}^{\infty}=v\oplus u[/itex]. Since [itex]a_{k}+b_{k}=b_{k}+a_{k}\in\mathbb N,\,\forall k[/itex], it follows that [itex]u\oplus v=v\oplus u\in\mathbb N^{\infty}[/itex]. So [itex](\mathbb N^{\infty},\oplus)[/itex] is both closed and commutative, as required.

- Let [itex]u,v,w\in\mathbb N^{\infty}[/itex] with [itex]u=(a_{k})_{k=1}^{\infty}[/itex],[itex]v=(b_{k})_{k=1}^{\infty}[/itex] and [itex]w=(c_{k})_{k=1}^{\infty}[/itex]. Then [itex](u\oplus v)\oplus w=((a_{k}+b_{k})+c_{k})_{k=1}^{\infty}=(a_{k}+(b_{k}+c_{k}))_{k=1}^{\infty}=u\oplus (v\oplus w)[/itex]. Thus [itex](\mathbb N^{\infty},\oplus)[/itex] is associative.

- Let [itex]1_{\mathbb N^{\infty}}\in\mathbb N^{\infty}[/itex] such that [itex]1_{\mathbb N^{\infty}}=(0,0,\dots)[/itex]. Then for any [itex]u=(a_{k})_{k=1}^{\infty}\in\mathbb N^{\infty}[/itex], we have [itex]1_{\mathbb N^{\infty}}\oplus u=(0+a_{k})_{k=1}^{\infty}=(a_{k})_{k=1}^{\infty}=(a_{k}+0)_{k=1}^{\infty}=u\oplus 1_{\mathbb N^{\infty}}[/itex]. Thus, [itex]1_{\mathbb N^{\infty}}[/itex] is the identity element of [itex](\mathbb N^{\infty},\oplus)[/itex].

Thus, [itex](\mathbb N^{\infty},\oplus)[/itex] is a commutative monoid.

Now I will show that [itex]\phi[/itex] is a commutative monoid homomorphism.

- Let [itex]a,b\in\mathbb N[/itex]. Then by the Fundamental Theorem of Arithmetic, both [itex]a[/itex] and [itex]b[/itex] have a unique factorization of primes. That is [itex]a=\prod_{k=1}^{\infty} p_{k}^{e_{k}}[/itex] and [itex]b=\prod_{k=1}^{\infty} p_{k}^{f_{k}}[/itex]. Then [itex]ab=\prod_{k=1}^{\infty} p_{k}^{e_{k}+f_{k}}[/itex]. So we have [itex]\phi (ab)=\phi (\prod_{k=1}^{\infty} p_{k}^{e_{k}+f_{k}})=(e_{k}+f_{k})_{k=1}^{\infty}=(e_{k})_{k=1}^{\infty}\oplus (f_{k})_{k=1}^{\infty}=\phi(a)\oplus\phi(b)[/itex], as required.

An interesting result to is to show that [itex]\phi[/itex] is actually a commutative monoid isomorphism. I leave this up to the reader.[EDIT: THIS IS INCORRECT IF WE'RE WORKING WITH [itex]\mathbb N^{\infty}[/itex]]

I've been messing around with this, and you can get some pretty cool new operations on numbers. Now that they are "basically" vectors, you can start making new operations like dot product, tensor product, their 'length' with respect to different norms. My favourite so far has been showing a homomorphism from [itex]\mathbb N^{\infty}\to\mathbb P[x]^{\infty}[/itex], the set of all polynomials with one variable. Once this is done, you can create a new binary operation such that you take two numbers from [itex]\mathbb N^{*}[/itex], convert both separately by [itex]\phi[/itex] to [itex]\mathbb N^{\infty}[/itex], then to [itex]\mathbb P[x]^{\infty}[/itex], and do normal polynomial multiplication, then convert that polynomial all the way back to [itex]\mathbb N^{*}[/itex].

I also feel (severely speculative), if you can figure out some way with all these new operations to define a homomorphic addition within [itex]\mathbb N^{\infty}[/itex], we could find a simpler solution to Fermat's Last Theorem.

You can also play around with it more finitely, which I can explain a bit later, but my fingers are tired from typing.

If you have input, or come up with your own results, or have criticisms, please contribute! I would like to see if these definitions bring out anything cool.

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