I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about Algebraic $K$-theory mention that we don't know $K(\mathbb{Z})$ and that somehow $K(\mathbb{Z})$ is worth computing because it contains lots of arithmetic information. I'd like to better understand what kinds of arithmetic information it contains. I've been unable to answer number theorists who've asked me this before. A related question is about what information is contained in $K(S)$ where $S$ is the sphere spectrum.

I am aware of Vandiver's Conjecture and that it is equivalent to the statement that $K_n(\mathbb{Z})=0$ whenever $4 | n$. I also know there's some connection between $K$-theory and Motivic Homotopy Theory, but I don't understand this very well (and I don't know if $K(\mathbb{Z})$ helps). It seems difficult to search for this topic on google. Hence my question:

Can you give me some examples of places where computations in $K(\mathbb{Z})$ or $K(S)$ would solve open problems in arithmetic or would recover known theorems with difficult proofs?

I'm hoping someone who has experience motivating this field to number theorists will come on and give his/her usual spiel. Here are some potential answers I might give a number theorist if I understood them better...The wikipedia page for Algebraic K-theory mentions non-commutative Iwasawa Theory, L-functions (and maybe even Birch-Swinnerton-Dyer?), and Bass's conjecture. I don't know anything about this, not even whether knowing $K(\mathbb{Z})$ would help. Quillen-Lichtenbaum seems related to $K(\mathbb{Z})$, but it seems it would tell us things about $K(\mathbb{Z})$ not the other way around. Milnor's Conjecture (or should we call it Voevodsky's Theorem?) is definitely an important application of $K$-theory, but it's the $K$-theory of field of characteristic $p$, not $K(\mathbb{Z})$.

There was a previous MO question about the big picture behind Algebraic K Theory but I couldn't see in those answers many applications to number theory. There's a survey written by Weibel on the history of the field, and that includes some problems it's solved (e.g. the congruence subgroup problem) but other than Quillen-Lichtenbaum I can't see anything which relies on $K(\mathbb{Z})$ as opposed to $K(R)$ for other rings. If $K(\mathbb{Z})$ could help compute $K(R)$ for general $R$ then that would be something I've love to hear about.

5more comments