Applications of number theory in physics?

Depends what you mean by number theory...

Modular forms are foundational in conformal field theory which is a theoretical physicists playground for a number of topics...
See the book “From Number Theory to Physics”.
I'm in the process of building a primary root acoustic diffuser (sometimes called a PRD or skyline for aid in googling). I've yet to really grasp *why* the design is the way it is, but it's built over number theory concepts. Reading over the patent, the cited math papers were sometimes within a year or two of the release of the patent itself in the 90s - so it was done with some cutting edge math for the time.

If anyone has a key intuition for how it works, please share!
Ok so i got my PhD in connections between string theory and number theory. I think my favorite example of this connection is the fact that black hole entropies are given by the Fourier coefficients of modular forms. This is particularly clear in the context of so-called small black holes, where literally the number of states in a black hole of mass \$n\$ is given by the \$n\$-th Fourier coefficient of 1/Delta(tau). where Delta is the weight-12 cusp form. In fact, the Fourier coefficients of this modular form admit an exact series expansion known as a Rademacher series, which takes the form of a sum over elements of (a quotient of) SL_2(Z). The *exact same series* appears in the evaluation of the gravitational path integral which is supposed to compute this entropy, so we have a highly nontrivial matching between the number theory result and actual string theory.

I can go on for hours about other connections between string theory and number theory (or, ya know, write a thesis about it :p), but I think this is the most striking demonstration.
Upvote and comment for visibility (I dont know of any, but I am curious as well.)
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I remember watching a youtube video where Ed witten said we were centuries away from getting a satisfactory integration of number theory with physics. Or at least that he thought he wouldn't get to see it in his lifetime.
I don't know exactly how, but my number theory/analysis prof once said there are connections between p-adic numbers and the realm of small particles. Unfortunately I don't know more about this.
Not contributing anything here but the thought of a physician caring about non-abelian extensions of the rational field made me chuckle :D
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A lot to be discovered yet. Imagine if ratios of particle masses, or force strengths, are ultimately based on some number properties only. That would mean universe could only be what it is. We are a long way off from proving or disproving that.
you seem to have accidentally typed “physicians” instead of “physicists” lol

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