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I was wondering if Fermat's Last Theorem can relate somehow to quantum numbers and energy spectrum in some theoretical system. Are there any examples for such systems?

And in general, Is there any use in physics for Fermat's Last Theorem?

Qmechanic
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ziv
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    Valid question, but I personally would approach it somehow similarly to Okham's Razor: as long as there are enough explanations for nature's phenomena that do not require something as complicated as a theorem that took almost 400 years to be proven true, it is very unlikely that Fermat's Last Theorem is related to physics. ;-) – oliver Mar 12 '21 at 23:16
  • And still, the universe is not the empty set so maybe you will be surprised :) – ziv Mar 12 '21 at 23:58
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    Related: https://physics.stackexchange.com/q/414/2451 , https://physics.stackexchange.com/q/26856/2451 – Qmechanic Mar 12 '21 at 23:59
  • Have you got any reason/example to suspect that Fermat's theorem might be relevant to physics? – oliver Mar 13 '21 at 00:00
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    http://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdf See here, ctrl f for "Fermat," it's related to what combination of chiral fermions you can have with no gravitational anomaly – user1379857 Mar 13 '21 at 00:04
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    Oliver, I know that sometimes quantum numbers relate to number theory. The first time I heard this was on lecture about Hearing the shape of a drum although I don't remember exactly how number theory was related in this system. You may look at Qmechanic comment – ziv Mar 13 '21 at 00:06
  • user1379857, Thanks! – ziv Mar 13 '21 at 00:09
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    The connection to Tong's anomaly equation is not so deep. The only relationship is that the equation is one of the infinitely many cases covered by FLT, and so the theorem tells you whether the anomaly equation can be satisfied... – Mitchell Porter Mar 13 '21 at 00:12
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    @oliver : counting from when humans first started to count, the time it took to prove fermat's last theorem was not significantly longer than the time it took to invent calculus. Yet calculus is often thought to have some applications in physics. – WillO Mar 13 '21 at 04:03
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    Also you don't need Wiles' proof of the full FLT for the case in Tong's notes. According to wikipedia a complete proof of this case dates back to 1760 by Euler. – Andrew Mar 13 '21 at 04:08
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    Though the FLT used in Lohitsiri and Tong's work https://arxiv.org/abs/1907.00514 is the version for n = 3, I found it fun to see that Euler's work published in 1770 is indeed cited in a modern physics paper. – tueda Jan 22 '22 at 11:10

1 Answers1

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There are some connections between physics and the last Fermat theorem.

  1. The Shimura-Taniyama Conjecture and Conformal Field Theory

  2. How arithmetic geometry emerges from string theory Arithmetic Spacetime Geometry from String Theory

  3. L-functions and string theory Emergent spacetime from modular motives

  4. The ABC conjecture can be reformulated in terms of [brane tillings]. THis is relevant for the Last Fermat theorem because the ABC conjecture imply the Fermat-Wiles theorem(https://arxiv.org/abs/0803.4474)Yang-Mills Theory and the ABC Conjecture.

Ramiro Hum-Sah
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    Thanks! I still don't mark this as answered question because I want to see if someone will come with more ideas. – ziv Mar 13 '21 at 12:22