I just finished my first undergrad course in Number Theory. I really like it, but now I'm wondering what applications of elementary number theory there are beyond cryptography. Specifically, are there any resources I can look at linking number theory to the physical sciences (phys, chem, bio, etc)? -- I'm still an undergrad so it would need to be relatively gentle for me to jump into it now (I've taken the standard courses in Abstract Algebra and Real Analysis, but nothing else really, yet). Is there anything that fits the bill?

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    $\begingroup$ I don't currently know of a reference for this, but elementary number theory is deceptively useful in the arrangement of long line telephone and data cables. This has to do with interference caused by proximity to other cables at points where cables are spliced together. So they end up using some elementary number theory to figure out optimal ways to distance cable splices from each other to minimize both cost and interference. Also, information theory, self-correcting codes in general, hamming codes. Some biological processes end up as recurrence relations, and thus loosely fall here too. $\endgroup$ – davidlowryduda Jun 15 '14 at 1:35
  • $\begingroup$ One of the recurring themes in Knuth's "Art of Computer Programing" is the appearance of all sorts of number theory related considerations (divisibility, congruences, primes) when analyzing algorithms. $\endgroup$ – vonbrand Jun 15 '14 at 2:16
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    $\begingroup$ Writing mathematical software. $\endgroup$ – J. W. Perry Jun 15 '14 at 3:37

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