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A pet peeve of mine is the statement that pops up every once in a while that number theory is not completely abstract and done for its own sake, and the example given is always cryptography. It is such a cliché phrase that it raises the question of other possible uses.

To specify, I refer to number theory as the study of integers and to applications as uses other than examples in other fields (for example, I understand modular arithmetic is an instance of a group, but it alone does not involve a number-theoretical finding being relevant in a non-number-theoretical context).

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    $\begingroup$ Quite closely related to what you're asking is that Riemann's hypothesis has quite a few connections with areas outside of number theory, especially in physics, as was asked & I answered in Applications of Riemann Hypothesis outside number theory. $\endgroup$ Apr 12, 2020 at 20:54
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    $\begingroup$ See e.g. here for applications to physics, communication, music, statistical mechanics, dynamical systems, random number generators, cyclotomy and cyclic codes, computer graphics, etc. $\endgroup$ Apr 12, 2020 at 21:03
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    $\begingroup$ I suggest you dip into Knuth's Seminumerical Algorithms (Volume 2 of The Art of Computer Programming). Given your definition of number theory as properties of the integers, you will find that everything your computer does is crucially dependent on number theory. $\endgroup$
    – Rob Arthan
    Apr 12, 2020 at 21:21
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    $\begingroup$ Number theory has numerous practical applications in acoustics. Manfred Schroeder did work along these lines. For example, see "Number theory in music, speech, and acoustics", Journal of the Acoustical Society of America, asa.scitation.org/doi/10.1121/1.2021921. $\endgroup$ Apr 13, 2020 at 4:23

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