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What practical use are prime numbers? Why do we emphasise the teaching of prime numbers?

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    $\begingroup$ Your question isn’t really very clear. What context do you have in mind when you say that we emphasize the teaching of prime numbers? $\endgroup$ Sep 3, 2013 at 9:31
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    $\begingroup$ Cryptography is the only "practical use" that comes to mind. $\endgroup$
    – daniel
    Sep 3, 2013 at 9:46
  • $\begingroup$ To continue Brian's question (into possibly a different direction what he had in mind): at what level are you being taught about prime numbers? Knowing that helps answering your question. $\endgroup$ Sep 3, 2013 at 9:51

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For real life applications see Real-world applications of prime numbers?. Why are prime numbers so important that we teach about prime numbers ? This question is not so easy as it seems. Of course, for elementary number theory, prime numbers are like the "atoms", and several questions involve prime numbers.
For mathematics in general, the value of prime numbers lies much deeper. For example, the distribution of prime numbers encodes very deep mathematical information in general (not only via the Riemann Hypothesis). Completions of the rational numbers naturally lead to $p$-adic fields, and the idea of being "prime" applies to many other structures (like prime ideals, prime geodesics etc.). So we emphasise teaching prime numbers because they lie at the very heart of mathematics.

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  • $\begingroup$ Actually, there's so much general number theory. Rather people's minds just happened to fixate on prime numbers because they just happened to be interested in studying the topic that treats all positive integers as being built from the prime number. There are all kinds of other advanced topics of number theory that are nothing to do with prime numbers. $\endgroup$
    – Timothy
    Apr 29, 2020 at 0:11
  • $\begingroup$ @Timothy There is certainly much more than just primes, but still, even for many very advanced structures in modern number theory, a "prime" is involved, e.g., Schulze's perfectoid rings and spaces over characteristic $p>0$, local Langlands correspondence for $GL(\ell,F)$, $\ell$ a prime and $F$ is a $p$-adic field etc. It is hard to name a topic, which really has "nothing to do with prime numbers". $\endgroup$ Apr 29, 2020 at 9:04
  • $\begingroup$ What about the function that assigns to each nonnegative integer the floor function of that integer times $\sqrt3$? Maybe 3 was the prime number that was chosen to pick the square root of but once that has already been chosen, the functions seems nothing to do with any of the other prime numbers. It can have quite interesting properties that are nothing to do with prime numbers. $\endgroup$
    – Timothy
    Apr 29, 2020 at 20:48
  • $\begingroup$ But I don't know if this qualifies as "other advanced topics of number theory", as you said. Yes, we can certainly find topics without prime numbers, but essentially primes are everywhere in modern number theory. $\endgroup$ Apr 30, 2020 at 8:48
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Primes are of prime importance in cryptography (take for example the RSA cryptosystem). See http://en.wikipedia.org/wiki/RSA_(algorithm)

However one does Mathematics simply for doing Mathematics. What Mathematicians think today, may find its application even more than 200 years later. Although many branches (like variational calculus) came up from practical issues, not every branch of mathematics can be even remotely related to some real-life problem.

Prime Number Theory is a very interesting topic. It not only consists of some exciting results and findings over the ages but it actually enhances our mathematical thinking and imagination.

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  • $\begingroup$ > one does Mathematics simply for doing Mathematics -- that's fabulous if you can get some generous person to pay you to work on stuff that (maybe) has some unintended benefit 200 years from now. That kind of generosity is getting increasingly rare, though. Presumably that's how Yitang Zhang ended up working at Subway. $\endgroup$
    – bubba
    Sep 3, 2013 at 10:40

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