Undergrad What should I say about elementary number theory?

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The discussion focuses on preparing an engaging talk on elementary number theory, particularly emphasizing the study of positive integers and primes, along with their applications in cryptography. A suggested approach is to simplify concepts for audiences unfamiliar with number theory by illustrating the difficulty of prime factorization, especially with large numbers. An example is provided to demonstrate how easy it is to multiply two primes but challenging to reverse the process. The RSA principle is highlighted as a key application in cryptography that relies on this difficulty. Overall, the conversation emphasizes the importance of tailoring the presentation to the audience's background in number theory.
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Motivating talk.
I need to give an option talk about elementary number theory module. I will discuss how it is study of positive integers particularly the primes and give some cryptography applications. What is a good hook to stipulate in this talk regarding an introduction to elementary number theory?
 
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What is your target audience? Did they have a number theory/discrete math course already?

Here is what you can do if they have not really a background in number theory:

Keep things simple!

Everyone can multiply two (big prime) numbers together (given enough time).

But, given a (big) number that is the product of two primes, give an example where it is hard to find the two primes that were multiplied together. Explain that this is in general a very hard problem. No algorithms exist (yet) to find prime decompositions efficiently. Proceed with the RSA principle.
 
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Thanks for this. I will definitely include this in my talk.
 

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