What to include in an introduction on number theory?

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  • Thread starter matqkks
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I am writing an introduction to a first course in elementary number theory. The topics are linear Diophantine equations, modular arithmetic including FLT and Euler's Generalization, quadratic residues and Non - linear Diophantine equations.
How can I write an introduction to this showing linkage between the various topics and hook potential students to do this course? What is the motivation on covering these topics?
 

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Here are some links I've searched yesterday on modular arithmetic and applications.

https://pdfs.semanticscholar.org/331c/f92e3155b765aede69ef8e6dedc3319f5eb6.pdf

http://www2.math.uu.se/~astrombe/talteori2016/lindahl2002.pdf
http://homepages.warwick.ac.uk/staff/J.E.Cremona/courses/MA257/ma257.pdf

Modular arithmetic alone is quite easy. During my search I came across some pages which provided a short and typical introduction:

http://www.acm.ciens.ucv.ve/main/entrenamiento/material/ModularArithmetic-Presentation.pdfhttps://euclid.ucc.ie/MATHENR/MathCircles_files/2nd and 3rd year Maths Circles/ModularArithmetic.pdf
I would take a few of those small examples and then turn to RSA as the major application here. Other interesting applications are error correcting codes or encryption in general.
 
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