What to include in an introduction on number theory?

In summary, the conversation discusses writing an introduction for a first course in elementary number theory, which covers topics such as linear and nonlinear Diophantine equations, modular arithmetic (including FLT and Euler's Generalization), and quadratic residues. The speaker is seeking advice on how to connect these topics and engage potential students. They also mention finding some resources on modular arithmetic and its applications, such as RSA, error correcting codes, and encryption. They plan to use small examples and applications to illustrate the usefulness of modular arithmetic.
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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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1. What is number theory?

Number theory is a branch of mathematics that focuses on the properties of integers and their relationships with one another. It is a fundamental and ancient field of study that has many applications in modern mathematics and other fields such as cryptography and computer science.

2. Why is number theory important?

Number theory is important because it helps us understand the basic building blocks of mathematics - the integers. It also has many practical applications in fields such as cryptography, coding theory, and computer science. Additionally, number theory has played a crucial role in the development of other branches of mathematics, such as algebra and geometry.

3. What are some key concepts in number theory?

Some key concepts in number theory include prime numbers, divisibility, modular arithmetic, and Diophantine equations. These concepts are fundamental to understanding the properties of integers and their relationships with one another.

4. What should be included in an introduction on number theory?

An introduction on number theory should provide a brief overview of the field, its history, and its importance. It should also introduce key concepts and definitions, such as prime numbers and modular arithmetic. Additionally, it may include some examples of real-world applications of number theory.

5. How can number theory be applied in real life?

Number theory has many practical applications in real life, particularly in fields such as cryptography, coding theory, and computer science. For example, number theory is used in the encryption of sensitive information, such as credit card numbers and passwords, to ensure their security. It is also used in coding theory to design efficient error-correcting codes for communication systems.

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