MHB What to include in an introduction?

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An effective introduction to a first course in elementary number theory should emphasize the interconnectedness of topics such as linear Diophantine equations, modular arithmetic, and quadratic residues. Highlighting historical context, like Fermat's Last Theorem and its implications for modern mathematics, can engage students by showcasing the evolution of mathematical thought. Additionally, linking number theory to practical applications in cryptography can illustrate its relevance in today's digital world. The introduction could also mention intriguing problems, such as Archimedes's cattle problem, to spark curiosity. Ultimately, the goal is to create a compelling narrative that motivates students to explore these foundational concepts in number theory.
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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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matqkks said:
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?
Hmmm...perhaps what "hooked" YOU...get my drift?
 
Possible topics that may fit your requirements:
1. Fermat's Last Theorem. Original statement by Fermat as a marginal note. Margins to small to contain Fermat's alleged proof. Attemps to solve led to advances in other ares of math. Large prize offered for a solution, leading to many "crackpot" solutions. Ultimate proof used math unknown in Fermat's time.

2. Number theory and cryptography. Number theory provides computationally complex problems (e.g., prime factorization, elliptic curve logarithm problem) that lead to codes that are unbreakable in practice. Used in credit card encryption and other situations.

3. Archimedes's cattle problem (see here). Leads to a non-linear Diophantine equation (i.e., Pell's Equation). The solution has more than 200 000 digits and wasn't written out until the advent of digital computers.
 
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Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
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