Undergrad Why certain topics in elementary number theory?

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Elementary number theory courses typically cover essential topics such as gcd, linear Diophantine equations, and modular arithmetic because they provide necessary tools for deeper mathematical exploration. These foundational concepts enable students to engage with advanced subjects like Computer Science and the prime number theorem. While some argue for a purely analytical approach, it risks narrowing the scope of the subject. The difficulty of solving Diophantine equations highlights the complexity of the field, as seen in conjectures like the Legendre conjecture. Overall, a solid grounding in these topics is crucial for advancing in number theory and related areas.
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Why do all elementary number theory courses have the following topics - gcd, linear Diophantine equations, Fundamental Theorem of Arithmetic, factorization, modular arithmetic, Fermat's Little Theorem, Euler's Theorem, primitive roots, quadratic residues and nonlinear Diophantine equations?
 
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Those are all tools which are necessary to do the real work. As long as they won't be taught at school, such courses will have to start with them. What is the alternative? Riemann and Chebyshev right from the start? One can approach the subject purely analytically, but this narrows the subject. With the theorems listed above, one can continue with Computer Science or the prime number theorem and other analytical results.
 
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Or perhaps prove or disprove the Riemann Hypothesis. One can always dream.

Diophantine equations can be devilishly hard to solve analytically.
 
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When I think of all these conjectures, which are more or less easily stated, and yet, are devilishly hard, then the entire field is an analytical nightmare. Just read today about the Legendre conjecture (unproven):
$$\text{ There is always a prime between }n^2 \text{ and }(n+1)^2$$
I mean, could it look more innocent?
 
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That’s a nice conjecture that I’ve not heard of before either.
 
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Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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