# Why certain topics in elementary number theory?

• I
• matqkks
In summary, elementary number theory courses cover topics such as 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 because they are necessary tools for understanding and working with number theory. Without these foundations, it would be difficult to delve into more complex concepts such as the Riemann Hypothesis or the prime number theorem. Additionally, many conjectures in number theory appear simple on the surface but are actually incredibly difficult to prove analytically, making the subject an "analytical nightmare."

#### matqkks

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?

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.

matqkks and Klystron
Or perhaps prove or disprove the Riemann Hypothesis. One can always dream.

Diophantine equations can be devilishly hard to solve analytically.

matqkks
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?

matqkks and jedishrfu
That’s a nice conjecture that I’ve not heard of before either.

matqkks
Pi-is-3 and fresh_42