# Why certain topics in elementary number theory?

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## Main Question or Discussion Point

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?

## Answers and Replies

fresh_42
Mentor
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
jedishrfu
Mentor
Or perhaps prove or disprove the Riemann Hypothesis. One can always dream.

Diophantine equations can be devilishly hard to solve analytically.

matqkks
fresh_42
Mentor
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
jedishrfu
Mentor
That’s a nice conjecture that I’ve not heard of before either.

matqkks
jedishrfu
Mentor
Pi-is-3 and fresh_42