Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

Number theory is concerned with the study of natural numbers. One of the main subjects is studying the behavior of prime numbers.

We know that by the prime number theorem, the number of primes less than $x$ is approximately $\frac{x}{\ln(x)}$. Another good approximation is $\operatorname{li}(x)$. Despite these estimates, we don't know much about the maximal prime gaps. The weaker conjectures, such as Legendre's conjecture, Andrica's conjecture and Opperman's conjecture, imply a gap of $O\left(\sqrt{p}\right)$. Stronger conjectures even imply a gap of $O(\ln^2(p))$. The Riemann Hypothesis implies a gap of $O\left(\sqrt{p} \ln(p)\right)$, though proving this is not sufficient to show the RH. The minimal gap is also a subject of research. It has been shown that gaps smaller than or equal to $246$ occur infinitely often. It is conjectured that gaps equal to $2$ occur infinitely often. This is known as the twin prime conjecture.

Another subject in number theory are Diophantine equations, which are polynomial equations in more than one variable, where variables are integer valued. Some equations can be solved by considering terms modulo some number or by considering divisors, prime factors or the number of divisors. Other equations, such as Fermat's Last Theorem, are much harder, and are or were famous open problems. Recent progress usually uses algebraic number theory and the related elliptic curves.

Another subject is the study of number theoretic functions, most notably $\tau(n)$, the number of divisiors of $n$, $\sigma(n)$, the sum of divisors of $n$ and $\varphi(n)$, the Euler-phi function, the number of numbers smaller than $n$ coprime with $n$.

For questions on congruences, linear Diophantine equations, greatest common divisors, etc. , please use the tag. This tag is for more advanced topics, such as questions about the distributions of prime numbers, non-linear Diophantine equations, quadratic residues, primitive roots and questions about number theoretic functions.

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