Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.
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 elementary topics such as congruences, linear Diophantine equations, greatest common divisors, quadratic and power residues, primitive roots, please use the elementary-number-theory tag. This tag is for more advanced topics such as higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebraic geometry, exponential and character sums, zeta and L-functions, multiplicative and additive number theory, etc.
