Number Theory
Number theory studies integers, primes, Diophantine equations, arithmetic functions, and their analytic and algebraic structure.
The subject runs from elementary arithmetic to its analytic and algebraic theories. Congruences and power residues build up to residue symbols and quadratic reciprocity. The distribution of primes, from the prime number theorem and explicit bounds, is approached through the Riemann zeta function and Dirichlet characters and L-functions, with exponential-sum estimates such as Vinogradov's mean value theorem as a tool.
Diophantine questions take several forms: equations such as Pell's equation, Diophantine approximation through Roth's theorem and linear forms in logarithms, and representation by sums of powers in Waring's problem. Algebraic number theory and p-adic numbers give the global and local structure behind them, and arithmetic geometry reaches elliptic curves, with Mazur's torsion theorem.
The largest field is linear recurrence sequences: Fibonacci and Lucas numbers, Pell, Jacobsthal and Tribonacci-type sequences, Lucas sequences and their primitive divisors, and Diophantine problems about the terms. Arithmetic functions, digits and numeration systems, modular forms and partitions, special numbers such as the Bernoulli and Catalan numbers, and integer sequences make up much of the rest.
- Congruences and power residues
- Primes and primality
- Distribution of primes
- Arithmetic functions
- Zeta and L-functions
- Exponential and character sums
- Diophantine equations
- Quadratic forms and sums of squares
- Diophantine approximation and transcendence
- Continued fractions and expansions of reals
- Heights and Mahler measure
- Algebraic number theory
- p-adic numbers and local fields
- Modular forms, q-series and partitions
- Linear recurrence sequences
- Special numbers and polynomials
- Digits and numeration systems
- Integer sequences
- Additive number theory
- Numerical semigroups and the Frobenius problem
- Arithmetic geometry
- Arithmetic of function fields
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