Number theory

Number theory studies integers and their properties, including divisibility, prime numbers, congruences, and equations with integer solutions.

Divisibility: Divisibility is the relation in which one integer is an integer multiple of another. It organizes integers into factors and multiples, the starting point for many questions in number theory.
Euclidean algorithm: The Euclidean algorithm computes the greatest common divisor by repeatedly replacing a pair with a smaller pair having the same gcd. It turns common-divisor questions into a finite sequence of exact divisions.
Diophantine equation: A Diophantine equation is a polynomial equation whose solutions are required to be integers or rational numbers. Integer-solution questions form a central strand of number theory, from simple equations to unsolved problems.
RSA cryptosystem: RSA is a public-key cryptosystem whose security relies on the practical difficulty of factoring large semiprimes. Its key construction uses modular exponentiation and properties of prime factors.
Riemann hypothesis: The Riemann hypothesis asserts that every nontrivial zero of the Riemann zeta function has real part one-half. Its truth would sharpen understanding of the distribution of prime numbers.
Prime number: A prime number is an integer greater than one whose only positive divisors are one and itself. Primes are the indivisible building blocks underlying integer factorization.
Bézout's identity: For integers a and b, their greatest common divisor is an integer linear combination of a and b. It links the Euclidean algorithm to solvability of linear Diophantine equations.
Fermat's Last Theorem: Fermat's Last Theorem states that no positive integers satisfy x^n + y^n = z^n for any integer n greater than two. Its long path from a simple-looking equation to proof illustrates the depth of Diophantine problems.
Elliptic-curve cryptography: Elliptic-curve cryptography uses the algebraic structure of elliptic-curve points over finite fields to build public-key systems. It applies finite-field arithmetic and hard discrete-logarithm problems to cryptographic keys.
abc conjecture: The abc conjecture relates the prime factors of coprime positive integers a, b, and c satisfying a + b = c. It predicts strong limits on how addition and prime factorization can interact.

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