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Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions.German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."
The Selberg trace formula is the analogue for these functions of the explicit formulas in prime number theory. Selberg proved that the Selberg zeta functions satisfy the analogue of the Riemann hypothesis, with the imaginary parts of their zeros related to the eigenvalues of the Laplacian operator of the Riemann surface.
Bernoulli number. Agoh–Giuga conjecture; Von Staudt–Clausen theorem; Dirichlet series; Euler product; Prime number theorem. Prime-counting function. Meissel–Lehmer algorithm; Offset logarithmic integral; Legendre's constant; Skewes' number; Bertrand's postulate. Proof of Bertrand's postulate; Proof that the sum of the reciprocals of the ...
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. [1] It is often said to have begun with Peter Gustav Lejeune Dirichlet 's 1837 introduction of Dirichlet L -functions to give the first proof of Dirichlet's theorem on arithmetic progressions .
Traditionally, number theory is the branch of mathematics concerned with the properties of integers and many of its open problems are easily understood even by non-mathematicians. More generally, the field has come to be concerned with a wider class of problems that arise naturally from the study of integers.
In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind U n (P, Q) with relatively prime parameters P, Q and positive discriminant, an element U n with n ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12) = U 12 (1, − ...
In Zermelo–Fraenkel (ZF) set theory, the natural numbers are defined recursively by letting 0 = {} be the empty set and n + 1 (the successor function) = n ∪ {n} for each n. In this way n = {0, 1, …, n − 1} for each natural number n. This definition has the property that n is a set with n elements.
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