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convergence of the geometric series with first term 1 and ratio 1/2; Integer partition; Irrational number. irrationality of log 2 3; irrationality of the square root of 2; Mathematical induction. sum identity; Power rule. differential of x n; Product and Quotient Rules; Derivation of Product and Quotient rules for differentiating. Prime number ...
Minkowski's theorem (geometry of numbers) Minkowski's second theorem (geometry of numbers) Minkowski–Hlawka theorem (geometry of numbers) Monsky's theorem (discrete geometry) Pick's theorem ; Pizza theorem ; Radon's theorem (convex sets) Separating axis theorem (convex geometry) Steinitz theorem (graph theory) Stewart's theorem (plane geometry)
The following famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that is a rational number. This proof uses that 2 {\displaystyle {\sqrt {2}}} is irrational (an easy proof is known since Euclid ), but not that 2 2 {\displaystyle {\sqrt {2}}^{\sqrt {2}}} is irrational (this is true, but the proof ...
The proof was completed by Werner Ballmann about 50 years later. Littlewood–Richardson rule. Robinson published an incomplete proof in 1938, though the gaps were not noticed for many years. The first complete proofs were given by Marcel-Paul Schützenberger in 1977 and Thomas in 1974. Class numbers of imaginary quadratic fields.
Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in R n , {\displaystyle \mathbb {R} ^{n},} and the study of these lattices provides fundamental information on algebraic numbers. [ 1 ]
Proof without words of the Nicomachus theorem (Gulley (2010)) that the sum of the first n cubes is the square of the n th triangular number. In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text.
Mathematical constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then ...
A proof of the necessity that a, b, c be expressed by Euclid's formula for any primitive Pythagorean triple is as follows. [6] All such primitive triples can be written as (a, b, c) where a 2 + b 2 = c 2 and a, b, c are coprime. Thus a, b, c are pairwise coprime (if a prime
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