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The sequence of numbers involved is sometimes referred to as the hailstone sequence, hailstone numbers or hailstone numerals (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), [5] or as wondrous numbers. [6] Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such ...
Lothar Collatz (German:; July 6, 1910 – September 26, 1990) was a German mathematician, born in Arnsberg, Westphalia. The "3x + 1" problem is also known as the Collatz conjecture, named after him and still unsolved. The Collatz–Wielandt formula for the Perron–Frobenius eigenvalue of a positive square matrix was also named after him.
The following other wikis use this file: Usage on ca.wikipedia.org Conjectura de Collatz; Usage on eu.wikipedia.org Collatzen aierua; Usage on fi.wikipedia.org
English: This is a graph, generated in bottom-up fashion, of the orbits of all numbers under the Collatz map with an orbit length of 20 or less. Created with Graphviz, with the help of this Python program: # This python script generates a graph that shows 20 levels of the Collatz Conjecture.
The 3x + 1 semigroup has been used to prove a weaker form of the Collatz conjecture. In fact, it was in such context the concept of the 3 x + 1 semigroup was introduced by H. Farkas in 2005. [ 2 ] Various generalizations of the 3 x + 1 semigroup have been constructed and their properties have been investigated.
Directed graph showing the orbits of the numbers less than 30 (with the exception of 27 because it would make it too tall) under the Collatz map. For a larger graph containing only odd numbers, see Image:Collatz-graph-300.svg. Created with Graphviz, with the help of this Python program:
The question is whether or not, for all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time), an algorithm can also find that solution quickly. Since the former describes the class of problems termed NP, while the latter describes P, the question is equivalent to asking whether all problems in NP are ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.