Collatz conjecture

Collatz conjecture - In reverse. There is another approach to prove the following conjecture, which considers the bottom-up method of growing the Collatz graph. The Collatz graph is defined by an inverse relation, So, instead of proving that all natural numbers eventually lead to 1, we can prove that 1 leads to all natural numbers. Also, the inverse relation forms a tree except for the 1-2 loop. Note that the relation being inverted here is (3n + 1) / 2 (see Optimizations below). ...

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Collatz conjecture, Collatz conjecture - Statement of the problem, Collatz conjecture - Examples, Collatz conjecture - Program to calculate Collatz sequences, Collatz conjecture - Supporting arguments, Collatz conjecture - Experimental evidence, Collatz conjecture - Probabilistic evidence, Collatz conjecture - Other ways of looking at it, Collatz conjecture - In reverse, Collatz conjecture - As rational numbers, Collatz conjecture - As an abstract machine, Collatz conjecture - As iterating a real or complex map, Collatz conjecture - Optimizations

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The Collatz conjecture is an unresolved conjecture in mathematics. It is named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), the Syracuse problem, as the hailstone sequence or hailstone numbers, or as Wondrous numbers as per Gödel, Escher, Bach. It asks whether a certain kind of number sequence always ...

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• Collatz conjecture - Statement of the problem
• Collatz conjecture - Examples
• Collatz conjecture - Program to calculate Collatz sequences
• Collatz conjecture - Supporting arguments
• Collatz conjecture - Experimental evidence
• Collatz conjecture - Probabilistic evidence
• Collatz conjecture - Other ways of looking at it
• Collatz conjecture - In reverse
• Collatz conjecture - As rational numbers
• Collatz conjecture - As an abstract machine
• Collatz conjecture - As iterating a real or complex map
• Collatz conjecture - Optimizations

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Until its proof in 1995, the most famous of all conjectures was the mis-named Fermat's last theorem - this conjecture became a true theorem only after its proof. In the process, a special case of the Taniyama-Shimura conjecture, itself a longstanding open problem, was proven; this conjecture has since been completely proven. Other famous conjectures include: There are no odd perfect numbers Goldbach's conjecture The twin prime conjecture The Collatz conjecture The Riemann hypothesis P ≠ NP The Poinca ...

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Conjecture, Conjecture - Famous conjectures, Conjecture - Counterexamples, Conjecture - Use of conjectures in conditional proofs, Conjecture - Undecidable conjectures, Conjecture - Usage outside of mathematics

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In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove. Once a conjecture has been proven, it becomes known as a theorem, and it joins the realm of known mathematical facts. Until that point in time, mathematicians must be extremely careful about their use of a conjecture within logical structures. Conjecture - Famous conjectures. Until its proof in 1995, the most famous of all conjectures was the m ...

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• Conjecture - Famous conjectures
• Conjecture - Counterexamples
• Conjecture - Use of conjectures in conditional proofs
• Conjecture - Undecidable conjectures
• Conjecture - Usage outside of mathematics

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This article describes some currently unsolved problems in mathematics. The seven Millennium Prize Problems set by the Clay Mathematics Institute are: P versus NP The Hodge Conjecture The Poincaré Conjecture The Riemann Hypothesis Yang-Mills Existence and Mass Gap Navier-Stokes Existence and Smoothness The Birch and Swinnerton-Dyer Conjecture Other still-unsolved problems: Twin prime conjecture Number of Magic squares Including:

• Unsolved problems in mathematics - Quotes
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• Unsolved problems in mathematics - Books discussing unsolved problems

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Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Today, the natural sciences, engineering, economics, and medici ...

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• Mathematics - History
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• Mathematics - Notation language and rigor
• Mathematics - Is mathematics a science?
• Mathematics - Overview of fields of mathematics
• Mathematics - Major themes in mathematics
• Mathematics - Quantity
• Mathematics - Structure
• Mathematics - Space
• Mathematics - Change
• Mathematics - Foundations and methods
• Mathematics - Discrete mathematics
• Mathematics - Applied mathematics
• Mathematics - Important theorems
• Mathematics - Important conjectures
• Mathematics - History and the world of mathematicians
• Mathematics - Mathematics and other fields
• Mathematics - Common misconceptions

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List of number theory topics - Primality tests. Prime factorization algorithm Trial division Sieve of Eratosthenes Probabilistic algorithm Fermat primality test Pseudoprime Carmichael number Euler pseudoprime Euler-Jacobi pseudoprime Fibonacci pseudoprime Probable prime Miller-Rabin primality test Lucas-Lehmer primality test Lucas-Lehmer test for Mersenne numbers AKS prim ...

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List of number theory topics, List of number theory topics - Factors, List of number theory topics - Fractions, List of number theory topics - Modular arithmetic, List of number theory topics - Arithmetic functions, List of number theory topics - Analytic number theory: additive problems, List of number theory topics - Algebraic number theory, List of number theory topics - Quadratic forms, List of number theory topics - L-functions, List of number theory topics - Diophantine equations, List of number theory topics - Diophantine approximation, List of number theory topics - Sieve methods, List of number theory topics - Named primes, List of number theory topics - Combinatorial number theory, List of number theory topics - Computational number theory, List of number theory topics - Primality tests, List of number theory topics - Integer factorization, List of number theory topics - Pseudo-random numbers, List of number theory topics - History

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List of combinatorics topics - 0-9. (0,1) matrix List of combinatorics topics - A. Abstract simplicial complex Addition chain Scholz conjecture Alternating sign matrix Almost disjoint sets Antichain Arrangement of hyperplanes Assignment problem Audioactive decay List of combinatorics topics - B. Barcode Matrix code ...

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List of combinatorics topics, List of combinatorics topics - General combinatorial principles and methods, List of combinatorics topics - Problem solving as an art, List of combinatorics topics - Some general theories, List of combinatorics topics - Living with large numbers, List of combinatorics topics - Topics in combinatorics: alphabetical list, List of combinatorics topics - 0-9, List of combinatorics topics - A, List of combinatorics topics - B, List of combinatorics topics - C, List of combinatorics topics - D, List of combinatorics topics - E, List of combinatorics topics - F, List of combinatorics topics - G, List of combinatorics topics - H, List of combinatorics topics - I, List of combinatorics topics - K, List of combinatorics topics - L, List of combinatorics topics - M, List of combinatorics topics - N, List of combinatorics topics - O, List of combinatorics topics - P, List of combinatorics topics - R, List of combinatorics topics - S, List of combinatorics topics - T, List of combinatorics topics - U, List of combinatorics topics - V, List of combinatorics topics - W, List of combinatorics topics - Y, List of combinatorics topics - Data structure concepts, List of combinatorics topics - People, List of combinatorics topics - Journals, List of combinatorics topics - Prizes, List of combinatorics topics - Publications

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Number theory - Elementary number theory. In elementary number theory, integers are studied without use of techniques from other mathematical fields. Questions of divisibility, use of the Euclidean algorithm to compute greatest common divisors, factorization of integers into prime numbers, investigation of perfect numbers and congruences belong here. Several important discoveries of this field are Fermat's little theorem, Euler's theorem, the Chinese remainder theorem and the law of quadratic reciprocity. ...

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Number theory, Number theory - Fields, Number theory - Elementary number theory, Number theory - Analytic number theory, Number theory - Algebraic number theory, Number theory - Geometric number theory, Number theory - Combinatorial number theory, Number theory - Computational number theory, Number theory - History, Number theory - Early history, Number theory - Beginnings of a systematic theory, Number theory - Prime number theory, Number theory - Nineteenth-century developments, Number theory - Quotations

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An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possible view. For a fuller treatment, see areas of mathematics or the list of mathematics lists. Mathematics - Quantity. This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. See also:

Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Common misconceptions

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Number theory - Elementary number theory. In elementary number theory, integers are studied without use of techniques from other mathematical fields. Questions of divisibility, use the Euclidean algorithm to compute greatest common divisors, factorization of integers into prime numbers, investigation of perfect numbers and congruences belong here. Several important discoveries of this field are Fermat's little theorem, Euler's theorem, the Chinese remainder theorem and the law of quadratic reciprocity. The ...

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Number theory, Number theory - Fields, Number theory - Elementary number theory, Number theory - Analytic number theory, Number theory - Algebraic number theory, Number theory - Geometric number theory, Number theory - Combinatorial number theory, Number theory - Computational number theory, Number theory - History, Number theory - Early history, Number theory - Beginnings of a systematic theory, Number theory - Prime number theory, Number theory - Nineteenth-century developments, Number theory - Quotations

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The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g., addition, subtraction, mul ...

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Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions

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Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict grammar (under the influence o ...

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Mathematics, Mathematics - History, Mathematics - Inspiration, pure and applied mathematics, and aesthetics, Mathematics - Notation, language, and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Common misconceptions

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Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that ins ...

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Mathematics, Mathematics - History, Mathematics - Inspiration, pure and applied mathematics, and aesthetics, Mathematics - Notation, language, and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Common misconceptions

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An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possible view. For a fuller treatment, see areas of mathematics or the list of mathematics lists. Mathematics - Quantity. Quantity starts with counting and measurement. Natural numbers ...

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Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions

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Old: Abacus Napier's bones, slide rule Ruler and compass Mental calculation New: Calculators and computers Programming languages Computer algebra systems (listing) Internet shorthand notation statistical analysis software SPSS SAS programming language R programming language ...

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Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions

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Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudosci ...

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Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions

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As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the emp ...

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Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions

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Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is < ...

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Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions

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Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict grammar (under the influence of ...

See also:

Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions

Read more here: » Mathematics: Encyclopedia II - Mathematics - Notation language and rigor