Conjectures
Collatz Conjecture
KOH-lahts (German surname Collatz, after Lothar Collatz)
Also Known As 3n + 1 problem (also called the Ulam conjecture, Kakutani's problem, the Thwaites conjecture, Hasse's algorithm, or the Syracuse problem)
Number Theory
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Perhaps the simplest-sounding unsolved problem in mathematics, proposed by the German mathematician Lothar Collatz in 1937 (not yet a live entity on this atlas). Take any positive whole number: if it is even, halve it; if it is odd, triple it and add one; repeat. The conjecture claims this process always eventually reaches the number one, however large the starting number and however long and unpredictable the path there. Every starting value ever tested by computer, into the astronomically large, reaches one, and mathematician Paul Erdos is widely quoted as having said that mathematics may not yet be ready for such a problem, offering a small cash prize for its solution; no proof exists for every possible starting number.
Facts
StatementStarting from any positive integer, repeatedly applying the rule (halve if even, triple and add one if odd) will always eventually reach the number one. 1 Proposed YearProposed by Lothar Collatz in 1937; not yet a live entity on this atlas. Prize StatusNot a Millennium Prize Problem; Paul Erdos offered a modest personal cash prize for its solution, remarking that mathematics might not yet be ready for such problems, but no major institutional prize exists. 1 Progress Toward ResolutionVerified by computer for an enormous range of starting values, published searches reaching past 2 to the 68th power, with no counterexample ever found; no general proof exists for every possible starting number, and even the weaker claim that almost all starting values eventually reach a bounded value has only partial results. 1 Cross-Tradition Connections
In Branch
Posed By
Proposed in 1937; never published by Collatz himself, transmitted through the mathematical community by word of mouth.
Sources
1. Wolfram MathWorld
Wolfram Research, Inc.
Collatz Conjecture (Wikipedia)
Wikimedia Foundationlead sectionQuote, lead section
It is also known as the 3n + 1 problem (or conjecture), the 3x + 1 problem (or conjecture), the Ulam conjecture, Kakutani's problem, the Thwaites conjecture, Hasse's algorithm, or the Syracuse problem.
Collatz Conjecture (Wikipedia)
Wikimedia Foundationlead section, named after Lothar CollatzQuote, lead section, named after Lothar Collatz
The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1.
Collatz Conjecture (Wikipedia)
Wikimedia FoundationPosed By: Lothar Collatz, Statement of the problem section, opening paragraphQuote, Posed By: Lothar Collatz, Statement of the problem section, opening paragraph
It is named after the mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate.
Collatz Conjecture (Wikipedia)
Wikimedia FoundationIn Branch: Number Theory, Footer categories, Unsolved problems in number theoryQuote, In Branch: Number Theory, Footer categories, Unsolved problems in number theory
Unsolved problems in number theory
Open Questions (1 open question)
Does the Collatz process really always reach one, from every possible starting number?
Every starting number ever tested by computer, into the astronomically large, reaches one, but no proof rules out either an unbounded starting number whose path never terminates or a hidden cycle other than the trivial one at 1-4-2; Paul Erdos himself is widely quoted as doubting mathematics currently has the tools to settle it.
What would resolve this A general proof covering every positive starting integer, or the discovery of a starting number that provably never reaches one (an unbounded trajectory or a second cycle).
Number theory / dynamical systemsWolfram MathWorld
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