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The Collatz conjecture is one of the most famous
unsolved problems in mathematics 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 Eucli ...
. The
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
asks whether repeating two simple arithmetic operations will eventually transform every
positive integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if a term is even, the next term is one half of it. If a term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. The conjecture has been shown to hold for all positive integers up to , but no general proof has been found. It is named after the mathematician
Lothar Collatz Lothar Collatz (; July 6, 1910 – September 26, 1990) was a German mathematician, born in Arnsberg, Province of Westphalia, Westphalia. The "3''x'' + 1" problem is also known as the Collatz conjecture, named after him and still unsolved. The Col ...
, who introduced the idea in 1937, two years after receiving his doctorate. 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), or as wondrous numbers.
Paul Erdős Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, g ...
said about the Collatz conjecture: "Mathematics may not be ready for such problems." Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics". However, though the Collatz conjecture itself remains open, efforts to solve the problem have led to new techniques and many partial results.


Statement of the problem

Consider the following operation on an arbitrary
positive integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
: * If the number is even, divide it by two. * If the number is odd, triple it and add one. In
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ...
notation, define the
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
as follows: f(n) = \begin n/2 &\text n \equiv 0 \pmod,\\ 3n+1 & \text n\equiv 1 \pmod .\end Now form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next. In notation: a_i = \beginn & \text i = 0, \\ f(a_) & \text i > 0 \end (that is: is the value of applied to recursively times; ). The Collatz conjecture is: ''This process will eventually reach the number 1, regardless of which positive integer is chosen initially. That is, for each'' n, there is some i with a_i = 1. If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. The smallest such that is called the stopping time of . Similarly, the smallest such that is called the total stopping time of . If one of the indexes or doesn't exist, we say that the stopping time or the total stopping time, respectively, is infinite. The Collatz conjecture asserts that the total stopping time of every is finite. It is also equivalent to saying that every has a finite stopping time. Since is even whenever is odd, one may instead use the "shortcut" form of the Collatz function: f(n) = \begin \frac &\text n \equiv 0 \pmod,\\ \frac & \text n\equiv 1 \pmod. \end This definition yields smaller values for the stopping time and total stopping time without changing the overall dynamics of the process.


Empirical data

For instance, starting with and applying the function without "shortcut", one gets the sequence . The number takes longer to reach 1: . The sequence for , listed and graphed below, takes 111 steps (41 steps through odd numbers, in bold), climbing as high as 9232 before descending to 1. : Numbers with a total stopping time longer than that of any smaller starting value form a sequence beginning with: :1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, ... . The starting values whose
maximum In mathematical analysis, the maximum and minimum of a function (mathematics), function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given Interval (ma ...
trajectory point is greater than that of any smaller starting value are as follows: :1, 2, 3, 7, 15, 27, 255, 447, 639, 703, 1819, 4255, 4591, 9663, 20895, 26623, 31911, 60975, 77671, 113383, 138367, 159487, 270271, 665215, 704511, ... Number of steps for to reach 1 are :0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14, 9, 9, 17, 17, 4, 12, 20, 20, 7, 7, 15, 15, 10, 23, 10, 111, 18, 18, 18, 106, 5, 26, 13, 13, 21, 21, 21, 34, 8, 109, 8, 29, 16, 16, 16, 104, 11, 24, 24, ... The starting value having the largest total stopping time while being :less than 10 is 9, which has 19 steps, :less than 100 is 97, which has 118 steps, :less than 1000 is 871, which has 178 steps, :less than 104 is 6171, which has 261 steps, :less than 105 is , which has 350 steps, :less than 106 is , which has 524 steps, :less than 107 is , which has 685 steps, :less than 108 is , which has 949 steps, :less than 109 is , which has 986 steps, :less than 1010 is , which has 1132 steps, :less than 1011 is , which has 1228 steps, :less than 1012 is , which has 1348 steps. (Note: "Delay records" are total stopping time records.) These numbers are the lowest ones with the indicated step count, but not necessarily the only ones below the given limit. As an example, has 1132 steps, as does . The starting values having the smallest total stopping time with respect to their number of digits (in base 2) are the
powers of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In the fast-growing hierarchy, is exactly equal to f_1^n(1). In the Hardy hi ...
, since is halved times to reach 1, and it is never increased.


Visualizations

File:Collatz orbits of the all integers up to 1000.svg, Directed graph showing the orbits of the first 1000 numbers. File:CollatzConjectureGraphMaxValues.jpg, The axis represents starting number, the axis represents the highest number reached during the chain to 1. This plot shows a restricted axis: some values produce intermediates as high as (for ) File:Collatz-max.png, The same plot as the previous one but on log scale, so all values are shown. The first thick line towards the middle of the plot corresponds to the tip at 27, which reaches a maximum at 9232. File:All Collatz sequences of a length inferior to 20.svg, The tree of all the numbers having fewer than 20 steps. File:Collatz Conjecture 100M.jpg, alt=Collatz Conjecture 100M, The number of iterations it takes to get to one for the first 100 million numbers. File:Collatz_conjecture_tree_visualization.png, Collatz conjecture paths for 5000 random starting points below 1 million.


Supporting arguments

Although the conjecture has not been proven, most mathematicians who have looked into the problem think the conjecture is true because experimental evidence and heuristic arguments support it.


Experimental evidence

The conjecture has been checked by computer for all starting values up to 271 ≈ . All values tested so far converge to 1. This computer evidence is still not rigorous proof that the conjecture is true for all starting values, as
counterexamples A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a co ...
may be found when considering very large (or possibly immense) positive integers, as in the case of the disproven
Pólya conjecture In number theory, the Pólya conjecture (or Pólya's conjecture) stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an ''odd'' number of prime factors. The conjecture was set forth by the Hungarian mathe ...
and
Mertens conjecture In mathematics, the Mertens conjecture is the statement that the Mertens function M(n) is bounded by \pm\sqrt. Although now disproven, it had been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1885 ...
. However, such verifications may have other implications. Certain constraints on any non-trivial cycle, such as
lower bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of . Dually, a lower bound or minorant of is defined to be an element of that is less th ...
s on the length of the cycle, can be proven based on the value of the lowest term in the cycle. Therefore, computer searches to rule out cycles that have a small lowest term can strengthen these constraints.


A probabilistic heuristic

If one considers only the ''odd'' numbers in the sequence generated by the Collatz process, then each odd number is on average of the previous one. (More precisely, the geometric mean of the ratios of outcomes is .) This yields a heuristic argument that every Hailstone sequence should decrease in the long run, although this is not evidence against other cycles, only against divergence. The argument is not a proof because it assumes that Hailstone sequences are assembled from uncorrelated probabilistic events. (It does rigorously establish that the 2-adic extension of the Collatz process has two division steps for every multiplication step for
almost all In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
2-adic starting values.)


Stopping times

As proven by Riho Terras, almost every positive integer has a finite stopping time. In other words, almost every Collatz sequence reaches a point that is strictly below its initial value. The proof is based on the distribution of parity vectors and uses the
central limit theorem In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. In 2019, Terence Tao improved this result by showing, using logarithmic
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
, that
almost all In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
(in the sense of logarithmic density) Collatz orbits are descending below any given function of the starting point, provided that this function diverges to infinity, no matter how slowly. Responding to this work, ''
Quanta Magazine ''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science. History ''Quanta Magazine'' was initially launched as ''Simons Science ...
'' wrote that Tao "came away with one of the most significant results on the Collatz conjecture in decades".


Lower bounds

In a
computer-aided proof Automation describes a wide range of technologies that reduce human intervention in processes, mainly by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machine ...
, Krasikov and Lagarias showed that the number of integers in the interval that eventually reach 1 is at least equal to for all sufficiently large .


Cycles

In this part, consider the shortcut form of the Collatz function f(n) = \begin \frac &\text n \equiv 0 \pmod,\\ \frac & \text n \equiv 1 \pmod. \end A cycle is a sequence of distinct positive integers where , , ..., and . The only known cycle is of period 2, called the trivial cycle.


Cycle length

The length of a non-trivial cycle is known to be at least (or without shortcut). If it can be shown that for all positive integers less than 3 \times 2^ the Collatz sequences reach 1, then this bound would raise to ( without shortcut). In fact, Eliahou (1993) proved that the period of any non-trivial cycle is of the form p = 301994 a + 17087915 b + 85137581 c where , and are non-negative integers, and . This result is based on the
simple continued fraction A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence \ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fr ...
expansion of .


-cycles

A -cycle is a cycle that can be partitioned into contiguous subsequences, each consisting of an increasing sequence of odd numbers, followed by a decreasing sequence of even numbers. For instance, if the cycle consists of a single increasing sequence of odd numbers followed by a decreasing sequence of even numbers, it is called a ''1-cycle''. Steiner (1977) proved that there is no 1-cycle other than the trivial . Simons (2005) used Steiner's method to prove that there is no 2-cycle. Simons and de Weger (2005) extended this proof up to 68-cycles; there is no -cycle up to . Hercher extended the method further and proved that there exists no ''k''-cycle with . As exhaustive computer searches continue, larger values may be ruled out. To state the argument more intuitively; we do not have to search for cycles that have less than 92 subsequences, where each subsequence consists of consecutive ups followed by consecutive downs.


Other formulations of the conjecture


In reverse

There is another approach to prove the conjecture, which considers the bottom-up method of growing the so-called ''Collatz graph''. The ''Collatz graph'' is a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
defined by the inverse
relation Relation or relations may refer to: General uses * International relations, the study of interconnection of politics, economics, and law on a global level * Interpersonal relationship, association or acquaintance between two or more people * ...
R(n) = \begin \ & \text n\equiv 0,1,2,3,5 \\ \left\ & \text n\equiv 4 \end \pmod 6. So, instead of proving that all positive integers eventually lead to 1, we can try to prove that 1 leads backwards to all positive integers. For any integer ,
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
. Equivalently, if and only if . Conjecturally, this inverse relation forms a
tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only ...
except for the 1–2–4 loop (the inverse of the 4–2–1 loop of the unaltered function defined in the Statement of the problem section of this article). When the relation of the function is replaced by the common substitute "shortcut" relation , the Collatz graph is defined by the inverse relation, R(n) = \begin \ & \text n\equiv 0,1 \\ \left\ & \text n\equiv 2 \end \pmod 3. For any integer , if and only if . Equivalently, if and only if . Conjecturally, this inverse relation forms a tree except for a 1–2 loop (the inverse of the 1–2 loop of the function f(n) revised as indicated above). Alternatively, replace the with where and is the highest power of 2 that divides (with no
remainder In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In a ...
). The resulting function maps from
odd number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The ...
s to odd numbers. Now suppose that for some odd number , applying this operation times yields the number 1 (that is, ). Then in
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two values (0 and 1) for each digit * Binary function, a function that takes two arguments * Binary operation, a mathematical op ...
, the number can be written as the concatenation of strings where each is a finite and contiguous extract from the representation of . The representation of therefore holds the repetends of , where each repetend is optionally rotated and then replicated up to a finite number of bits. It is only in binary that this occurs. Conjecturally, every binary string that ends with a '1' can be reached by a representation of this form (where we may add or delete leading '0's to ).


As an abstract machine that computes in base two

Repeated applications of the Collatz function can be represented as an
abstract machine In computer science, an abstract machine is a theoretical model that allows for a detailed and precise analysis of how a computer system functions. It is similar to a mathematical function in that it receives inputs and produces outputs based on p ...
that handles strings of
bit The bit is the most basic unit of information in computing and digital communication. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented as ...
s. The machine will perform the following three steps on any odd number until only one remains: # Append to the (right) end of the number in binary (giving ); # Add this to the original number by binary addition (giving ); # Remove all trailing s (that is, repeatedly divide by 2 until the result is odd).


Example

The starting number 7 is written in base two as . The resulting Collatz sequence is:
111 1111 10110 10111 100010 100011 110100 11011 101000 1011 10000


As a parity sequence

For this section, consider the shortcut form of the Collatz function f(n) = \begin \frac &\text n \equiv 0 \\ \frac & \text n \equiv 1 \end \pmod. If is the parity of a number, that is and , then we can define the Collatz parity sequence (or parity vector) for a number as , where , and . Which operation is performed, or , depends on the parity. The parity sequence is the same as the sequence of operations. Using this form for , it can be shown that the parity sequences for two numbers and will agree in the first terms if and only if and are equivalent modulo . This implies that every number is uniquely identified by its parity sequence, and moreover that if there are multiple Hailstone cycles, then their corresponding parity cycles must be different. Applying the function times to the number will give the result , where is the result of applying the function times to , and is how many increases were encountered during that sequence. For example, for there are 3 increases as 1 iterates to 2, 1, 2, 1, and finally to 2 so the result is ; for there is only 1 increase as 1 rises to 2 and falls to 1 so the result is . When is then there will be rises and the result will be . The power of 3 multiplying is independent of the value of ; it depends only on the behavior of . This allows one to predict that certain forms of numbers will always lead to a smaller number after a certain number of iterations: for example, becomes after two applications of and becomes after four applications of . Whether those smaller numbers continue to 1, however, depends on the value of .


As a tag system

For the Collatz function in the shortcut form f(n) = \begin \frac &\text n \equiv 0 \\ \frac & \text n \equiv 1. \end \pmod Hailstone sequences can be computed by the 2-tag system with production rules :, , . In this system, the positive integer is represented by a string of copies of , and iteration of the tag operation halts on any word of length less than 2. (Adapted from De Mol.) The Collatz conjecture equivalently states that this tag system, with an arbitrary finite string of as the initial word, eventually halts (see ''
Tag system In the theory of computation, a tag system is a deterministic model of computation published by Emil Leon Post in 1943 as a simple form of a Post canonical system. A tag system may also be viewed as an abstract machine, called a Post tag machine ( ...
'' for a worked example).


Extensions to larger domains


Iterating on all integers

An extension to the Collatz conjecture is to include all integers, not just positive integers. Leaving aside the cycle 0 → 0 which cannot be entered from outside, there are a total of four known cycles, which all nonzero integers seem to eventually fall into under iteration of . These cycles are listed here, starting with the well-known cycle for positive : Odd values are listed in large bold. Each cycle is listed with its member of least absolute value (which is always odd) first. The generalized Collatz conjecture is the assertion that every integer, under iteration by , eventually falls into one of the four cycles above or the cycle 0 → 0.


Iterating on rationals with odd denominators

The Collatz map can be extended to (positive or negative) rational numbers which have odd denominators when written in lowest terms. The number is taken to be 'odd' or 'even' according to whether its numerator is odd or even. Then the formula for the map is exactly the same as when the domain is the integers: an 'even' such rational is divided by 2; an 'odd' such rational is multiplied by 3 and then 1 is added. A closely related fact is that the Collatz map extends to the ring of 2-adic integers, which contains the ring of rationals with odd denominators as a subring. When using the "shortcut" definition of the Collatz map, it is known that any periodic parity sequence is generated by exactly one rational. Conversely, it is conjectured that every rational with an odd denominator has an eventually cyclic parity sequence (Periodicity Conjecture). If a parity cycle has length and includes odd numbers exactly times at indices , then the unique rational which generates immediately and periodically this parity cycle is For example, the parity cycle has length 7 and four odd terms at indices 0, 2, 3, and 6. It is repeatedly generated by the fraction \frac = \frac as the latter leads to the rational cycle \frac \rightarrow \frac \rightarrow \frac \rightarrow \frac \rightarrow \frac \rightarrow \frac \rightarrow \frac \rightarrow \frac . Any cyclic permutation of is associated to one of the above fractions. For instance, the cycle is produced by the fraction \frac = \frac . For a one-to-one correspondence, a parity cycle should be ''irreducible'', that is, not partitionable into identical sub-cycles. As an illustration of this, the parity cycle and its sub-cycle are associated to the same fraction when reduced to lowest terms. In this context, assuming the validity of the Collatz conjecture implies that and are the only parity cycles generated by positive whole numbers (1 and 2, respectively). If the odd denominator of a rational is not a multiple of 3, then all the iterates have the same denominator and the sequence of numerators can be obtained by applying the " " generalization of the Collatz function T_d(x) = \begin \frac &\text x \equiv 0 \pmod,\\ \frac & \text x\equiv 1 \pmod. \end


2-adic extension

The function T(x) = \begin \frac &\text x \equiv 0 \pmod\\ \frac & \text x\equiv 1 \pmod \end is well-defined on the ring \mathbb_2 of 2-adic integers, where it is continuous and measure-preserving with respect to the 2-adic measure. Moreover, its dynamics is known to be
ergodic In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies th ...
. Define the ''parity vector'' function acting on \mathbb_2 as Q(x) = \sum_^ \left( T^k (x) \mod 2 \right) 2^k . The function is a 2-adic
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
. Consequently, every infinite parity sequence occurs for exactly one 2-adic integer, so that
almost all In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
trajectories are acyclic in \mathbb_2. An equivalent formulation of the Collatz conjecture is that Q\left(\mathbb^\right) \subset \tfrac13 \mathbb.


Iterating on real or complex numbers

The Collatz map can be extended to the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
by choosing any function which evaluates to x/2 when x is an even integer, and to either 3x + 1 or (3x + 1)/2 (for the "shortcut" version) when x is an odd integer. This is called an interpolating function. A simple way to do this is to pick two functions g_1 and g_2, where: :g_1(n) = \begin1, &n\text\\ 0, &n\text\end :g_2(n) = \begin0, &n\text\\1, &n\text\end and use them as switches for our desired values: :f(x) \triangleq \frac\cdot g_1(x) \,+\, \frac\cdot g_2(x). One such choice is g_1(x) \triangleq \cos^2\left(\tfrac x\right) and g_2(x) \triangleq \sin^2\left(\tfrac x\right). The iterations of this map lead to a
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
, further investigated by Marc Chamberland. He showed that the conjecture does not hold for positive real numbers since there are infinitely many
fixed points Fixed may refer to: * ''Fixed'' (EP), EP by Nine Inch Nails * ''Fixed'' (film), an upcoming animated film directed by Genndy Tartakovsky * Fixed (typeface), a collection of monospace bitmap fonts that is distributed with the X Window System * Fi ...
, as well as
orbits In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificia ...
escaping monotonically to infinity. The function f has two attracting cycles of period 2: (1;\,2) and (1.1925...;\,2.1386...). Moreover, the set of unbounded orbits is conjectured to be of measure 0. Letherman, Schleicher, and Wood extended the study to the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
. They used Chamberland's function for complex sine and cosine and added the extra term \tfrac\left(\tfrac12 - \cos(\pi z)\right)\sin(\pi z)\,+ h(z)\sin^2(\pi z), where h(z) is any
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any ...
. Since this expression evaluates to zero for real integers, the extended function :\beginf(z) \triangleq \;&\frac\cos^2\left(\frac z\right) + \frac\sin^2\left(\frac z\right) \, + \\ &\frac\left(\frac12 - \cos(\pi z)\right)\sin(\pi z) + h(z)\sin^2(\pi z)\end is an interpolation of the Collatz map to the complex plane. The reason for adding the extra term is to make all integers critical points of f. With this, they show that no integer is in a Baker domain, which implies that any integer is either eventually periodic or belongs to a wandering domain. They conjectured that the latter is not the case, which would make all integer orbits finite. Most of the points have orbits that diverge to infinity. Coloring these points based on how fast they diverge produces the image on the left, for h(z) \triangleq 0. The inner black regions and the outer region are the Fatou components, and the boundary between them is the
Julia set In complex dynamics, the Julia set and the Classification of Fatou components, Fatou set are two complement set, complementary sets (Julia "laces" and Fatou "dusts") defined from a function (mathematics), function. Informally, the Fatou set of ...
of f, which forms a
fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
pattern, sometimes called a "Collatz fractal". There are many other ways to define a complex interpolating function, such as using the complex exponential instead of sine and cosine: :f(z) \triangleq \frac + \frac14(2z + 1)\left(1 - e^\right), which exhibit different dynamics. In this case, for instance, if \operatorname(z) \gg 1, then f(z) \approx z + \tfrac14. The corresponding Julia set, shown on the right, consists of uncountably many curves, called ''hairs'', or ''rays''.


Optimizations


Time–space tradeoff

The section '' As a parity sequence'' above gives a way to speed up simulation of the sequence. To jump ahead steps on each iteration (using the function from that section), break up the current number into two parts, (the least significant bits, interpreted as an integer), and (the rest of the bits as an integer). The result of jumping ahead is given by :. The values of (or better ) and can be precalculated for all possible -bit numbers , where is the result of applying the function times to , and is the number of odd numbers encountered on the way. For example, if , one can jump ahead 5 steps on each iteration by separating out the 5 least significant bits of a number and using : (0...31, 5) = , : (0...31, 5) = . This requires
precomputation In algorithms, precomputation is the act of performing an initial computation before Run time (program lifecycle phase), run time to generate a lookup table that can be used by an algorithm to avoid repeated computation each time it is executed. ...
and storage to speed up the resulting calculation by a factor of , a
space–time tradeoff space–time trade-off, also known as time–memory trade-off or the algorithmic space-time continuum in computer science is a case where an algorithm or program trades increased space usage with decreased time. Here, ''space'' refers to the d ...
.


Modular restrictions

For the special purpose of searching for a counterexample to the Collatz conjecture, this precomputation leads to an even more important acceleration, used by Tomás Oliveira e Silva in his computational confirmations of the Collatz conjecture up to large values of . If, for some given and , the inequality : holds for all , then the first counterexample, if it exists, cannot be modulo . For instance, the first counterexample must be odd because , smaller than ; and it must be 3 mod 4 because , smaller than . For each starting value which is not a counterexample to the Collatz conjecture, there is a for which such an inequality holds, so checking the Collatz conjecture for one starting value is as good as checking an entire congruence class. As increases, the search only needs to check those residues that are not eliminated by lower values of . Only an exponentially small fraction of the residues survive. For example, the only surviving residues mod 32 are 7, 15, 27, and 31. Integers divisible by 3 cannot form a cycle, so these integers do not need to be checked as counter examples.


Syracuse function

If is an odd integer, then is even, so with odd and . The Syracuse function is the function from the set of positive odd integers into itself, for which . Some properties of the Syracuse function are: * For all , . (Because .) * In more generality: For all and odd , . (Here is function iteration notation.) * For all odd , The Collatz conjecture is equivalent to the statement that, for all in , there exists an integer such that .


Undecidable generalizations

In 1972,
John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many b ...
proved that a natural generalization of the Collatz problem is algorithmically undecidable. Specifically, he considered functions of the form \text , where are rational numbers which are so chosen that is always an integer. The standard Collatz function is given by , , , , . Conway proved that the problem : Given and , does the sequence of iterates reach ? is undecidable, by representing the
halting problem In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run for ...
in this way. Closer to the Collatz problem is the following ''universally quantified'' problem: : Given , does the sequence of iterates reach , for all ? Modifying the condition in this way can make a problem either harder or easier to solve (intuitively, it is harder to justify a positive answer but might be easier to justify a negative one). Kurtz and Simon A
PDF
/ref> proved that the universally quantified problem is, in fact, undecidable and even higher in the
arithmetical hierarchy In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define th ...
; specifically, it is -complete. This hardness result holds even if one restricts the class of functions by fixing the modulus to 6480. Iterations of in a simplified version of this form, with all b_i equal to zero, are formalized in an
esoteric programming language An esoteric programming language (sometimes shortened to esolang) is a programming language designed to test the boundaries of computer programming language design, as a proof of concept, as software art, as a hacking interface to another language ...
called FRACTRAN.


In computational complexity

Collatz and related conjectures are often used when studying computational complexity. The connection is made through the
busy beaver In theoretical computer science, the busy beaver game aims to find a terminating Computer program, program of a given size that (depending on definition) either produces the most output possible, or runs for the longest number of steps. Since an ...
function, where BB(n) is the maximum number of steps taken by any n state
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
that halts. There is a 15 state Turing machine that halts if and only if a conjecture by
Paul Erdős Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, g ...
(closely related to the Collatz conjecture) is false. Hence if BB(15) was known, and this machine did not stop in that number of steps, it would be known to run forever and hence no counterexamples exist (which proves the conjecture true). This is a completely impractical way to settle the conjecture; instead it is used to suggest that BB(15) will be very hard to compute, at least as difficult as settling this Collatz-like conjecture.


See also

* semigroup *
Arithmetic dynamics Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex dynamics, the study of the Iterated function, iteration of self-maps of the complex plane or o ...
* Juggler sequence *
Modular arithmetic In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ...
* Residue-class-wise affine group


Notes


References


External links

* * An ongoing
volunteer computing Volunteer computing is a type of distributed computing in which people donate their computers' unused resources to a research-oriented project, and sometimes in exchange for credit points. The fundamental idea behind it is that a modern desktop ...
br>project
by David Bařina verifies Convergence of the Collatz conjecture for large values. (furthest progress so far) * (
BOINC The Berkeley Open Infrastructure for Network Computing (BOINC, pronounced rhymes with "oink") is an open-source middleware system for volunteer computing (a type of distributed computing). Developed originally to support SETI@home, it became the ...
) volunteer computin
project
that verifies the Collatz conjecture for larger values. * An ongoing volunteer computin

by Eric Roosendaal verifies the Collatz conjecture for larger and larger values. * Another ongoing volunteer computin

by Tomás Oliveira e Silva continues to verify the Collatz conjecture (with fewer statistics than Eric Roosendaal's page but with further progress made). * * . * * * * {{cite AV media , medium=short video , people= Alex Kontorovich (featuring) , title=The simplest math problem no one can solve , date=30 July 2021 , series=Veritasium , via=YouTube , url=https://www.youtube.com/watch?v=094y1Z2wpJg
Are computers ready to solve this notoriously unwieldy math problem?
Conjectures Arithmetic dynamics Integer sequences Unsolved problems in number theory