The Collatz conjecture is an unsolved 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 ends in the same way, regardless of the starting number. In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Lothar Collatz (July 6, 1910 — September 26, 1990) was a German mathematician. ... 1937 (MCMXXXVII) was a common year starting on Friday (link will take you to calendar). ... Stanisław Marcin Ulam (April 13, 1909–May 13, 1984) was a Polish-American mathematician who helped develop the key theory behind the hydrogen bomb. ... GEB cover Gödel, Escher, Bach: an Eternal Golden Braid (commonly GEB) is a Pulitzer Prize-winning book by Douglas Hofstadter, published in 1979 by Basic Books. ...

Paul Erdős said about the Collatz conjecture, "Mathematics is not yet ready for such problems." He offered $500 for its solution. (Lagarias 1985) Paul ErdÅ‘s, also Pál ErdÅ‘s, in English Paul Erdos or Paul Erdös (March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian mathematician who, with hundreds of collaborators, worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set...

Statement of the problem

Consider the following operation on an arbitrary positive integer: The integers are commonly denoted by the above symbol. ...

• If the number is even, divide it by two.
• If the number is odd, triple it and add one.

For example, if this operation is performed on 3, the result is 10; if it is performed on 28, the result is 14.

In modular arithmetic notation, define the function f as follows: Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... Partial plot of a function f. ...

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:

The Collatz conjecture is: This process will eventually reach the number 1, regardless of which positive integer is chosen initially. Or, more formally:

That smallest i such that the above holds is called the total stopping time of n. The conjecture asserts that every n has a well-defined stopping time. If, for some n, such an i doesn't exist, we say that n has infinite total stopping time and the conjecture is false.

If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence which does not contain 1. Such a sequence might enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found.

Examples

For instance, starting with n = 6, one gets the sequence 6, 3, 10, 5, 16, 8, 4, 2, 1.

Starting with n = 11, the sequence takes longer to reach 1: 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

If the starting value n = 27 is chosen, the sequence takes 111 steps, climbing above 9,000 before descending to 1.

{ 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1 }

Program to calculate Collatz sequences

A specific Collatz sequence can be easily computed, as is shown by this pseudocode example: Pseudocode (derived from pseudo and code) is a compact and informal high-level description of a computer programming algorithm that uses the structural conventions of programming languages, but omits detailed subroutines, variable declarations or language-specific syntax. ...

 function collatz(n) while n > 1 show n if n is odd set n to 3n + 1 else set n to n / 2 show n 

This program halts when the sequence reaches 1, in order to avoid printing an endless cycle of 4, 2, 1. If the Collatz conjecture is true, the program will always halt no matter what positive starting integer is given to it. (See Halting problem for a discussion of the relationship between open-ended computer programs and unsolved mathematics problems.) In computability theory the halting problem is a decision problem which can be informally stated as follows: Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input. ...

Supporting arguments

Although the conjecture has not been proven, most mathematicians who have looked into the problem believe intuitively that the conjecture is true. Here are two reasons for expecting this.

Experimental evidence

The conjecture has been checked by computer for all start values up to 10 × 2⁵⁸ ≈ 2.88 × 10¹⁸[1]. While impressive, such computer bounds are of very limited evidential value, with several conjectures having turned out to have exceptionally large-valued counterexamples (e.g. the Pólya conjecture). The Pólya Conjecture is an resolved and disproved conjecture in mathematics named after the Hungarian mathematician George Pólya, who stated it in 1919. ...

Probabilistic evidence

If one considers only the odd numbers in the sequence generated by the Collatz process, then one can argue that on average (specifically, the geometric mean of the ratios) the next odd number should be about ¾ of the previous one [2], which suggests that they should decrease in the long run (although this is not evidence against cycles, only against divergence).

Other ways of approaching the problem

In reverse

There is another approach to prove the conjecture, which considers the bottom-up method of growing the Collatz graph. The Collatz graph is defined by an inverse relation, In mathematics, a finitary relation is defined by one of the formal definitions given below. ...

So, instead of proving that all natural numbers eventually lead to 1, we can prove that 1 leads to all natural numbers. For any integer n, 3n + 1 ≡4 (mod 6) iff n ≡1 (mod 2) and thus n ≡1, 3 or 5 (mod 6). Also, the inverse relation forms a tree except for the 1-2-4 loop (the inverse of the 1-4-2 loop of the unaltered function f defined in the statement of the problem above). When the relation 3n + 1 of the function f(n) is replaced by the common substitute "shortcut" relation (3n + 1)/2 (see Optimizations below), the Collatz graph is defined by the inverse relation, IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

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).

As rational numbers

The natural numbers can be converted to rational numbers in a certain way. To get the rational version, find the highest power of two less than or equal to the number, use it as the denominator, and subtract it from the original number for the numerator (527 → 15/512). To get the natural version, add the numerator and denominator (255/256 → 511).

The Collatz conjecture then says that the numerator will eventually equal zero. The Collatz function changes to:

(n = numerator; d = denominator)

This works because 3x + 1 = 3(d + n) + 1 = (2d) + (3n + d + 1) = (4d) + (3n - d + 1). Reducing a rational before every operation is required to get x as an odd.

As an abstract machine

Repeated applications of the Collatz function can be represented as an abstract machine that handles strings of bits. The machine will perform the following two steps on any odd number until only one "1" remains: An abstract machine, also called an abstract computer, is a theoretical model of a computer hardware or software system used in Automata theory. ... In computer programming and some branches of mathematics, strings are sequences of various simple objects. ... This article is about the unit of information. ...

1. Add the original with a "1" appended to the end to the original (interpreting the string as a binary integer), i.e. 3n + 1 = (2n + 1) + n
2. Remove all trailing "0"s.

As a parity sequence

For this section, consider the Collatz function in the slightly modified form

This can be done because when n is odd, 3n + 1 is always even.

If P(…) is the parity of a number, that is P(2n) = 0 and P(2n + 1) = 1, then we can define the Collatz parity sequence for a number n as p_i = P(a_i), where a₀ = n, and a_i+1 = f(a_i).

Using this form for f(n), it can be shown that the parity sequences for two numbers m and n will agree in the first k terms if and only if m and n are equivalent modulo 2^k. This implies that every number is uniquely identified by its parity sequence, and moreover that if there are multiple Collatz cycles, then their corresponding parity cycles must be different.

The Collatz Conjecture can be rephrased as stating that the Collatz parity sequence for every number eventually enters the cycle 0 → 1 → 0.

Extensions to larger domains

Iterating on all integers

For any integer n, we map it to the integer f(n), where

 f(n) = 3n + 1 if n is odd; f(n) = n/2 if n is even. 

Interestingly, there are in this case a total of 5 known cycles, which all integers seem to eventually fall into under iteration of f. These cycles are listed here, starting with the well-known cycle for positive n.

To save steps, we list only the odd numbers of each cycle (except for the trivial cycle {0}). Each odd number n, when f is applied repeatedly, will next reach an odd number at (3n+1) / (the largest power of 2 that divides 3n+1); each cycle is listed with its member of least absolute value first. We follow each cycle with the size of the full cycle (in parentheses): the number of members, odd or even, belonging to a cycle, counted without repetition.

 a) 1 → 1 (size 3) 

 b) 0 → 0 (size 1) 

 c) -1 → -1 (size 2) 

 d) -5 → -7 → -5 (size 5) 

 e) -17 → -25 → -37 → -55 → -41 → -61 → -91 → -17 (size 18) 

We may define the Generalized Collatz Conjecture as the assertion that every integer, under iteration by f, eventually falls into one of these five cycles a), b), c), d), or e).

Iterating on rational numbers with odd denominators

The standard 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.

The parity sequences as defined above are no longer unique for fractions. However, it can be shown that any possible parity cycle is the parity sequence for exactly one fraction: if a cycle has length n and includes odd numbers exactly m times at indices k₀, …, k_m-1, then the unique fraction which generates that parity cycle is

.

For example, the parity cycle (1 0 1 1 0 0 1) has length 7 and has 4 odd numbers at indices 0, 2, 3, and 6. The unique fraction which generates that parity cycle is

.

In this context, the Collatz conjecture is equivalent to saying that (0 1) is the only cycle which is generated by positive whole numbers (i.e. 1 and 2).

Iterating on real or complex numbers

Cobweb plot of the orbit 10-5-8-4-2-1-2-1-2-1-etc. in the real extension of the Collatz map (optimized by replacing "3n + 1" with "(3n + 1)/2" )

The Collatz map can be viewed as the restriction to the integers of the smooth real and complex map Image File history File links Download high resolution version (724x604, 24 KB) Summary Cobweb plot of an orbit of the real extension of the collatz map. ... Image File history File links Download high resolution version (724x604, 24 KB) Summary Cobweb plot of an orbit of the real extension of the collatz map. ... A cobweb plot of the logistic map, showing an attracting fixed point. ...

,

which simplifies to .

If the standard Collatz map defined above is optimized by replacing "3n + 1" with "(3n + 1)/2" (see Optimizations below), it can be viewed as the restriction to the integers of the smooth real and complex map

,

which simplifies to .

Iterating the above optimized map in the complex plane produces the Collatz fractal. In mathematics, iterated functions are the objects of study in fractals and dynamical systems. ... The boundary of the Mandelbrot set is a famous example of a fractal. ...

Collatz map fractal in a neighbourhood of the real line

Image File history File links Download high resolution version (996x597, 390 KB) Summary Fractal of the Collatz map. ... Image File history File links Download high resolution version (996x597, 390 KB) Summary Fractal of the Collatz map. ... The boundary of the Mandelbrot set is a famous example of a fractal. ...

Optimizations

• If n is a multiple of 4, it can be divided by 4.

Reason: It is initially even. When it is divided by two, it is still even.

Example: If n = 20, then the sequence goes

20 → 10 → 5 (= 20/4).

• More generally, one can write n as the product of its factors, and change the power of 2 to 2⁰.

Reason: If the power of 2 of the prime factorization is greater than 0, the number is even, and the following step would produce the same factorization with one less power of 2.

Example: Instead of

15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1 (17 steps),

one can go

15, 46 (2¹×23), 23, 70 (2¹×35), 35, 106 (2¹×53), 53, 160 (2⁵×5), 5, 16 (2⁴), 1 (11 steps).

• If n is odd, one step can be skipped by finding (3n + 1) / 2

Reason: an odd times an odd is always an odd (neither contributes a factor of 2 and without a 2, it cannot be even), so 3n is odd. Hence after two steps we have (3n + 1) / 2

Example: Take n = 35. Then the normal Collatz sequence is

35 → 3 × 35 + 1 = 106 → 53 (= (3 × 35 + 1)/2).

• If n ≡ 1 mod 4, then we can replace it repeatedly by (n - 1)/4 to obtain an intermediate k not equivalent to 1 modulo 4, and obtain 3k + 1.

Reason: n = 4m + 1 is always odd, hence becomes 3n + 1 = 12m + 4 = 4(3m + 1), and by the first optimization, we may drop the 4. Note that if m is odd, 3m + 1 is its Collatz successor, so that to continue the Collatz sequence of n, we can instead continue that of m. Hence we have effectively replaced n by m = (n - 1)/4. We may make this reduction repeatedly, until we obtain a number k which is not equivalent to 1 modulo 4. Finally, we are now looking at 3k + 1, as desired. Note that k may be even, so we cannot necessarily divide by 2.

Example: When n = 405 the optimized sequence is

405 → 101 → 25 → 6 → 19.

The normal Collatz sequence is:

405 → 1216 → 608 → 304 → 152 → 76 → 38 → 19.

The above facts can be used to create a new version of the Collatz function :

where

Syracuse Function

If k is an odd integer, then 3k + 1 is even, so we can write 3k + 1 = 2^ak′, with k' odd and a ≥ 1. We define a function f from the set I of odd integers into itself, called the Syracuse Function, by taking f (k) = k′.

Some properties of the Syracuse function are:

• f (4k + 1) = f (k) for all k in I.
• For all p ≥ 2 and h odd, f ^{p - 1}(2 ^p h - 1) = 2 3 ^{p - 1}h - 1 (see here for the notation).
• For all odd h, f (2h - 1) ≤ (3h - 1)/2

The Syracuse Conjecture is that for all k in I, there exists an integer n ≥ 1 such that f ⁿ(k) = 1. Equivalently, let E be the set of odd integers k for which there exists an integer n ≥ 1 such that f ⁿ(k) = 1. The problem is to show that E = I. The following is the beginning of an attempt at a proof by induction: In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

We know that 1, 3, 5, 7, and 9 are in E. Let k be an odd integer greater than 9. Suppose that the odd numbers up to and including k - 2 are in E and let us try to prove that k is in E. As k is odd, k + 1 is even, so we can write k + 1 = 2^ph for p ≥ 1, h odd, and k = 2^ph-1. Now we have:

• If p = 1, then k = 2h - 1. It is easy to check that f (k) < k , so f (k) ∈ E; hence k ∈ E.
• If p ≥ 2 and h is a multiple of 3, we can write h = 3h′. Let k′ = 2^{p + 1}h′ - 1; we have f (k′) = k , and as k′ < k , k′ is in E; therefore k = f (k′) ∈ E.
• If p ≥ 2 and h is not a multiple of 3 but h ≡ (-1)^p mod 4, we can still show that k ∈ E. (Cf.)

The problematic case is that where p ≥ 2 , h not multiple of 3 and h ≡ (-1)^p+1 mod 4. Here, if we manage to show that for every odd integer k′, 1 ≤ k′ ≤ k-2 ; 3k′ ∈ E we are done. (Cf.).

See also

• Residue class-wise affine groups
• Modular arithmetic

In mathematics, residue class-wise affine groups are certain permutation groups acting on , whose elements are bijective residue class-wise affine mappings. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

References and external links

• Jeffrey C. Lagarias. The 3x + 1 problem: An annotated bibliography (1963--2000).
• Jeffrey C. Lagarias. The 3x + 1 problem: An annotated bibliography, II (2001--).
• Jeffrey C. Lagarias. The 3x + 1 problem and its generalizations, American Mathematical Monthly Volume 92, 1985, pp. 3 - 23.
• Günther J. Wirsching. The Dynamical System Generated by the 3n + 1 Function. Number 1681 in Lecture Notes in Mathematics. Springer-Verlag, 1998.
• An ongoing distributed computing project by Eric Roosendaal verifies the Collatz conjecture for larger and larger values.
• Another ongoing distributed computing project by Tomás Oliveira e Silva continues to verify the Collatz conjecture (with fewer statistics than Eric Roosendaal's page but with further progress made).
• Eric W. Weisstein, Collatz Problem at MathWorld.
• Collatz Problem at PlanetMath.
• J.C. Lagarias: Syracuse problem at Springer Online Encyclopaedia of Mathematics.
• Hailstone Patterns discusses different resonators along with using important numbers in the problem (like 6 and 3^5) to discover patterns.
• Review of progress, plus various programs
• Ohira, Reiko & Yamashita, Michinori A generalization of the Collatz problem
• URATA, Toshio Some Holomorphic Functions connected with the Collatz Problem