Unsolved problem in mathematics:

Does the Collatz sequence eventually reach 1 for all positive integer initial values?

- Statement of the problem
- Empirical data
- Visualizations
- Supporting arguments
- Experimental evidence
- A probabilistic heuristic
- Stopping times
- Lower bounds
- Cycles
- Cycle length
- k-cycles
- Other formulations of the conjecture
- In reverse
- As an abstract machine that computes in base two
- As a parity sequence
- As a tag system
- Extensions to larger domains
- Iterating on all integers
- Iterating on rationals with odd denominators
- 2-adic extension
- Iterating on real or complex numbers
- Optimizations
- Time–space tradeoff
- Modular restrictions
- Syracuse function
- Undecidable generalizations
- In popular culture
- See also
- Further reading
- References
- External links

The **Collatz conjecture** is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. If the previous 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.

It is named after mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate.^{ [1] } It is also known as the **3 n + 1 problem**, the

Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems."^{ [8] } He also offered US$500 for its solution.^{ [9] } Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics".^{ [10] }

Consider the following operation on an arbitrary positive integer:

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

In modular arithmetic notation, define the function f as follows:

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:

(that is: *a _{i}* is the value of f applied to n recursively i times;

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

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 i such that *a _{i}* <

The Collatz conjecture asserts that the total stopping time of every n is finite. It is also equivalent to saying that every *n* ≥ 2 has a finite stopping time.

Since 3*n* + 1 is even whenever n is odd, one may instead use the "shortcut" form of the Collatz function:

This definition yields smaller values for the stopping time and total stopping time without changing the overall dynamics of the process.

For instance, starting with *n* = 12 and applying the function *f* without "shortcut", one gets the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1.

The number *n* = 19 takes longer to reach 1: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

The sequence for *n* = 27, listed and graphed below, takes 111 steps (41 steps through odd numbers, in bold), climbing as high as 9232 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**(sequence A008884 in the OEIS )

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, ... (sequence A006877 in the OEIS ).

The starting values whose maximum 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, ... (sequence A006884 in the OEIS )

Number of steps for n 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, ... (sequence A006577 in the OEIS )

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 10
^{4}is 6171, which has 261 steps, - less than 10
^{5}is 77031, which has 350 steps, - less than 10
^{6}is 837799, which has 524 steps, - less than 10
^{7}is 8400511, which has 685 steps, - less than 10
^{8}is 63728127, which has 949 steps, - less than 10
^{9}is 670617279, which has 986 steps, - less than 10
^{10}is 9780657630, which has 1132 steps,^{ [11] } - less than 10
^{11}is 75128138247, which has 1228 steps, - less than 10
^{12}is 989345275647, which has 1348 steps.^{ [12] }

These numbers are the lowest ones with the indicated step count, but not necessarily the only ones below the given limit. As an example, 9780657631 has 1132 steps, as does 9780657630.

The starting values having the smallest total stopping time with respect to their number of digits (in base 2) are the powers of two since 2^{n} is halved n times to reach 1, and is never increased.

- Directed graph showing the orbits of the first 1000 numbers.
- The x axis represents starting number, the y axis represents the highest number reached during the chain to 1. This plot shows a restricted y axis: some x values produce intermediates as high as 2.7×10
^{7}(for*x*= 9663) - The tree of all the numbers having fewer than 20 steps.
- The number of iterations it takes to get to one for the first 100 million numbers.

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.

As of 2020^{ [update] }, the conjecture has been checked by computer for all starting values up to 2^{68} ≈ 2.95×10^{20}. All initial values tested so far eventually end in the repeating cycle (4; 2; 1) of period 3.^{ [13] }

This computer evidence is not sufficient to prove that the conjecture is true for all starting values. As in the case of some disproved conjectures, like the Pólya conjecture, counterexamples might be found when considering very large numbers.

However, such verifications may have other implications. For example, one can derive additional constraints on the period and structural form of a non-trivial cycle.^{ [14] }^{ [15] }^{ [16] }

If one considers only the *odd* numbers in the sequence generated by the Collatz process, then each odd number is on average 3/4 of the previous one.^{ [17] } (More precisely, the geometric mean of the ratios of outcomes is 3/4.) 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 2-adic starting values.)

As proven by Riho Terras, almost every positive integer has a finite stopping time.^{ [18] } 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 2019, Terence Tao improved this result by showing, using logarithmic density, that almost all 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 * wrote that Tao "came away with one of the most significant results on the Collatz conjecture in decades".^{ [19] }^{ [20] }

In a computer-aided proof, Krasikov and Lagarias showed that the number of integers in the interval [1,*x*] that eventually reach 1 is at least equal to *x*^{0.84} for all sufficiently large x.^{ [21] }

In this part, consider the shortcut form of the Collatz function

A cycle is a sequence (*a*_{0}, *a*_{1}, ..., *a _{q}*) of distinct positive integers where

The only known cycle is (1,2) of period 2, called the trivial cycle.

The length of a non-trivial cycle is known to be at least 17087915. In fact, Eliahou (1993) proved that the period p of any non-trivial cycle is of the form

where a, b and c are non-negative integers, *b* ≥ 1 and *ac* = 0. This result is based on the continued fraction expansion of ln 3/ln 2.^{ [15] }

A k-cycle is a cycle that can be partitioned into 2*k* contiguous subsequences: k increasing sequences of odd numbers alternating with k decreasing sequences of even numbers.^{ [16] } 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 (1; 2).^{ [22] } Simons (2005) used Steiner's method to prove that there is no 2-cycle.^{ [23] } Simons & de Weger (2005) extended this proof up to 68-cycles: there is no k-cycle up to *k* = 68.^{ [16] } For each *k* beyond 68, this method gives an upper bound for the smallest term of a k-cycle: for example, if there is a 77-cycle, then at least one element of the cycle is less than 38137×2^{50}.^{ [16] } Along with the verification of the conjecture up to 2^{68}, this implies the nonexistence of a non-trivial k-cycle up to *k* = 77.^{ [13] } As exhaustive computer searches continue, larger *k* values may be ruled out. To state the argument more intuitively: we need not look for cycles that have at most 77 circuits, where each circuit consists of consecutive ups followed by consecutive downs.

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 defined by the inverse relation

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 n, *n* ≡ 1 (mod 2) if and only if 3*n* + 1 ≡ 4 (mod 6). Equivalently, *n* − 1/3 ≡ 1 (mod 2) if and only if *n* ≡ 4 (mod 6). Conjecturally, this inverse relation forms a tree except for the 1–2–4 loop (the inverse of the 4–2–1 loop of the unaltered function f defined in the Statement of the problem section of this article).

When the relation 3*n* + 1 of the function f is replaced by the common substitute "shortcut" relation 3*n* + 1/2, the Collatz graph is defined by the inverse relation,

For any integer n, *n* ≡ 1 (mod 2) if and only if 3*n* + 1/2 ≡ 2 (mod 3). Equivalently, 2*n* − 1/3 ≡ 1 (mod 2) if and only if *n* ≡ 2 (mod 3). 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 3*n* + 1 with *n*′/*H*(*n*′) where *n*′ = 3*n* + 1 and *H*(*n*′) is the highest power of 2 that divides *n*′ (with no remainder). The resulting function f maps from odd numbers to odd numbers. Now suppose that for some odd number n, applying this operation k times yields the number 1 (that is, *f*^{k}(*n*) = 1). Then in binary, the number n can be written as the concatenation of strings *w*_{k}*w*_{k−1} ... *w*_{1} where each *w*_{h} is a finite and contiguous extract from the representation of 1/3^{h}.^{ [24] } The representation of n therefore holds the repetends of 1/3^{h}, where each repetend is optionally rotated and then replicated up to a finite number of bits. It is only in binary that this occurs.^{ [25] } Conjecturally, every binary string s that ends with a '1' can be reached by a representation of this form (where we may add or delete leading '0's to s).

Repeated applications of the Collatz function can be represented as an abstract machine that handles strings of bits. The machine will perform the following three steps on any odd number until only one 1 remains:

- Append 1 to the (right) end of the number in binary (giving 2
*n*+ 1); - Add this to the original number by binary addition (giving 2
*n*+ 1 +*n*= 3*n*+ 1); - Remove all trailing 0s (that is, repeatedly divide by 2 until the result is odd).

The starting number 7 is written in base two as 111. The resulting Collatz sequence is:

11111110111~~0~~1011100011~~0~~1000111011~~00~~11011011~~000~~10111~~0000~~

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

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

If P(...) is the parity of a number, that is P(2*n*) = 0 and P(2*n* + 1) = 1, then we can define the Collatz parity sequence (or parity vector) for a number n as *p _{i}* = P(

Which operation is performed, 3*n* + 1/2 or *n*/2, depends on the parity. The parity sequence is the same as the sequence of operations.

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 Hailstone cycles, then their corresponding parity cycles must be different.^{ [3] }^{ [18] }

Applying the f function k times to the number *n* = 2^{k}*a* + *b* will give the result 3^{c}*a* + *d*, where d is the result of applying the f function k times to b, and c is how many increases were encountered during that sequence. For example, for 2^{5}*a* + 1 there are 3 increases as 1 iterates to 2, 1, 2, 1, and finally to 2 so the result is 3^{3}*a* + 2; for 2^{2}*a* + 1 there is only 1 increase as 1 rises to 2 and falls to 1 so the result is 3*a* + 1. When b is 2^{k} − 1 then there will be k rises and the result will be 2 × 3^{k}*a* − 1. The factor of 3 multiplying a is independent of the value of a; it depends only on the behavior of b. 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, 4*a* + 1 becomes 3*a* + 1 after two applications of f and 16*a* + 3 becomes 9*a* + 2 after 4 applications of f. Whether those smaller numbers continue to 1, however, depends on the value of a.

For the Collatz function in the form

Hailstone sequences can be computed by the extremely simple 2-tag system with production rules

*a*→*bc*,*b*→*a*,*c*→*aaa*.

In this system, the positive integer n is represented by a string of n copies of a, 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 a as the initial word, eventually halts (see * Tag system#Example: Computation of Collatz sequences * for a worked example).

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 f. These cycles are listed here, starting with the well-known cycle for positive n:

Odd values are listed in large bold. Each cycle is listed with its member of least absolute value (which is always odd) first.

Cycle | Odd-value cycle length | Full cycle length |
---|---|---|

1 → 4 → 2 → 1... | 1 | 3 |

−1 → −2 → −1... | 1 | 2 |

−5 → −14 → −7 → −20 → −10 → −5... | 2 | 5 |

−17 → −50 → −25 → −74 → −37 → −110 → −55 → −164 → −82 → −41 → −122 → −61 → −182 → −91 → −272 → −136 → −68 → −34 → −17... | 7 | 18 |

The generalized Collatz conjecture is the assertion that every integer, under iteration by f, eventually falls into one of the four cycles above or the cycle 0 → 0. (The 0 → 0 cycle is only included for the sake of completeness.)

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.^{ [26] } Conversely, it is conjectured that every rational with an odd denominator has an eventually cyclic parity sequence (Periodicity Conjecture^{ [3] }).

If a parity cycle has length n and includes odd numbers exactly m times at indices *k*_{0} < ⋯ < *k*_{m−1}, then the unique rational which generates immediately and periodically this parity cycle is

**(1)**

For example, the parity cycle (1 0 1 1 0 0 1) has length 7 and four odd terms at indices 0, 2, 3, and 6. It is repeatedly generated by the fraction

as the latter leads to the rational cycle

Any cyclic permutation of (1 0 1 1 0 0 1) is associated to one of the above fractions. For instance, the cycle (0 1 1 0 0 1 1) is produced by the fraction

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 (1 1 0 0 1 1 0 0) and its sub-cycle (1 1 0 0) are associated to the same fraction 5/7 when reduced to lowest terms.

In this context, assuming the validity of the Collatz conjecture implies that (1 0) and (0 1) are the only parity cycles generated by positive whole numbers (1 and 2, respectively).

If the odd denominator d 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 "3*n* + *d* " generalization^{ [27] } of the Collatz function

The function

is well-defined on the ring 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.^{ [3] }

Define the *parity vector* function Q acting on as

The function Q is a 2-adic isometry.^{ [28] } Consequently, every infinite parity sequence occurs for exactly one 2-adic integer, so that almost all trajectories are acyclic in .

An equivalent formulation of the Collatz conjecture is that

The Collatz map (with shortcut) can be viewed as the restriction to the integers of the smooth map

The iterations of this map on the real line lead to a dynamical system, further investigated by Chamberland.^{ [29] } He showed that the conjecture does not hold for positive real numbers since there are infinitely many fixed points, as well as orbits 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, where most of the points have orbits that diverge to infinity (colored region on the illustration).^{ [30] } The boundary between the colored region and the black components, namely the Julia set of f, is a fractal pattern, sometimes called the "Collatz fractal".

The section * As a parity sequence * above gives a way to speed up simulation of the sequence. To jump ahead k steps on each iteration (using the f function from that section), break up the current number into two parts, b (the k least significant bits, interpreted as an integer), and a (the rest of the bits as an integer). The result of jumping ahead k steps can be found as:

*f*^{k}(2^{k}*a*+*b*) = 3^{c(b)}*a*+*d*(*b*).

The c (or better 3^{c}) and d arrays are precalculated for all possible k-bit numbers b, where *d*(*b*) is the result of applying the f function k times to b, and *c*(*b*) is the number of odd numbers encountered on the way.^{ [31] } For example, if *k* = 5, one can jump ahead 5 steps on each iteration by separating out the 5 least significant bits of a number and using:

- c(0...31) = { 0, 3, 2, 2, 2, 2, 2, 4, 1, 4, 1, 3, 2, 2, 3, 4, 1, 2, 3, 3, 1, 1, 3, 3, 2, 3, 2, 4, 3, 3, 4, 5 }
- d(0...31) = { 0, 2, 1, 1, 2, 2, 2, 20, 1, 26, 1, 10, 4, 4, 13, 40, 2, 5, 17, 17, 2, 2, 20, 20, 8, 22, 8, 71, 26, 26, 80, 242 }.

This requires 2^{k} precomputation and storage to speed up the resulting calculation by a factor of k, a space–time tradeoff.

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 n. If, for some given b and k, the inequality

*f*^{k}(2^{k}*a*+*b*) = 3^{c(b)}*a*+*d*(*b*) < 2^{k}*a*+*b*

holds for all a, then the first counterexample, if it exists, cannot be b modulo 2^{k}.^{ [14] } For instance, the first counterexample must be odd because *f*(2*n*) = *n*, smaller than 2*n*; and it must be 3 mod 4 because *f*^{2}(4*n* + 1) = 3*n* + 1, smaller than 4*n* + 1. For each starting value a which is not a counterexample to the Collatz conjecture, there is a k 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 k increases, the search only needs to check those residues b that are not eliminated by lower values of k. Only an exponentially small fraction of the residues survive.^{ [32] } For example, the only surviving residues mod 32 are 7, 15, 27, and 31.

If k is an odd integer, then 3*k* + 1 is even, so 3*k* + 1 = 2^{a}*k*′ with *k*′ odd and *a* ≥ 1. The **Syracuse function** is the function f from the set I of odd integers into itself, for which *f*(*k*) = *k*′(sequence A075677 in the OEIS ).

Some properties of the Syracuse function are:

- For all
*k*∈*I*,*f*(4*k*+ 1) =*f*(*k*). (Because 3(4*k*+ 1) + 1 = 12*k*+ 4 = 4(3*k*+ 1).) - In more generality: For all
*p*≥ 1 and odd h,*f*^{p − 1}(2^{p}*h*− 1) = 2 × 3^{p − 1}*h*− 1. (Here*f*^{p − 1}is function iteration notation.) - For all odd h,
*f*(2*h*− 1) ≤ 3*h*− 1/2

The Collatz conjecture is equivalent to the statement that, for all k in I, there exists an integer *n* ≥ 1 such that *f*^{n}(*k*) = 1.

In 1972, John Horton Conway proved that a natural generalization of the Collatz problem is algorithmically undecidable.^{ [33] }

Specifically, he considered functions of the form

and *a*_{0}, *b*_{0}, ..., *a*_{P − 1}, *b*_{P − 1} are rational numbers which are so chosen that *g*(*n*) is always an integer.

The standard Collatz function is given by *P* = 2, *a*_{0} = 1/2, *b*_{0} = 0, *a*_{1} = 3, *b*_{1} = 1. Conway proved that the problem:

- Given g and n, does the sequence of iterates
*g*(^{k}*n*) reach 1?

is undecidable, by representing the halting problem in this way.

Closer to the Collatz problem is the following *universally quantified* problem:

- Given g does the sequence of iterates
*g*(^{k}*n*) reach 1, for all*n*> 0?

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^{ [34] } proved that the above problem is, in fact, undecidable and even higher in the arithmetical hierarchy, specifically Π^{0}_{2}-complete. This hardness result holds even if one restricts the class of functions g by fixing the modulus P to 6480.^{ [35] }

In the movie * Incendies *, a graduate student in pure mathematics explains the Collatz conjecture to a group of undergraduates. She puts her studies on hold for a time to address some unresolved questions about her family's past. Late in the movie, the Collatz conjecture turns out to have foreshadowed a disturbing and difficult discovery that she makes about her family.^{ [36] }^{ [37] }

Wikimedia Commons has media related to Collatz conjecture .

*The Ultimate Challenge: The 3x + 1 Problem*,^{ [10] }published in 2010 by the American Mathematical Society and edited by Jeffrey Lagarias, is a compendium of information on the Collatz conjecture, methods of approaching it, and generalizations. It includes two survey papers by the editor and five by other authors concerning the history of the problem, generalizations, statistical approaches, and results from the theory of computation. It also includes reprints of early papers on the subject, including the paper by Lothar Collatz.

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In additive number theory, **Fermat's theorem on sums of two squares** states that an odd prime *p* can be expressed as:

**Gauss's lemma** in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity.

**Quartic** or **biquadratic reciprocity** is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence *x*^{4} ≡ *p* is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence *x*^{4} ≡ *p* to that of *x*^{4} ≡ *q*.

In algebra, the **3 x + 1 semigroup** is a special subsemigroup of the multiplicative semigroup of all positive rational numbers. The elements of a generating set of this semigroup are related to the sequence of numbers involved in the still open Collatz conjecture or the "3

In mathematics, specifically in number theory, **Newman's conjecture** is a conjecture about the behavior of the partition function modulo any integer. Specifically, it states that for any integers m and r such that , the value of the partition function satisfies the congruence for infinitely many non-negative integers n. It was formulated by mathematician Morris Newman in 1960. It is unsolved as of 2020.

- ↑ O'Connor, J.J.; Robertson, E.F. (2006). "Lothar Collatz". St Andrews University School of Mathematics and Statistics, Scotland.
- ↑ Maddux, Cleborne D.; Johnson, D. Lamont (1997).
*Logo: A Retrospective*. New York: Haworth Press. p. 160. ISBN 0-7890-0374-0.The problem is also known by several other names, including: Ulam's conjecture, the Hailstone problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and the Collatz problem.

- 1 2 3 4 5 6 7 Lagarias, Jeffrey C. (1985). "The 3
*x*+ 1 problem and its generalizations".*The American Mathematical Monthly*.**92**(1): 3–23. doi:10.1080/00029890.1985.11971528. JSTOR 2322189. - ↑ According to Lagarias (1985),
^{ [3] }p. 4, the name "Syracuse problem" was proposed by Hasse in the 1950s, during a visit to Syracuse University. - ↑ Pickover, Clifford A. (2001).
*Wonders of Numbers*. Oxford: Oxford University Press. pp. 116–118. ISBN 0-19-513342-0. - ↑ "Hailstone Number".
*MathWorld*. Wolfram Research. - ↑ Hofstadter, Douglas R. (1979).
*Gödel, Escher, Bach*. New York: Basic Books. pp. 400–2. ISBN 0-465-02685-0. - ↑ Guy, Richard K. (2004). ""E16: The 3x+1 problem"".
*Unsolved Problems in Number Theory*(3rd ed.). Springer-Verlag. pp. 330–6. ISBN 0-387-20860-7. Zbl 1058.11001. - ↑ Guy, R. K. (1983). "Don't try to solve these problems".
*Amer. Math. Monthly*.**90**(1): 35–41. doi:10.2307/2975688. JSTOR 2975688. By this Erdos means that there aren't powerful tools for manipulating such objects. - 1 2 Lagarias, Jeffrey C., ed. (2010).
*The Ultimate Challenge: The 3*x*+ 1 Problem*. American Mathematical Society. ISBN 978-0-8218-4940-8. Zbl 1253.11003. - ↑ Leavens, Gary T.; Vermeulen, Mike (December 1992). "3
*x*+ 1 search programs".*Computers & Mathematics with Applications*.**24**(11): 79–99. doi: 10.1016/0898-1221(92)90034-F . - ↑ Roosendaal, Eric. "3x+1 delay records" . Retrieved 14 March 2020. (Note: "Delay records" are total stopping time records.)
- 1 2 Barina, David (2020). "Convergence verification of the Collatz problem".
*The Journal of Supercomputing*.**77**(3): 2681–2688. doi:10.1007/s11227-020-03368-x. S2CID 220294340. - 1 2 Garner, Lynn E. (1981). "On the Collatz 3
*n*+ 1 algorithm".*Proceedings of the American Mathematical Society*.**82**(1): 19–22. doi: 10.1090/S0002-9939-1981-0603593-2 . JSTOR 2044308. - 1 2 Eliahou, Shalom (1993). "The 3
*x*+ 1 problem: new lower bounds on nontrivial cycle lengths".*Discrete Mathematics*.**118**(1): 45–56. doi: 10.1016/0012-365X(93)90052-U . - 1 2 3 4 Simons, J.; de Weger, B. (2005). "Theoretical and computational bounds for
*m*-cycles of the 3*n*+ 1 problem" (PDF).*Acta Arithmetica*.**117**(1): 51–70. Bibcode:2005AcAri.117...51S. doi: 10.4064/aa117-1-3 . - ↑ Lagarias (1985),
^{ [3] }section "A heuristic argument". - 1 2 Terras, Riho (1976). "A stopping time problem on the positive integers" (PDF).
*Acta Arithmetica*.**30**(3): 241–252. doi: 10.4064/aa-30-3-241-252 . MR 0568274. - ↑ Tao, Terence (10 September 2019). "Almost all Collatz orbits attain almost bounded values".
*What's new*. Retrieved 11 September 2019. - ↑ Hartnett, Kevin. "Mathematician Proves Huge Result on 'Dangerous' Problem".
*Quanta Magazine*. Retrieved 26 December 2019. - ↑ Krasikov, Ilia; Lagarias, Jeffrey C. (2003). "Bounds for the 3
*x*+ 1 problem using difference inequalities".*Acta Arithmetica*.**109**(3): 237–258. arXiv: math/0205002 . Bibcode:2003AcAri.109..237K. doi:10.4064/aa109-3-4. MR 1980260. S2CID 18467460. - ↑ Steiner, R. P. (1977). "A theorem on the syracuse problem".
*Proceedings of the 7th Manitoba Conference on Numerical Mathematics*. pp. 553–9. MR 0535032. - ↑ Simons, John L. (2005). "On the nonexistence of 2-cycles for the 3
*x*+ 1 problem".*Math. Comp*.**74**: 1565–72. Bibcode:2005MaCom..74.1565S. doi: 10.1090/s0025-5718-04-01728-4 . MR 2137019. - ↑ Colussi, Livio (9 September 2011). "The convergence classes of Collatz function".
*Theoretical Computer Science*.**412**(39): 5409–5419. doi: 10.1016/j.tcs.2011.05.056 . - ↑ Hew, Patrick Chisan (7 March 2016). "Working in binary protects the repetends of 1/3
^{h}: Comment on Colussi's 'The convergence classes of Collatz function'".*Theoretical Computer Science*.**618**: 135–141. doi: 10.1016/j.tcs.2015.12.033 . - ↑ Lagarias, Jeffrey (1990). "The set of rational cycles for the 3x+1 problem".
*Acta Arithmetica*.**56**(1): 33–53. doi: 10.4064/aa-56-1-33-53 . ISSN 0065-1036. - ↑ Belaga, Edward G.; Mignotte, Maurice (1998). "Embedding the 3x+1 Conjecture in a 3x+d Context".
*Experimental Mathematics*.**7**(2): 145–151. doi:10.1080/10586458.1998.10504364. - ↑ Bernstein, Daniel J.; Lagarias, Jeffrey C. (1996). "The 3
*x*+ 1 conjugacy map".*Canadian Journal of Mathematics*.**48**(6): 1154–1169. doi: 10.4153/CJM-1996-060-x . ISSN 0008-414X. - ↑ Chamberland, Marc (1996). "A continuous extension of the 3
*x*+ 1 problem to the real line".*Dynam. Contin. Discrete Impuls Systems*.**2**(4): 495–509. - ↑ Letherman, Simon; Schleicher, Dierk; Wood, Reg (1999). "The (3
*n*+ 1)-problem and holomorphic dynamics".*Experimental Mathematics*.**8**(3): 241–252. doi:10.1080/10586458.1999.10504402. - ↑ Scollo, Giuseppe (2007). "Looking for class records in the 3
*x*+ 1 problem by means of the COMETA grid infrastructure" (PDF).*Grid Open Days at the University of Palermo*. - ↑ Lagarias (1985),
^{ [3] }Theorem D. - ↑ Conway, John H. (1972). "Unpredictable iterations".
*Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder*. pp. 49–52. - ↑ Kurtz, Stuart A.; Simon, Janos (2007). "The undecidability of the generalized Collatz problem". In Cai, J.-Y.; Cooper, S. B.; Zhu, H. (eds.).
*Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007*. pp. 542–553. doi:10.1007/978-3-540-72504-6_49. ISBN 978-3-540-72503-9. As PDF - ↑ Ben-Amram, Amir M. (2015). "Mortality of iterated piecewise affine functions over the integers: Decidability and complexity".
*Computability*.**1**(1): 19–56. doi:10.3233/COM-150032. - ↑ Emmer, Michele (2012).
*Imagine Math: Between Culture and Mathematics*. Springer Publishing. pp. 260–264. ISBN 978-8-847-02426-7. - ↑ Mazmanian, Adam (19 May 2011). "MOVIE REVIEW: 'Incendies'".
*The Washington Times*. Retrieved 7 December 2019.

- Matthews, Keith. "3 x + 1 page".
- An ongoing distributed computing project by David Bařina verifies Convergence of the Collatz conjecture for large values. (furthest progress so far)
- Distributed computing (BOINC) project that verifies the Collatz conjecture for larger values.
- 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).
- Weisstein, Eric W. "Collatz Problem".
*MathWorld*. - Collatz Problem at PlanetMath ..
- Nochella, Jesse. "Collatz Paths".
*Wolfram Demonstrations Project*. - Eisenbud, D. (8 August 2016).
*Uncrackable? The Collatz conjecture*(short video). Numberphile. Archived from the original on 2021-12-11 – via YouTube. - Eisenbud, D. (August 9, 2016).
*Uncrackable? Collatz conjecture*(extra footage). Numberphile. Archived from the original on 2021-12-11 – via YouTube. - Alex Kontorovich (featuring) (30 July 2021).
*The simplest math problem no one can solve*(short video). Veritasium – via YouTube. - Are computers ready to solve this notoriously unwieldy math problem?

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