This page contains the following sections:  •  Definition of the problem  •  The Glide  •  Delay and Residue  •  Completeness and Gamma  •  Class Records  •  Strength and Levels  •  Path Records  •  Links to related pages  •  Current status of the problem  •  Join the distributed search for new class records!  •  Watch the progress of the distributed search project  •  Find pages quickly on the Site Map

Convergence Between 1999 and 2011 all numbers up to 260 (± 1153E+15) were checked for convergence. During this process all Path records were recorded as well. A complete list of Path records is on the Path records page

For any positive integer N a sequence S_i can be defined by putting This latter formula usually gives the sequence its name, the 3x + 1 problem, sometimes also referred to as the Collatz problem, the Syracuse problem or some other name. Thus for N = 13 we find and from now on the sequence will repeat itself. We now define

Mx(N)

Mx(N) = lim k → ∞ Max(S₀, S₁, ..., S_k)

Mn(N)

Mn(N) = lim k → ∞ Min(S₀, S₁, ..., S_k)

Thus we will call any positive integer

Convergent

if Mn(N) = 1

Divergent

if Mx(N) does not exist

Cyclic

otherwise

No positive integer N is known which may possibly be divergent, and we therefore reach the widely believed yet still unproven:

Conjecture 1a : (weak 3x+1 conjecture)

No integer N is divergent.

Serious indications exist however that no other cycle but the simple 4-2-1 cycle exists. In fact it is already known that any other cycle must contain at least a very large number of elements. See the cycles page to find out the exact numbers and how they follow from the current status of the problem.

It is therefore not unreasonable to state

Conjecture 1b : (strong 3x+1 conjecture)

All positive integers N are convergent.

This conjecture remains unproven to this day.

While most authors on the subject have concerned themselves mainly with proving conjecture 1b, on these pages we will from here on assume that conjecture 1b holds and look into a number of other properties of the 3x+1 sequences.

Glide

for any positive integer N > 1 let k be the lowest index for which S_k < N.
We shall call k the Glide of N and write this as G(N).

Calculating the Glide for any positive integer is tantamount to showing it is convergent, as long as all smaller numbers are known to be convergent as well.
Thus we may restate conjecture 1b as follows:

Conjecture 1b : (strong 3x+1 conjecture, restated)

All positive integers N > 1 have a finite Glide

This conjecture is obviously unproven as well. However Terras proved a very important though weaker theorem:

Theorem (Terras) :

Almost all positive integers N have finite Glide.

For an informal proof of this theorem look at the Terras theorem page

It may not be immediately obvious from the definition of the sequence that arbitrarily high Glides must exist. But observe that any number of the form N = 2^p - 1 with p > 1 will have S₂ = 3 * 2^p-1 - 1 and, since S₂ is odd : S₄ = 3 * 3 * 2^p-2 - 1 and eventually S_2p = 3^p - 1.
Hence S_i will not be below N for at least 2p steps, and therefore G(N) > 2p.

Although Glides are thus unbounded it is a far different proposition to find the smallest number for which G(N) > k for any given k.

Glide Record

a positive integer N is called a Glide Record
if for all M < N we have G(M) < G(N)

Skipping the trivial numbers 1 and 2 the first Glide Records are and so on. The author calculated Glides over a substantial interval, and the Glide Records found so far are given in the Glide Records Table.

Delay

for any positive integer N let k be the lowest index for which S_k = 1.
We shall call k the Delay of N and write this as D(N).

Delay Record

A positive integer N is called a Delay Record
if for all M < N we have D(M) < D(N).

Lengthy calculations have been made in trying to find Delay Records.
All currently known Delay Records are given in the Delay Records Table.
Note: to calculate the delay of any number yourself try the 3x + 1 calculator page.

Residue

for any positive integer N let E(N) and O(N) denote the number of elements from S₀ to S_D(N)-1 which are even or odd, respectively. Obviously O(N) + E(N) = D(N).
Now due to the construction of the sequence we may write

2^E(N) = 3^O(N) * N * Res(N)

where Res(N) is a factor equal to the product of
(1 + 1 / ( 3 * S_i ) ) taken over the odd elements S_i for 0 ≤ i < D(N).
We will call Res(N) the Residue of N.

It turns out that Res(N) is usually small and typically lies between 1.10 and 1.25. Therefore it appears reasonable to propose

Conjecture 2a : (weak residue conjecture)

A number Res_max exists such that for every positive integer

Res(N) < Res_max.

There are serious indications though that a much more explicit conjecture can be made:

Conjecture 2b : (strong residue conjecture)

For all N : Res(N) ≤ Res(993).

For the evidence to support this conjecture look at the residues page.
Whether conjecture 2b is true or not, we will from now on assume that at least conjecture 2a holds, and also that Res_max << 6.

Completeness

for all integers N > 1 the Completeness of N, C(N), is defined by C(N) = O(N) / E(N).

Theorem 1 :

For all N > 1: C(N) < ln(2)/ln(3) = 0.63092975...

Proof :

From the previous paragraph we had, by definition
   2^E(N) = 3^O(N) * N * Res(N)
Taking the logarithm and regrouping yields
   O(N) / E(N) = ln(2) / ln(3) - (ln(N) + ln(Res(N))) / (E(N).ln(3) )
The final term is often small, but always > 0.

This leads to the rather more complex question of how close C(N) can get to its theoretical limit. If the final term could get infinitesimally close to zero then the completeness would get infinitesimally close to this limit. For large N the final term is proportional to ln(N) / E(N). The reciprocal of this quantity is usually defined as Gamma.

Gamma

for all integers N > 1 Gamma(N) is defined by E(N) / ln(N).

There is ample evidence to suggest that Gamma(N) will not reach arbitrary large values, and so we reach

Conjecture 3 :

A number C_max exists such that for all N > 1 : C(N) < C_max.
with C_max < ln(2) / ln(3).

Or, in simpler terms: C(N) will not get infinitesimally close to its theoretical limit. For the evidence to support this conjecture look at the completeness page.

Completeness Record

A number N > 1 is called a Completeness Record
if for all M < N we have C(M) < C(N).

Similarly we have

Gamma Record

A number N > 1 is called a Gamma Record
if for all M < N we have Gamma(M) < Gamma(N).

From the similarity of their definitions it should come as no surprise that the tables of known Completeness and Gamma records are virtually the same.

All Completeness and Gamma Records currently known are given in the Completeness and Gamma Records Table.

Assuming Conjecture 1b holds we can partition the positive integers into Delay Classes DC_k (k=0,1,2,3...) where an integer N is said to belong to class D(N). We note that no class is empty as for N = 2^k we have D(N) = k.

Class Record

The lowest element of a Delay Class DC_k is called its Class Record, indicated by R_k.

Class Records of consecutive classes are often 'related', the lower one occurring in the sequence of the higher one. Assuming Conjecture 2a holds, with Res_max < 6 we have:

Theorem 2 :

Let R_k and R_k+1 be the records of Delay Classes k and k + 1, respectively.
Then O(R_k) <= O(R_k+1) and E(R_k) <= E(R_k+1).

Proof :

Let R_k and R_k+1 be the records of classes k and k+1 respectively.
Assume O(R_k) - O(R_k+1) = x, x > 0. Then R_k+1 / R_k is equal to 2^x+1 * 3^x * Res(R_k) / Res(R_k+1) which for positive x is definitely >> 2.
Since N = 2 * R_k has a delay of k + 1 and is smaller than R_k+1, this would imply that R_k+1 would be no class record.
Therefore O(R_k) <= O(R_k+1).

Assume that E(R_k) - E(R_k+1) = x, x > 0. Then R_k / R_k+1 is equal to 2^x * 3^x+1 * Res(R_k+1) / Res(R_k) which for positive x is definitely >> 3.
Since either N = 3 * R_k+1 + 1 or N = R_k+1 / 2 has delay k, and both are smaller than R_k this would imply that R_k would be no class record.
Therefore E(R_k) <= E(R_k+1).

From the above result we may easily derive

Theorem 3 :

Let N be any Completeness Record.
Then N is a Class Record as well.

Proof :

Assume a number M < N exists with D(M) = D(N).

 •  If O(M) = O(N), and thus E(M) = E(N), then C(M) = C(N) and thus N would be no Completeness Record.
 •  If O(M) > O(N) then C(M) > C(N) and again N would be no Completeness Record.
 •  Finally if O(M) < O(N) then, still assuming conjecture 2a holds, M would not be < N. Therefore such an M does not exist and N is therefore a Class Record.

Theoretically a Completeness Record does not necessarily have to be a Delay Record as well, since a lower number with a higher Delay but lower Completeness might exist. All currently known Completeness Records are Delay Records as well though.

All Class Records currently known are given in the Class Records Table.

If Conjecture 2a holds, with Rmax << 6, this also implies that the elements of a Delay Class occur at discrete levels, which roughly lie a factor 6 apart. The elements of class 13 for example are 34, 35, 192, 208, 212, 213, 226, 227, 1280, 1344, 1360, 1364, 1365 and 8192. The grouping into four different "subclasses" is obvious.
In order to work with levels we first define a more basic parameter called the Strength:

Strength

For all positive integers N with D(N) = k the Strength S(N) is defined by
S(N) = 5 * O(N) - 3 * E(N).

The Strength of most numbers is well below zero, and positive Strengths are quite rare. It would appear though that arbitrarily high Strengths should exist.

Strength Record

A positive integer N is called a Strength Record
if for all M < N we have S(M) < S(N).

Strength records are extremely rare. Only five non-trivial records are known, and the highest of those has 15 digits already. They can be found on the Strength page together with a few numbers that are currently the best known candidates for the next records.
Based on the Strength it is easy to define the Level. Note that for all numbers N with D(N) = k we have necessarily S(N) ≡ 5k (mod 8)

Level

For all positive integers N with D(N) = k the Level L(N) is defined by
L(N) = - [S(N) / 8 ]
(where [x] is meant to be the largest integer ≤ x)

Thus S(34) = 5 * 3 - 3 * 10 = -15. Therefore L(34) = -[-15/8] = 2. Similarly L(192) = 3, L(1280) = 4 and L(8192) = 5. For any delay k the highest possible level occurs at N = 2^k, as this is the highest number with this delay. Level 0 can be seen as the highest level for which C(N) >= 0.60.

With lower numbers the 'limited availability' of numbers makes statistics a risky approach, but with higher numbers we may say that the level yields information about the relative 'exclusiveness' of the record. From a statistical point of view with levels 'close to 0' we find that the lower the level, the less numbers are found at that level. Although numbers with negative levels do exist they are quite rare. The only number below 10,000,000,000 for which L(N) < 0 occurs at N = 63,728,127, for which L(N) = -1.

The next ones are L(12,235,060,455) = -1 (D(N) = 1184) and L(13,371,194,527) = -1 (D(N) = 1210). Up to 21,000,000,000 three more level -1 numbers exist, but then surprisingly the next one does not appear until R₁₄₀₈ = 1,444,338,092,271, a number almost 70 times larger than its predecessor. Only 9 numbers of level -1 exist before the first number of level -2 occurs: R₁₅₄₉ = 3,743,559,068,799. This number shows a surprisingly high delay, surpassing the previous Delay Record by no less than over one hundred.

Up to 100,000,000,000,000 (10¹⁴) one finds approximately 100 numbers of level -1 and just a single one of level -2, which gives a fair indication of their rarity. Thus it is rather unexpected the first number of level -3 is found just outside this interval at N = 100,759,293,214,567. With a Delay of 1820 this number is not only a Delay Record, surpassing the previous record with no fewer than 158 steps, but a Strength and Completeness Record as well. Remarkably a further level -3 number pops up almost immediately as R₁₇₈₉ is found at N = 104,797,092,792,063. A total of 15 level -3 numbers exist between 100 * 10¹² and 531 * 10¹² but no others appear quickly thereafter. In fact no further level -3 numbers exist below 10¹⁶. The next one is N = 12769,884180,266527, over 20 times larger than the previous one.

Still lower levels are ever harder to come by. As the numbers increase searches become slower and search intervals get wider. The lowest level -4 number, now confirmed as a Delay and Strength record, has 18 digits. The smallest number of level -5 found so far has 21 digits and the lowest number currently known of level -10 has no less than 38 digits. Reactions from anybody who might have found stronger candidates are very welcome!

Any positive integer N can be completely defined by the delay, the level and the residue, i.e. from these three data N can be calculated exactly. Neither of these however give any information about the path of the sequence S_i. In particular no information is available of Mx(N), the highest number occurring in the 3x+1-sequence. This number is of practical importance as well, since it indicates the minimum number of bits needed for a computer algorithm to calculate a particular sequence without overflows occurring.
Note: to calculate the maximum of any number yourself try the 3x + 1 calculator page.

Path Record

a positive integer N is called a Path Record
if for all M < N we have Mx(M) < Mx(N)

The first Path Records are and so on. Path records can be conveniently numbered in the order in which they occur. Thus P₁ = 2, P₂ = 3, etc.
A curious observation about the values of Mx(N) is

Theorem 4 :

Let N be any odd number > 2. If Mx(N) > 3N + 1 then Mx(N) ≡ 16 (mod 36).

Proof :

Let x be Mx(N), where x ≡ a (mod 36). We note that a can not be odd, since x must be even. We also note that we can not have a ≡ 2 (mod 4) or else x/2 would be odd which would yield a higher number than x. Therefore a ≡ 0 (mod 4).
Since x must be produced by a 3n+1 iteration we have a ≡ 1 (mod 3). This leaves just 4, 16 and 28 as possible values for a.
Assume a = 4. Then let x = 36n+4. The predecessor of x must have been 12n+1. Then its predecessor was 24n+2. Since this number is not ≡ 4 (mod 6) it must have had 48n+4 as its predecessor. But this number is greater than x, which contradicts the assumption that x is Mx(N).
Similarly for a = 28 we find predecessors 12n+9, 24n+18 and 48n+36, which again contradicts the assumption that x is Mx(N).
Therefore we can only have a = 16, and therefore Mx(N) ≡ 16 (mod 36) (and its predecessors in the sequence are 12n+5, 24n+10 and 8n+3).

For all Path Records P_i ≥ 3 we indeed have Mx(P_i ) > 3.P_i + 1, therefore for i ≥ 2 all Mx(P_i ) ≡ 16 (mod 36).

A list of all currently known Path Records can be found in the Path Records Table.

Expansion

The Expansion with respect to Q, X_Q of N is defined as
X_Q(N) = Mx(N) / N^Q

It is easily seen that X₁(N) is unbounded, since if N = 2^p - 1 then, as we saw earlier, S_2p = 3^p - 1 and Mx(N) / N is at least approximately equal to 3^p / 2^p.

A more interesting question arises when we ask for which p X_p might stay bounded. The following result is well known:

Theorem 5 :

X_Q is unbounded for Q < ln(3) / ln(2)

Proof :

We saw earlier that for N = 2^p - 1 Mx(N) will be at least 3^p - 1.
Thus for p towards infinity ln(X_Q( N )) = ln( Mx(N) ) - Q * ln( N ) >= p * ( ln(3) - Q * ln(2) )
So for Q < ln(3) / ln(2) the limit does not exist.

When ²log( Mx(P_i )) is depicted graphically against ²log( P_i ) the curve shows a tendency to become linear with an approximate slope of 2. It is not currently known whether or not this tendency is likely to persist. If it does it implies that X_Q(N) is bounded for Q > 2.

Open Question 1 :

Does a number C exist for which X₂(N) < C for all N ?
And if it does, what is the value of C ?

For many years the highest X₂(N) known was that of P₆₂ (± 15.054) and it looked unlikely a higher one would be found in the near future. In 2005 however Tomás Oliveira e Silva found P₈₈ which has an expansion of ± 16.315. As can be seen from the Path Records Table record expansions are very rare. The function X₂(N) reaches a temporary maximum of 12.66 at N = 27, which holds until P₄₃ = 319,804,831 is reached, a number over 10,000,000 times larger. Again, the value of 13.83 reached by P₄₃ is not surpassed until P₆₂, which scores 15.05. P₆₂ is over 10,000 times larger than P₄₃. Finally the highest expansion currently known is that of P₈₈ which reaches 16.32. P₈₈ is well over 500,000 times larger than its predecessor.

If successive gaps between ever higher values of the function X₂(N) continue to be of this magnitude it looks unlikely we will ever know more than a handful of Expansion Records.

 •   In August 1999 the first conference on the 3x+1 problem took place in Eichstätt.
 •   Here's a nice picture of the Eichstätt conference participants.
 •   Lagarias overview of the 3x+1 problem.
 •   Super-Index of the 3x+1 Problem and its generalizations.
 •   Gary T. Leavens and Mike Vermeulen : 3x+1 search programs
     Computers Math. Applic. Vol. 24, No 11, pp. 79-99 (1992)
 •   Tomás Oliveira e Silva was the first to find all Path and Glide Records up to 100.2⁵⁰.
     He has extended his search well beyond 2⁶⁰.
 •   MathWorld page on the collatz Problem
 •   The On-Line Encyclopedia of Integer Sequences contains many sequences related to the 3x+1 problem;
     for instance A6370, A6884, A600412, A6577, A6877 and A6878.
 •   This French page allows you to do calculations yourself.
 •   Eric Farin has developed a number of very interesting models.
 •   Ken Conrow's 3x+1 problem structure page.
 •   This excellent online 3x+1 calculator by Alfred Wassermann can handle really big numbers as well.
 •   Marc Lapierre's page to calculate the delay and paths of arbitrary big numbers.
 •   Darrell Cox is researching 3x+1 and 3x-1 loops.
 •   Peter Schorer is looking very hard for a possible proof of the 3x+1 Conjecture. Take a look at his papers here.
 •   Thanks to some hard working Chinese translators you can also view this page in Chinese.

All numbers up to 260 ( ± 1153 * 1015 ) have been double checked for convergence.
All numbers up to 166,200 * 1012 have been checked for class records. ( View progress )
Number of Glide Records	32
Highest known Glide Record	1575	at N =	180352,746940,718527
Highest known Residue	1.253142	at N =	993
Number of Class Records	2072
Number of Delay Records	131
Highest known Delay Record	2254	at N =	104899,295810,901231
Number of Completeness Records	16
Highest known Completeness Record	0.605413	at N =	104899,295810,901231
Number of Gamma Records	15
Highest known Gamma Record	35.823841	at N =	104899,295810,901231
Number of confirmed Path Records	87
Highest confirmed Path Record	± 319,391 E+30	at N =	1,038743,969413,717663
Highest known Expansion (for Q=2)	16.315	at N =	1,980976,057694,848447

Whenever (computer) time is available the quest for new records still continues!
But with numbers getting ever larger, more and more CPU time is needed to make any progress.
The program runs totally unattended, so anyone with lots of idle time on a PC can join the 3x+1 search! Just look at the 3x+1 search page to see how to join, or take a look at the Class record progress map.

Special thanks to Luigi Morelli from Italy who was first to join and to Kevin Hebb from Canada for the time and effort he took for many years to make two entire classrooms of PC's work day and night on the problem.