The Collatz Conjecture (also known as the 3n + 1 conjecture, the Ulam conjecture, Kakutani's problem, the Thwaites conjecture, Hasse's algorithm, the Syracuse problem and the Hailstone problem), concerns itself with the properties of the series of numbers which is generated when you start from any integer greater than zero, and repeat the following steps:-

• If the current number is even, divide by 2 to generate the next member.
• If the current number is odd, multiply by 3 and add 1 to generate the next member.
Examples.
• The Collatz series for 1:
`1  4  2  1`

• The Collatz series for 3:
`3  10  5 16  8  4  2  1`

• The Collatz series for 7:
`7  22  11  34  17  52  26  13  40  20  10  5  16  8  4  2  1`

• The Collatz Series for 27:
```  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```
Note that in all of these examples, the numbers in the series vary up and down for a time, but finally decay to a value of 1. Over time, researchers have tested all odd numbers up to 1020, as well as a great many much larger numbers and in every case studied to date, the series continues until it reaches 1. This explains the title Collatz Conjecture, the Conjecture being that ALL numbers will ultimately suffer this fate. Ever since 1937, a proof of this conjecture has been lacking. In the end, the contents of this article may not provide the long anticipated proof, but it will provide the most convincing yet simple to understand evidence you will ever see in support of its truth.

The Collatz Series for 1, 3, and 7 are quite easy to corelate with the operation of the Collatz process, but as the series becomes longer, and especially as the numbers within it become bigger, it becomes more difficult to gain a clear picture of what is happening. Imagine how it would look if we were dealing with numbers of 100 or more digits! This article introduces a more concise method of presenting the series. Each number is replaced by a letter O or E depending on whether the number is odd or even, and the resulting string of Os and Es is referred to as the Signature of the number. The Signature for 27 then has the following appearance:-

OEOEEOEOEOEOEOEEOEEOEOEEOEOEOEEOEOEOEOEEOEEEOEOEOEEOEOEE OEOEOEOEOEOEEEOEOEOEOEEEEOEEOEEOEEEEOEEEOEOEOEEEEEOEEEEO

A further change in the presentation is to insert a space immediately before each O in the series. This breaks the series up into fragments called Syllables. The number of Syllables in a Signature is a very important factor in the analysis of the Collatz process as you will see in subsequent sections of this article.

OE OEE OE OE OE OE OEE OEE OE OEE OE OE OEE OE OE OE OEE OEEE OE OE OEE OE OEE OE OE OE OE OE OEEE OE OE OE OEEEE OEE OEE OEEEE OEEE OE OE OEEEEE OEEEEO

It should be noted that all Signatures in this article begin with O, which implies that only odd numbers are of interest. This is because application of the Collatz process to an even number immediately reveals an underlying (and smaller) odd number.

Signature Categories
As a matter of interest only, Signatures fall into two categories, namely Final and Partial:-

• Final Signature.
If you apply the Collatz process repetitively to a number, adding the letters E or O as appropriate to the Signature string as you go until you encounter the digit 1, then the Signature string which results will be a Final Signature.

• Partial Signature.
A Partial Signature results when the Collatz process is interrupted before a digit 1 has been encountered. Alternatively, you can create a Partial Signature using a text editor by typing a string consisting only of Os and Es. The first and last letters must both be O, and there must not be any examples of consecutive Os. It is most unlikely that a Signature created in this way will be of the Final variety.
Important Notes.
• The Signature of a series is a string of letters which correspond to the numbers of the series, with an O for each odd number and an E for each even number.

• The members of most Collatz series go up and down in what appears to be a totally random fashion in much the same way as a Hailstone rises and falls in a storm cell until it grows to a such a size and weight that it has no alternative but to fall to the ground. Because of this, the numbers considered here are often referred to as Hailstone Numbers

• Most commentaries on the Collatz conjecture warn of the unpredictable and haphazard behavior of the number series it produces. This is undoubtedly a fact, but by the time you have finished studying this article, you will see that some of the behaviors are reassuringly regular and very predictable and that the Normal or Gaussian Distribution of statistics is a remarkably accurate predictor for some of this behavior.

• The first of the above series appears to be unique. It is the only one which returns to its starting value.

• The number 10 appears in both the second and third examples. This will happen to any number which is half of some even number, whilst also being one more than three times some odd number.

• Theoretically, there are two ways in which a number could fail to return to 1, and thereby fail the conjecture. It could continue its up and down behavior forever, or it could enter a loop, and circulate around that loop forever. In either event, an infinitely long Signature would ensue, and this would quickly become apparent to the Rule-of-8 process (see later) if ever it should happen to encounter such a number.

• Signatures can vary greatly in length. The longest Signature for any number less than 1017 has 2091 letters. But after you have studied the Tutorial on Signatures and Syllables you will be able to design and build numbers which have vastly longer signatures than this. There is no upper limit to the length of a Signature.

• The algorithm for calculating a Signature from a Number is defined quite simply by the two rules stated at the beginning of this page. The algorithm for performing the reverse operation is considerably more intricate, and although an understanding of its operation is not essential, a complete listing of it is provided in  Appendix 2.

• It will be very much to your advantage if you have a working copy of the Collatz software on your computer as you study this section of the article, so before you proceed any further, please refer to  Appendix 4.  and complete the Download and Installation instructions. Try to have the program showing on the screen at the same time as you read the contents of this tutorial.

• Calculate a Signature from a Number. Type the number directly into the Number field, and click the Number to Signature button. The Signature will appear in the Signature field, and some explanatory notes will appear in the Collatz Results field.

• Calculate a Number from a Signature. Type the signature directly into the Signature field, and click the Signature to Number button. The Number will appear in the Number field, and the Collatz Results field will be populated with an abbreviated description of the algorithm used to calculate the number. Note especially the format of the number display. It represents an infinite series of numbers, all of which have signatures which begin with a common set of characters.

• How to design a number which has very special characteristics. For example say you wanted a number which would start its Collatz series with ten consecutive odd numbers, followed by ten consecutive even numbers, followed by another ten odd numbers. All you need to do is to create a signature with ten consecutive OE syllables followed by a syllable containing ten consecutive Es, followed again by ten consecutive OE syllables. In other words, the complete signature would be
OEOEOEOEOEOEOEOEOEOE
EEEEEEEEEE
OEOEOEOEOEOEOEOEOEOE
Type it into the Signature field, and click the Signature to Number button. You should receive the number 451306495. Use a calculator to extract the Collatz series for this number, and you will find that the series begins with the odd and even numbers appearing in exactly the order you specified.

• Here is a number for you to ponder. It is quite ordinary except for the fact that it has 3,011 digits. An interesting experiment for you to try, is to copy the array of digits into the clipboard of your computer and then paste them into the Number window of the Collatz program. Clicking the Number to Signature button will start the calculation of the Signature. Depending on your computer it may take quite a bit of time (think in terms of minutes rather than seconds), but the interesting result will be well worth the wait. You will find that the first 20,000 characters of the signature will consist of 10,000 OE Syllables!

```19950631168807583848837421626835850838234968318861924548520089498529438830221946631919961
68403619459789933112942320912427155649134941378111759378593209632395785573004679379452676
52465512660598955205500869181933115425086084606181046855090748660896248880904898948380092
53941633257850621568309473902556912388065225096643874441046759871626985453222868538161694
31577562964076283688076073222853509164147618395638145896946389941084096053626782106462142
73333940365255656495306031426802349694003359343166514592977732796657756061725820314079941
98179607378245683762280037302885487251900834464581454650557929601414833921615734588139257
09537976911927780082695773567444412306201875783632550272832378927071037380286639303142813
32414016241956716905740614196543423246388012488561473052074319922596117962501309928602417
08340807605932320161268492288496255841312844061536738951487114256315111089745514203313820
20293164095759646475601040584584156607204496286701651506192063100418642227590867090057460
64178569519114560550682512504060075198422618980592371180544447880729063952425483392219827
07404473162376760846613033778706039803413197133493654622700563169937455508241780972810983
29131440357187752476850985727693792643322159939987688666080836883783802764328277517227365
75727447841122943897338108616074232532919748131201976041782819656974758981645312584341359
59862784130128185406283476649088690521047580882615823961985770122407044330583075869039319
60460340497315658320867210591330090375282341553974539439771525745529051021231094732161075
34748257407752739863482984983407569379556466386218745694992790165721037013644331358172143
11791398222983845847334440270964182851005072927748364550578634501100852987812389473928699
54083434615880704395911898581514577917714361969872813145948378320208147498217185801138907
12282509058268174362205774759214176537156877256149045829049924610286300815355833081301019
87675856234343538955409175623400844887526162643568648833519463720377293240094456246923254
35040067802727383775537640672689863624103749141096671855705075909810024678988017827192595
33812824219540283027594084489550146766683896979968862416363133763939033734558014076367418
77711055384225739499110186468219696581651485130494222369947714763069155468217682876200362
77725772378136533161119681128079266948188720129864366076855163986053460229787155751794738
52463694469230878942659482170080511203223654962881690357391213683383935917564187338505109
70271613915439590991598154654417336311656936031122249937969999226781732358023111862644575
29913575817500819983923628461524988108896023224436217377161808635701546848405862232979285
38756234865564405369626220189635710288123615675125433383032700290976686505685571575055167
27518899194129711337690149916181315171544007728650573189557450920330185304847113818315407
32405331903846208403642176370391155063978900074285367219628090347797453332046836879586858
02379522186291200807428195513179481576244482985184615097048880272747215746881315947504097
32115080498190455803416826949787141316063210686391511681774304792596709375```

This means that if you were to perform the Collatz process on this number, then each time you multiplied by 3, added 1 and divided by 2 you would get another odd number, and this would continue for an amazing 10,000 times. Thereafter, the pattern would cease, and the normal mixture of odd and even numbers would return, until the number 1 is finally encountered as it always is.

Some additional results will appear in the Collatz Results window as shown below:-
The initial number has 3011 digits.
The number of Signature letters is 134405
The number of Signature Syllables is 48126
The biggest number encountered in the Collatz series is 3262700 - - - - 4400000
This Number contains 4772 digits

• Try some signatures of your own choosing. You should be able to create numbers with some very unlikely characteristics. Relax about the size of the numbers involved. The program should have no problem whatever handling numbers with at least a few thousand digits. There is no end to the interesting games you can play in this way.

• In case you are wondering... Well I hope you are wondering about how one finds a number of over 3000 digits which begins with a string of 10,000 OE syllables. Clearly, this is not the result of any sort of a search routine. It is done by crafting a signature which consists of 10,000 OEs and using the Signature / Number conversion program to obtain the desired number.

• Towards an infinitely long signature It has already been mentioned that an exception to the Collatz Conjecture demands the existence of an infinitely long Signature, so you may be thinking that as long as you can isolate a Number which produces a Signature designed to your specification, you might be able to isolate one which generates an infinitely long signature. The facts however are that although you can create signatures of any given length, infinitely long signatures will apparently always remain beyond your grasp.

In articles which discuss the Collatz conjecture, you will often find comments to the effect that some numbers should continually increase rather than eventually decrease to 1, based on the fact that odd numbers are multiplied by 3 (and incremented by 1), but even numbers are divided only by 2. Viewing the problem in that way is simplistic and misleading. It is better to break the Signature up into Syllables as defined in the introduction.

Every Syllable begins with OE, which implies:-

• Multiplication by 3.
• Division by 2.

Following the division by 2, our number may be odd or even with equal probability.

This implies that:-
It will terminate at OE or extend beyond OE, with equal probability of 1/2.

If it extends beyond OE
It will terminate at OEE or extend beyond OEE, with equal probability of 1/4.

If it extends beyond OEE
It will terminate at OEEE or extend beyond OEEE, with equal probability of 1/8.

And so on...

Clearly, the probability decreases by a factor of 2 for each E added to the Syllable, so longer Syllables are progressively less likely. However they do have a greater impact on the Collatz process due to the greater number of divisions by 2.

The reasoning presented above is captured in tabular form in the following:-.

 Signature Syllables [D]ivisions by 2 [P]robability [D]x[P] [D]x[P] (normalised) OE 1 1 / 2 1 / 2 32768 / 65536 OEE 2 1 / 4 2 / 4 32768 / 65536 OEEE 3 1 / 8 3 / 8 24576 / 65536 OEEEE 4 1 / 16 4 / 16 16384 / 65536 OEEEEE 5 1 / 32 5 / 32 10240 / 65536 OEEEEEE 6 1 / 64 6 / 64 6144 / 65536 OEEEEEEE 7 1 / 128 7 / 128 3584 / 65536 OEEEEEEEE 8 1 / 256 8 / 256 2048 / 65536 OEEEEEEEEE 9 1 / 512 9 / 512 1152 / 65536 OEEEEEEEEEE 10 1 / 1024 10 / 1024 640 / 65536 OEEEEEEEEEEE 11 1 / 2048 11 / 2048 352 / 65536 OEEEEEEEEEEEE 12 1 / 4096 12 / 4096 192 / 65536 OEEEEEEEEEEEEE 13 1 / 8192 13 / 8192 104 / 65536 OEEEEEEEEEEEEEE 14 1 / 16384 14 / 16384 56 / 65536 OEEEEEEEEEEEEEEE 15 1 / 32768 15 / 32768 30 / 65536 OEEEEEEEEEEEEEEEE 16 1 / 65536 16 / 65536 16 / 65536

About the contents of this table.
• Signature Syllables.
A list of possible Signature Syllables which contain varying numbers of Es from 1 up to 16. Syllables longer than this will of course be encountered when very large numbers are submitted to the Collatz process. Numbers having thousands of digits will be met and processed in later sections of this article.

• [D]ivisions by 2.
This is simply the number of Es contained within the Syllable.

• [P]robability.
The probability of a random Signature Syllable being of this type. As discussed previously, each Syllable type has a probability of half that of the previous Syllable type. The sum of the probabilities in this column will approach a value of 1 as the list is extended.

• [D]x[P].
The product of the number of divisions by 2 and the probability of this Syllable being generated. The resulting number provides a measure of the overall likelihood of achieving a division by 2 by means of this Syllable when a Syllable is generated.

• [D]x[P] (normalized)
The numbers in this column have exactly the same values as the numbers in the previous column, but they have been normalized so that each number has a denominator of 65536. When we see a number such as 6144 / 65536, it tells us that when 65536 Syllables of a Signature are generated, 6144 of the divisions by 2 will be generated by Syllables which have the form OEEEEEE.

Adding all of the items in column 5 gives us the sum 131054 / 65536 which equals 1.9997. This is the average number of divisions by 2 generated by each Signature Syllable. The fact that this number is so close to 2 is significant. In fact, adding additional lines to the table would move it even closer to 2. Summing up then, each Signature Syllable provides one multiplication by 3 (and an addition of 1) as well as an average of two divisions by 2. Reducing this thought to the simplest possible form implies that on average, calculating one additional Signature Syllable for the number multiplies that number by a factor of 3/4. This relationship is highly significant, and is worthy of a title. I propose that it should be called The rule of 3/4.

To see some experimental evidence which supports this rule, refer to  Result 2: Profile of the Signature Syllable data of 100 digit numbers.

A very interesting circumstance arises when we calculate a series of 8 consecutive Signature Syllables. On each of the eight occasions the subject number will be multiplied by a factor which, in the long run, will average out at 3/4. What actually happens is encapsulated in the following mathematical statement.

38 / 48 = .10011

In short it gives us a division by a number very close to 10. This in turn translates to a reduction of one in the number of digits in the number being processed. To a first approximation then, an n digit number will generate n*8 Signature Syllables on its journey to the expected concluding value of 1. This is the basis for what I call The Rule of 8 which will be demonstrated quite soon. When you study this topic, I believe you will be pleasantly surprised at how closely numbers right across the vast number spectrum obey this rule.

Three important caveats.
This Rule of 8 is only an approximation (although a remarkably precise one), and as a result small departures from it will be caused by the following:-
• The 3 in the mathematical statement is always accompanied by the addition of 1. This is not expected to cause a big departure in the operation of the Rule, but it is always present, and the departure is always in the same direction.

• The 4 is the average calculated in the Signature Syllable analysis discussed previously. On any given Syllable calculation it will in fact be some power of 2, but averaged over a large number of calculations the probabilities involved will dictate that the outcome will be a division by very close to 4.

• The mathematical statement above doesn't give us exactly one tenth, although it is, fortuitously, remarkably close to that figure. As a result, we are entitled to be quietly confident that the Rule of 8 will be closely observed.

It will be very much to your advantage if you have a working copy of the Collatz software on your computer as you study this section of the article, so before you proceed any further, please refer to  Appendix 4  and complete the Download and Installation instructions.

On the page dealing with  Signature / Syllable Analysis.  mention was made of the fact that as the Collatz process operates on a number, it will reduce the length of that number by roughly one digit for each group of 8 Signature Syllables which it calculates for that number. For a single number, this is of course just an approximation, but the Rule-of-8 demonstration discussed here allows you to execute the Collatz process on a large group of numbers all of the same length. When the results are averaged over a group of some (or many) thousands of numbers, a quite remarkable result emerges as you will soon see.

The demonstration is part of the Rule-of-8 program, and is best described by reference to the typical output shown in the following graphic. As the demonstration runs using the defaults supplied, the program applies the collatz process to one million 30 digit numbers, and in so doing produces a set of one million signatures for those numbers. The important information here is not the content of the signature, but its length. The program maintains a list of signature lengths, and as the signature length of each number is determined, the list item for that length is incremented by one. When all one million numbers have been processed this list can be used to display the histogram you see in the graphic. Naturally, as the program runs, it provides an indication of progress by redrawing the histogram at regular intervals. This can be quite spectacular to watch, and is recommended for your entertainment.

It is hoped that the following dot points will add meaning to what you see in the above graphic:-

• The X axis of this histogram is calibrated in terms of signature lengths which range between 64 and 654. These figures are also included in tabular form above the histogram.

• The Y axis is calibrated between 0 and 9937, indicating that the longest signature achieved by any of the one million 30 digit number was in fact achieved by a total 9937 of them.

• A downward pointing red arrow below the x axis points to the location of the average value of Signature Syllables predicted by the Rule of 8, and an upward pointing blue arrow labeled Average points to the location of the current average for the numbers processed so far. It is quite entertaining to watch this indicator as the program runs. Initially it wanders up and down but quickly settles down to a number very close to 240 as predicted by the Rule of 8.

• Although 64 is the shortest signature length encountered in this particular demonstration, it would be quite wrong to assume that 64 is the shortest possible signature for 30 digit numbers. In fact, there is always at least one, and often two numbers which will result in a signature having only one syllable. In the case of 30 digit numbers, the numbers 422550200076076467165567735125 and 105637550019019116791391933781 will collapse to 1 with a single Syllable Signature. These numbers look impressively large, but considerably less so when expressed as (2100 - 1) / 3 and (298 - 1) / 3.

• Similarly, although 654 was the longest signature encountered in the demonstration, it is almost certainly not the longest possible signature. Unlike the shortest signature, finding the longest signature seems not to be a trivial matter. This could be fertile ground for people who can't resist a mathematical challenge.

• The shape of this histogram will strike a chord with anyone who has more than a passing interest in the subjects of probability and statistics. The graph is immediately recognizable as a bell curve, or "normal" distribution with some obvious differences. The main difference is that it is very far from being the smooth curve normally expected. It seems that certain Signature Syllable count values are favored by the Collatz process while others are disadvantaged. Why this is so may be another interesting question for additional research in the future.

• Another point worth noting is the way in which the right extreme of the curve seems to extend out much further than the left extreme. For the time being we will just assume this is because there is an absolute lower limit of 1 for the numbers which can appear in the left portion this histogram, while the right portion might possibly extend indefinitely, as would be the case if an exception to the Collatz Conjecture were to be encountered.

A more detailed look at the Rule of 8 results.
Greater insights into the Rule of 8 can be obtained by running a much more ambitious test. The next graphic shows the result of running the program with one million ninety digit numbers. This means that we are testing numbers in the vicinity of 1090 which is approximately the number of atoms in ten billion universes. This is obviously a very big number, but successful trials have been performed using numbers having 1,000 digits and even up to 10,000 digits. Regardless of the enormity of the numbers being tested, the results always conform very closely with the following description.

The Collatz curve for ninety digit numbers. • The Normal Distribution curve is again firmly in evidence with the great majority of data points in this histogram crowded very compactly around the value of 720, which is the value suggested by the Rule of 8.

• The curve of the histogram thins out very substantially as it approaches both the left and the right extremes of the graphic.

A closer look at the extremes.
The tapering of the histogram at the extremes finally results in little more than a single pixel being displayed. Naturally we would like to know what is actually happening there. This is taken care of by printing the bars of the histogram in two passes. The first pass prints only the short bars ... the ones which represent signature lengths which were achieved by 10 or less numbers. These bars are stretched so that the longest of them occupy the entire height of the graph. Also they are dawn using a distinctive colour to distinguish them from the rest of the graph. The second pass is drawn using black, and the scaling is organized so that the longest bar occupies the entire height of the graph.

The extreme left of the curve.
• The thinning of the curve mentioned above continues to the left, to the extent that the last of the results shrink to only a single pixel.

• The shortest recorded Signature has a Syllable length of 364. This is most certainly not the shortest possible signature. It is mentioned elsewhere in this article that, for numbers having a given number of digits (90 in this case), there will always be at least one example of a signature length of just one syllable. The likelihood of encountering such a number in a run of the Rule of 8 program on 90 digit numbers is as good as zero. It is the same as the likelihood of selecting one particular atom out of 1090 atoms or, to put it another way, one particular atom out of all the atoms in 10 billion copies of the observable universe. I venture to suggest that this is not very likely.

• The stretched signatures are drawn using red, and clearly show the continued reduction in frequency.

• The Y axis of the histogram is calibrated in terms of frequency. The most frequently encountered signature length was 719, and was encountered 5598 times.

The extreme right of the curve.
• The thinning of the curve also continues to the right, with the last Signature having a Syllable length of 1295. This is very likely not the longest signature possible for numbers having 90 digits. Finding longer signatures than this is not such a simple matter as finding the shortest possible signature. More research is required to settle this question.

• In the Introduction to the Collatz problem it was mentioned that any number which fails the Collatz conjecture will be characterized by an infinite length signature. The rapid thinning out of longer signatures here doesn't bode well for the prospects of finding an infinitely long one which is required to disprove the Collatz conjecture.

• In order to obtain a more detailed view of the extreme right hand end of the Normal Distribution Curve for Numbers having exactly 100 digits, the program was modified to provide an output of the Signature Length, and the Number which produced that Signature for any cases in which the Signature Length was greater than 1400.

The following paragraph contains a list of forty 100 digit numbers, all of which yield signatures having lengths of more than 1400 syllables. It may appear from this that they are rather plentiful, and we shouldn't be surprised about this since they are selected randomly from a collection of 10100 100 digit numbers. Although they are quite plentiful, they are fiendishly hard to find. You can find them using the Rule of 8 function of the Collatz program if you use the option function of that program to request a run of say 10 million 100 digit numbers, and request that any number which produces a signature of over 1400 syllables be displayed in the Collatz Report section of the screen. Typically, a run of 10 million numbers will produce two or three examples of signature lengths in excess of 1400. On the other hand it is not unusual to find no such examples at all.

Patience is a virtue.

1402 : 6842757495773267586887806151688807436727743186126349007201127828468961393836806410856342616825894625
1403 : 2180801148686324953439462939778882626303731853654111339462437156761016480513749135384422020443688379
1406 : 2678311937747823955428233908126798043266098510223891544662249648949944961907723760460734814766910333
1407 : 3823718535532971207517904536989829309675030479388977545081310923773973862966462501763453945111079159
7622170809671124244743551875404553634800556574478143656625786196519428409570986806220345366335963493

1408 : 4820216256912984513189605847117956512837519700474367403251546673836856799328899781794800128254180253
2506365190376979278615060456758297938410847350707104358935686789427934817493845788432634124228325519

1409 : 6060257294094424836930858444496460904397254379087841820258975656997038526771140399033637812395623249
1410 : 4686973975765635616838998517650308280055079164945267339841401598957946602817094876336812022569995391
9374073467186332319437831785343186847740210300395202900173031197209535622358506974005191163376365815

1412 : 3732465449176743594224707380191993481984089791756450392918583200376536728650534763667263757341603203
7154264313251634209772562532969336146398156934218428580194249436324043572067810466236232627824661951

1414 : 3659640113614688382853914953076440177360139842033070807131266614715044469239717140998327646571334199
1415 : 8995089998187912595026462086517046267339660140522371736764191610482759529859248935170505760178449767
1417 : 4043559559806976359551087111494822836155102541571017178364399858575755907192497135991522484779844111
1419 : 7131190755481947195107934275281318476355170881457672715019446987921830936264922416160406395592394129
6771146834247137694732961971255838525322742172319874644186854495776487064109198768529832871573782059
7856528451007070433975984903163448123727975214790085824317098724864610725742702890346177760390827071

1420 : 9309794760299155642628929637431924875904944644155461315035051583979900098373752828867489265705838575
2334693199694858085441325298114605202857326117892976383782195097461485535727656770536332159567954119

1423 : 2830287823686069734541803020779532604253613699343910521381354929520209221476192189132279145735299901
1424 : 7438269916262190780119924450170867585002538104826307458335788281293123333618941445385862130534123371
7697272880142611622999325454520993661086991386431588693804692081387250383617266411700217074288672211

1427 : 4518015638758504762454896952014173200458087802783061434847829915557389908484527774750244585945495709
1428 : 2987351939502758017968763002262795094030303442986575672495787377421193383546038031368008177191767481
1429 : 8092646192449084550390721847739445404832588938966339379318593425997002227885473491409252483725167387
1432 : 9123695420409529171114400592587784174324640783510261967307613794037332553017495609149175866329395983
1436 : 3706105287219384670033366475152581904264055815421809436915465735952194777186386907699933009131046089
1447 : 5517046498299205737900588092793379349801960574321693647391635231648009790605397094833200279046740291
1449 : 2484469091877287715642028352428745726373577826592312905854489492569789134629962447426019564086370415
1454 : 5366017916635901776062517542587112469857321682017825015324268581233285357636552379577372694849721765
1455 : 6739376114482765766373616813230128205294275852107565774722510864761543950500853450866031186171124015
6793689273525437379018565305878252468658627531172429794461157003568686480121401099229595020365282249

1458 : 8098659249748649346462339440788399673519597757016037607323163102355838625354820422426232514262620585
1459 : 2735671485420203313402614296821931151400352116648886211126918787671682198087559807028241063050176535
1471 : 1305049607908580325169640257013784663287258107157267305227843138837988758167021876344271894149757407
1480 : 2114906264399229703377091102898189426947788654789037024114887199131871912693467009113804621622760153
1483 : 9922131373936600124796146073324036161679659546472207392368908204560210671900600300025651470071977871
1485 : 9192617866236964186795541944302165469471124072578069540398093355392057707815135005706757256462122995
1501 : 1925325427301238693436974716463324231743360748827222059797465213996993658480666367192272714798797665
7200774491333522318966513865183887923813564533295043070788032954829439786486921553433708195977833007

1504 : 8981501245210358466941063069963587676629346197118428970148956094519761258248051502546615028160501531
1564 : 7987221001644153042341078535366145131744118543385943691991226902014405396825535417707018213937349053

Even a cursory examination of these forty signatures provides some revealing facts about the Collatz sequence:-
• The first 20 signatures are associated with signature lengths which range between 1402 and 1420, a span of only 19 lengths.

• The range of lengths occupied by the second group of 20 signatures extends from 1420 up to 1504, a span of 84. This demonstrate clearly that the density of signatures is decreasing quite rapidly.

• In the first group of signatures there are six examples of signature lengths which play host to multiple signatures, while in the second group of 20 there are only two such examples.

• The highest signature length discovered was 1504. There are very likely longer signature lengths than this, but searching for them would be a very time consuming and probably unrewarding task.

• It would be foolish to insist that there are no examples of 100 digit numbers with signature lengths greater than 1504, but it is surely unlikely that there exists a 100 (or less) digit number with an infinitely long signature.

• Which brings us to the crucial point, (mentioned in several other places in this report), that an exception to the Collatz Conjecture demands the existence of a finite number which has an infinitely long signature.

• The above results apply to numbers having "only" 100 digits. I have used the Rule of 8 program to perform the same process on 1000 digit numbers, and also to a lesser extent on 10,000 digit numbers, and the result is the same in all cases.

How maximum Signature Length changes with Digits per Number.

The following graphic image serves an an illustration for the data contained in the table which immediately follows it. It is a graph of the signature length values of 100,000 thirty digit numbers. The key items of data are the center point of the curve as suggested by the Rule of 8 and indicated by the values on the x axis of the graph, and the length of the longest signature for each number length. The ratio between these two values is recorded in the fourth column of the table, and it is this ratio which provides a very telling story about the operation of the Collatz series. You will observe that the value of the ratio decreases as the number of digits in the subject numbers increases. Not only does it show that the prospects of finding an infinitely long signature are not very good at the outset, but those prospects become even less hopeful as the number of digits in the numbers being processed increases from 30 in the first line of the table up to the maximum 10,000 digits in the final entry.

Anyone armed with this information who continues a search for an exception to the Collatz Conjecture really is embarking on a forlorn quest!

Number of
Digits
Center of curve
as determined by
Rule of 8
Longest
Signature
Length
Ratio of
Signature Length
to Center of Curve
30  240      587     2.45
40  320      697     2.18
50  400      788     1.92
60  480      895     1.86
70  560      984     1.76
80  640      1135     1.77
90  720      1184     1.64
100  800      1285     1.61
150  1200      1792     1.49
200  1600      2250     1.41
1000  8000      9303     1.16
10000  80000      83324     1.04

Some illuminating results generated by the Rule of 8 function.

Result 1: Accuracy of the Rule of 8 function.
The following table contains the results of a series of tests performed by the Rule of 8 function. The columns contain the following data:-
• The Number Length is the number of digits in the numbers used for each test in the series. A minimum of one thousand numbers were processed in each test.

• Note that in every case, the Average Number of Syllables per Signature is very close to the (8 times the Number Length) figure predicted by the Rule of 8, and shown in square brackets.

• The Highest Number of Syllables per Signature is to be interpreted as the highest number encountered on this run. There will probably be a few higher numbers but these can safely be categorized as outliers.

• The Lowest Number of Syllables per Signature is not recorded. This is because there will always be at least one number for any given number length which will have a Signature with only one Syllable.
For example:-
• 2 digit numbers :- 21 and 85
• 3 digit numbers :- 341
• 4 digit numbers :- 1365 and 5461

Number
Length
Average number
of Syllables
per Signature
Highest number
of Syllables
per Signature
Number
Length
Average number
of Syllables
per Signature
Highest number
of Syllables
per Signature
14         43       200  1602      2201
22         65       300  2397      3250
31         96       400  3200      3963
38         129       500  4001      4857
47         176       600  4804      5553
55         203       700  5600      6401
63         218       800  6397      7458
71         265       900  7200      8213
10  78         272       1000  8000      9183
20  158       406       2000  16001   17429
30  238       589       3000  24017   26084
40  319       636       4000  32038   33767
50  398       723       5000  40044   42091
60  479       955       6000  48015   50528
70  559       909       7000  55984   58473
80  638       1121      8000  64013   66737
90  718       1118      9000  72002   74303
100  799       1237      10000 80047   83017

In the topic Signature / Syllable Analysis it was confidently predicted that the Rule of 8 would be closely observed as the Collatz Process is applied to numbers in general. This table demonstrates that the prediction was indeed justified.

The progressive results provided by the Rule of 8 function include a profile of the entire set of Signature Syllables generated during a batch processing run. The following table was generated during a typical run using one million 100 digit numbers. The Rule of 8 predicts that the total number of Signature Syllables should be 8 x 100 x 1,000,000 = 800,000,000. The figure that is reported by the program is 799,108,217. This adds quite significantly to the confidence we can feel in the validity of the Rule. Of equal interest are the numbers of each of the Syllable types. Note that almost exactly half of all the Syllables are of type OE, and that all of the subsequent types progressively decrease by a factor of close to 2. This justifies the equal probability assumptions made in the topic  Signature / Syllable Analysis.

Behavior 1.
At first sight, it is quite surprising to study the Colltz sequence for the number 27, and observe that it reaches a high point of 9232 which is over 340 times greater than its starting point. Perhaps it is natural to wonder how this compares to numbers in general, and especially to much larger numbers. A small adjustment was made to the Rule of 8 function of the Collatz program so that it could create a list of the biggest numbers encountered during a typical run. Each time a number bigger than the last big number on the list was encountered, it was added to the end of that list. The following list resulted when a sequence of one million randomly chosen 30 digit numbers were put to the test.

0     521864777009781972171399170839     1565594331029345916514197512518         
0     521864777009781972171399170839     2348391496544018874771296268778         
0     521864777009781972171399170839     3522587244816028312156944403168         
1     323420776575166478500093050537     3683964783176505669165122403790         
1     323420776575166478500093050537     5525947174764758503747683605686         
1     323420776575166478500093050537     8288920762147137755621525408530         
1     323420776575166478500093050537     12433381143220706633432288112796        
1     323420776575166478500093050537     13987553786123294962611324126898        
1     323420776575166478500093050537     20981330679184942443916986190348        
1     323420776575166478500093050537     23603997014083060249406609464144        
57     916855648438295922460724587307     24472957408552401667321569784324        
87     621623765988457188136100529945     28709147380980211756617196305502        
87     621623765988457188136100529945     43063721071470317634925794458254        
87     621623765988457188136100529945     64595581607205476452388691687382        
87     621623765988457188136100529945     96893372410808214678583037531074        
87     621623765988457188136100529945     145340058616212322017874556296612       
87     621623765988457188136100529945     183946011686143720053872485312906       
87     621623765988457188136100529945     275919017529215580080808727969360       
357     126481844367082466405697580063     384123720446478525524194192690150       
357     126481844367082466405697580063     576185580669717788286291289035226       
357     126481844367082466405697580063     864278371004576682429436933552840       
997     930817431324349698729704998367     1242071656860735704986768744595224      
3759     833314741525573657655193087439     1687177372739226871344116090500816      
4790     802879304996710367566694233327     2002334859286487864932913145951352      
4790     802879304996710367566694233327     2534205056284461204055718200344688      
12498     724912411722761965880444096681     3478991944441262688411679665259144      
13198     288842135708910560684485601487     4995984046651086558655411738847260      
13198     288842135708910560684485601487     5620482052482472378487338206203170      
13198     288842135708910560684485601487     8430723078723708567731007309304756      
18895     337595270660394052978947869065     9122282670150866104209365939507014      
18895     337595270660394052978947869065     13683424005226299156314048909260522     
18895     337595270660394052978947869065     20525136007839448734471073363890784     
46838     877683494233739625091193455399     23082073760820664932033630627369574     
46838     877683494233739625091193455399     34623110641230997398050445941054362     
46838     877683494233739625091193455399     51934665961846496097075668911581544     
56900     808023609957127685989776095391     58102268660421757049325755468362498     
56900     808023609957127685989776095391     87153402990632635573988633202543748     
121196     305282207143663775670173201499     123597536981034324928986406162487428    
121196     305282207143663775670173201499     160725630931863703306657628708270410    
121196     305282207143663775670173201499     241088446397795554959986443062405616    
121196     305282207143663775670173201499     257451382945302186864008960438223202    
121196     305282207143663775670173201499     386177074417953280296013440657334804    
159448     620544853285226825440776099267     557672052424846300330533042985763488    

• Column 1: The number of numbers which had been processed before processing was begun on this one.

• Column 2: The thirty digit number currently being processed.

• Column 3: The value of the current number in the Collatz series.

• Column 4: The number of additional digits in the Collatz series number, compared to the number being processed.
As you can see, in this case the Collatz series can contain numbers which are over a million times larger than the number which started the series.

What follows is the basis of an algorithm which can be used to construct a program which accepts a complete Collatz Signature and computes the number which generated it. Note that this algorithm works equally well for both partial and final signatures.

The Signature used as the input in this description will be OEOEOEEEEEOEEEEO.

We begin with two equal odd numbers called Α (the Greek letter alpha) and Ω (the Greek letter omega). Both will be initialized to the odd number 2n+1 before processing begins. Ω will change its value as processing proceeds, as dictated by the contents of the Signature which is listed vertically in the first column. The changes to the value of Ω are effected by carefully controlled changes to the value of n. Whatever changes are made to the value of n in Ω will also be made to the value of n in Α. The result of all this will be that at the completion of the algorithm, Α will contain the required number.

 Ω Α Ω 2n+1 2n+1 O 2n+1 This is odd so Multiply by 3 and add 1. 6n+4 2n+1 E 6n+4 This is even so Divide by 2. 3n+2 2n+1 O 3n+2 For this to be odd, n must be odd, so set n to 2n+1. 6n+5 4n+3 6n+5 is odd so Multiply by 3 and add 1. 18n+16 4n+3 E 18n+16 This is even so Divide by 2. 9n+8 4n+3 O 9n+8 For this to be odd, n must be odd, so set n to 2n+1. 18n+17 8n+7 18n+17 is odd so Multiply by 3 and add 1. 54n+52 8n+7 E 54n+52 This is even so Divide by 2. 27n+26 8n+7 E 27n+26 For this to be even, n must be even, so set n to 2n. 54n+26 16n+7 54n+26 is even so Divide by 2. 27n+13 16n+7 E 27n+13 For this to be even, n must be odd, so set n to 2n+1. 54n+40 32n+23 54n+40 is even so Divide by 2. 27n+20 32n+23 E 27n+20 For this to be even, n must be even, so set n to 2n. 54n+20 64n+23 54n+20 is even so Divide by 2. 27n+10 64n+23 E 27n+10 For this to be even, n must be even, so set n to 2n. 54n+10 128n+23 54n+10 is even so Divide by 2. 27n+5 128n+23 O 27n+5 For this to be odd, n must be even, so set n to 2n. 54n+5 256n+23 54n+5 is odd so Multiply by 3 and add 1. 162n+16 256n+23 E 162n+16 This is even so Divide by 2. 81n+8 256n+23 E 81n+8 For this to be even, n must be even, so set n to 2n. 162n+8 512n+23 162n+8 is even so Divide by 2. 81n+4 512n+23 E 81n+4 For this to be even, n must be even, so set n to 2n. 162n+4 1024n+23 162n+4 is even so Divide by 2. 81n+2 1024n+23 E 81n+2 For this to be even, n must be even, so set n to 2n. 162n+2 2048n+23 162n+2 is even so Divide by 2. 81n+1 2048n+23 O 81n+1 For this to be odd, n must be even, so set n to 2n. 162n+1 4096n+23 162n+1 is odd as required.

So, numbers having the given Signature will have the form 4096n+23
```    When n=0, the initial number is 23.
O   E   O   E    O   E    E   E   E   E   O   E   E   E   E   O
23  70  35  106  53  160  80  40  20  10  5   16  8   4   2   1

When n=1, the initial number is 4096 + 23 = 4119.
O    E     O    E     O    E     E     E    E    E    O    E
4119 12358 6179 18538 9269 27808 13904 6952 3476 1738 869  2608
E    E     E    O
1304 652   326  163```
The algorithm provides us with an infinite series of numbers all of which have a Signature beginning with OEOEOEEEEEOEEEEO. Explore the Signatures which appear when you substitute other values of n. When you do you will find that the first 16 characters of the Signature will correspond exactly to the Signature of 23. Thereafter, all bets are off as to how the Signatures will progress except for the fact that they will all terminate with an O at number 1.

It has been mentioned several times in this report that it is an easy matter to derive large numbers which have quite short signatures. However the inverse operation of finding the longest possible signature for numbers which have a (relatively) small number of digits is far from simple. This appendix will describe a method of obtaining relatively long examples.

Consider the following diagram of a number map. Virtually everyone who knows anything about the Collatz conjecture could start at 1442 and calculate their way down to the 1. It is simply a matter of applying the Collatz process to generate the familiar sequence of numbers, but how do you get back to 1442 if you start from, for example, 5. The difficulty is that at each number you have a choice of two operations. You can always multiply by 2, but if the current number is 1 modulo 3, you can also subtract 1 and divide by 3. How do you decide which choice to make?

The aim of the process is to get the longest possible signature, and to achieve this you must take the divide by 3 option as often as possible.

^^^^
1442
^^^^                          721
1624 - 541 - 1082 - 2164 >
812
406
^^^             203
916 - 305 - 610 >
458
^^^                                    229
130 - 43 - 86 - 172 - 344 - 688 >
^^^                    65
148 - 49 - 98 - 196 >
74
^^                         37
22 - 7 - 14 - 28 - 112 >
^^           11
52 - 17 - 34  >
26
^^                       13
16 - 5 - 10 - 20 - 40 >
4
2
1

The Collatz program which supports the principles discussed in this article was generated over a period of some years while the principles themselves were emerging from my research. During this time, a program called Crossword Express was also being developed, and at the time it seemed appropriate to incorporate the much shorter Collatz program into the main body of Crossword Express. As a result, if you go the extra mile to download and install this program you will receive a significant bonus of an extensive puzzle generation platform which I believe surpasses the performance of all competing products, and it won't cost you a cent.

The entire package is written in the Java programming language, and so is able to be installed and run on both Apple and Windows computers. There are some differences in the steps involved between these two systems, and so what follows are two separate paragraphs which provide all of the details you will need to achieve a satisfactory result.

• Unzip the file by means of a double click on the icon of the .zip file. This will give you a folder called CrosswordExpress. You can now safely delete the CrosswordExpress.zip file.

• Move the CrosswordExpress folder to the location on your hard drive where you would like it to be stored permanently. The Applications folder would be a good choice, but the decision is up to you.

• Double click the CrosswordExpress folder in the new location you chose for it, right click the Crossword-Express.jar file, and select Open With followed by Jar Launcher.app. This is a once only operation which in effect tells the computer that the application being opened is a trusted application.

• Double click the CrosswordExpress folder in the new location you chose for it, then right click the Crossword-Express.jar file, and select Make Alias. This will give you a file called Crossword-Express.jar alias.

• Drag the Crossword-Express.jar alias file to an easily accessible location, so that you can start Crossword Express and make use of the Collatz program whenever the need arises. The Desktop would be a good choice, but once again, the choice is yours.

• This is where the recommended procedure for Apple and Windows diverge. Thirty years of experience with Crossword Express has revealed that there are many unzip utilities available for use with Windows, and many of these have a faithful band of regular users. Consequently, I make no firm recommendation on the subject of unzipping this file, except to say that when you are finished, the Desktop must contain a new folder called CrosswordExpress. You can now safely delete the CrosswordExpress.zip file.

• Move the CrosswordExpress folder to the location on your hard drive where you would like it to be stored permanently. The Programs folder would be a good choice, but the decision is up to you.

• Double click the CrosswordExpress folder in the new location you chose for it, right click the Crossword-Express.jar file, and select Open With followed by Java (TM) Platform SE binary. This is a once only operation which in effect tells the computer that the application being opened is a trusted application.

• Double click the CrosswordExpress folder in the new location you chose for it, then right click the Crossword-Express.jar file, and select Create shortcut. This will give you a file called Crossword-Express.jar-Shortcut.

• Drag the Crossword-Express.jar-Shortcut file to an easily accessible location, so that you can start Crossword Express and make use of the Collatz program whenever the need arises. The choice is yours.

Possible Installation Difficulties.
The executable file of Crossword Express is CrosswordExpress.jar Such files (having a .jar extension) will run on any computer which has a Java Runtime Environment (JRE) installed. If it fails to run on your computer, it simply means that your computer has not yet been equipped with a JRE. This is easily remedied:-

• For Apple
Select Settings / Java / Update The resulting dialog will tell you the version of Java you have installed. If this is not the recommended version, an Update button will appear, and you can use this to download the correct version. For detailed assistance on the subject, visit https://java.com/en/download/help/mac_install.xml

• For Windows.
Select Start / Control Panel / Java / Update. Clicking the Update Now button will either tell you that you have the correct version already installed, or it will automatically install the correct version. For detailed assistance on the subject, visit https://java.com/en/download/faq/java_win64bit.xml

And here is the payoff for your care and attention. To start the Collatz program, select Collatz from the list of puzzle options, and then click and GO. This will present you with a window similar to this:- The window that you see will be larger than this, but all of the components of the window are resizeable using simple mouse operations.

When you are studying Number / Signature Conversions, you will be working with the Number window, the Signature window, and the associated green arrow buttons.

When you are studying The Rule-of-8 you will be using the Rule-of-8 Demonstration menu option.