What sort of maths is this?

[James Magan]

+
M

I started a string a while ago by asking some questions about the number 153.
I will first recap on a couple of observations made there, then ask a question to which I would be very happy to have the answer – or perhaps several answers.
So to recap:
a) 153 has an unsual property because if you sum the cube of its digits 1³ (1) + 5³ (125) + 3³ (27), you arrive back at the original number, 153. The only other numbers that have this property are 370, 371 and 407 (and 1).
b) 153 is unique inasmuch as not only does it revert to itself if you sum the cube of its digits, but every number that is divisible by three in the entire numerical system is reducible to 153 if you sum the cube of its digits, then repeat the process on the product, and keep repeating it ... eventually you will arrive at the irreducible 153. The example that I gave was the number 3 itself, but it works on any multiple of 3. Here is 3 being “reduced” to 153:
3³ = 27;
2³ (8) + 7³ (343) = 351;
3³ (27) + 5³ (125) + 1³ (1) = 153.

Now here is my question. I discovered this information from various sources on the internet, and it was provided by people who were trained mathematicians: but what sort of maths is it? I cannot even formulate the question I am trying to ask very clearly, but what I mean is this: 3³ = 27 is alright – even I understand that because it is simple arithmetic ... but what sort of a process is it when you take the result and instead of calling it 27, you simply call it a 2 and a 7?

I would understand it still to be arithemtic if you decided to treat 27 as a 20 and a 7 ... and if you added 20³ and 7³ together you would have 8343, or if you simply cubed 27 you would have 19683.

But in the process that I have been describing the base 10 system is collapsed into just a sequence of digits between 1 and 9, and irrespective of their place in the sequence, the digits are treated as being of equal “dignity” with one another. E.g. if the number 5 is in the penultimate place in the sequence, it is treated just as a 5 and not as 50. All the trailing zeroes disappear.

Is this valid mathematics? (Is there such a thing as valid and invalid mathematics?) What are the principles that it is based on? Or is this process of reducing any multiple of 3 to 153 simply a “conjuring trick”? I expect that a lot of conjuring is based on mathematics, but it seems to me that this process is at least in some way “serious maths”, because there must be some solid underlying numerical principle that produces this very consistent effect.

I guess that having a clear understanding (which I don’t) of the base 10 numbering system and number-base systems in general might help to clarify the answer for me. I also suspect that the relationship between 3 and 153 probably has something to do with a property of the number 3 which I have just discovered for myself with some amazement, namely that when you add the digits of any multiple of 3 together, the result is always divisible by 3 (so for example, 27 is 2+7 = 9; 99 is 9+9 = 18; 111 is 1+1+1 = 3, etc.) Therefore if you start the process described above with any multiple of 3, each new product in the process will also be divisible by 3 (because if the sum of the digits of a number is divisible by 3, so is the sum of its cubed digits) ... until things eventually boil down to 153. [It seems that 9 also has the same property as 3 if you sum the digits of any of its multiples (e.g. 63 is 6+3 = 9; 108 is 1+8 = 9; etc.), but none of the other numbers in the decimal system (except 1 of course) share this property.]

It has taken me a while to pose the question, but by exposing the extent of my ignorance I hope it will make it easier for the experts kindly to tailor their answer(s) in such a way that I might be able to understand it (/them)!

Any suggestions for further reading specific to this question would also be gratefully received. Many thanks.

 

[Dale10101]

What sort of math is this?

What sort of math is this?

Maybe the answer is ... any type you can invent based on clearly stated fundamental assumptions and defined logical deductions.

Take your discovery:

"I have just discovered for myself with some amazement, namely that when you add the digits of any multiple of 3 together, the result is always divisible by 3 (so for example, 27 is 2+7 = 9; 99 is 9+9 = 18; 111 is 1+1+1 = 3, etc.) Therefore if you start the process described above with any multiple of 3, each new product in the process will also be divisible by 3 (because if the sum of the digits of a number is divisible by 3, so is the sum of its cubed digits) ... until things eventually boil down to 153.

You might invent a function that disassembles a number string into an unordered list, say Q(x) ,so that Q(27) = (2,7) or (7,2) since a list has no specified order. That, I think, would capture your sense of "equal dignity" mentioned earlier in your post.

You could use an ADD() function to sum the list elements. You could use a Modulo() function to test the result to find out if the the result of ADD(Q()) is divisible by 3 .....None of which gives you a key as to why things add up the way that they do but does perhaps register the idea that math is made "valid" by definition and design, and thereby becomes a tool for exploring, essentially puzzles, that may or may not have an analog in nature.

Your discovery about the nature of 153 does indeed seem a puzzle, but you seem to make perhaps too much of the fact that such a puzzle exists. What I am saying is that if you took all possible combinations of all symbols and could comprehend the result as a whole you would see all sort of remarkable co-incidences (in the strict sense) as artifacts of the symbols of your set and the relations you have defined between them. For example, I find it remarkable that Euler's number "e", which does describe the real world growth of populations, the accumulation of interest, the decay of radioactive particles, "add's up" two different ways:

\[e = \sum\limits_{n = 0}^\infty {\frac{1}{{n!}}} \]

and,

\[e = \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{1}{n}} \right)^n}\]

Never mind that, I assume, they can be derived from one another, just the fact that two different phenomena might lead to the genesis of each and only later discovered equal, that is the astonishing thing!

Or, consider the discovery by the Greeks of "amicable numbers", which is a puzzle that reads much like the one that you find so interesting.

Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.) A pair of amicable numbers constitutes an aliquot sequence of period 2. A related concept is that of a perfect number, which is a number that equals the sum of its own proper divisors, in other words a number which forms an aliquot sequence of period 1. Numbers that are members of an aliquot sequence with period greater than 2 are known as sociable numbers.
For example, the smallest pair of amicable numbers is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220.
The first few amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368) (sequence A063990 in OEIS).

It is also interesting that "Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties."

Perhaps if you want to understand your puzzle deep in the soul you need to study "Number Theory".

Number theory (or arithmetic^{[note 1]}) is a branch of pure mathematics devoted primarily to the study of theintegers, sometimes called "The Queen of Mathematics" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers).

Having said all of that to put into perspective your puzzle as one of many, many interesting math puzzles that are found within math, and to recognize it as a "type" of puzzle with brothers and sisters that can be treated on a rational basis with possibly, but not necessarily, far ranging results .... having said that, well, it is an interesting puzzle and worth, at least playing with to "wet ones whistle", and/or "sharpen ones teeth" as the case may be. I find myself playing around with it ... searching for what (?) ... the sort of understanding that would allow me to predict prime numbers in advance or a computer search. Yow! desperate, sleepless nights ahead.

Hmmm you seem to be say if n is a positive integer then letting J = (n)(3^n) will generate a number whose constituent digits will add up to a quantity that will be evenly divisible by 3. If that is correct I suppose the next step is to build a function that expresses, K(J), as the sum of the digits of J. Well ... maybe, I am signing off, other things to do at the moment. Good luck with your adventure.

 

[lookagain]

For example, I find it remarkable that Euler's number "e",
which does describe the real world growth of populations, the accumulation of interest,
the decay of radioactive particles, "add's up" two different ways:
\[e = \sum\limits_{n = 1}^\infty {\frac{1}{{n!}}} \], and
\[e = \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{1}{n}} \right)^n}\]

Dale10101 I didn't read your whole post, but your first formula caught my eye.

It is supposed to be \(\displaystyle \ e \ = \sum\limits_{n = 0}^\infty {\frac{1}{{n!}}}

\)


Source: http://en.wikipedia.org/wiki/E_(mathematical_constant)

Your first formula has a sum of \(\displaystyle \ e - 1.\)

 

[James Magan]

+
M

Thank you. I couldn't cope with the algebra (except the VERY simple bits), but I think I understood most of the rest. I am sure you are right that number theory is the place to look, because I would like to discover, as far as possible, the mathematical laws that govern this particular phenomenon. (I would prefer to call it a mystery rather than a puzzle, since this is a case which I am sure has far-reaching analogues in the nature-of-things / the structure-of-reality.)

So for me it is not a question of inventing a function that turns a decimal number into a string of digits (and back again); it is more about discovering why and in what sense doing this is a valid mathematical (or at least numerical) operation ... and perhaps also what the applications of this type of operation are.

If any number-theorist can shed further light on the mystery of the relation of 3 to 153, or on what it means, mathematically, to regard a numerical sequence alternately as a) an ordinary number in base 10 and b) as a string of decimal digits, I would be grateful for the insights.

 

[daon2]

If any number-theorist can shed further light on the mystery of the relation of 3 to 153, or on what it means, mathematically, to regard a numerical sequence alternately as a) an ordinary number in base 10 and b) as a string of decimal digits, I would be grateful for the insights.

The entire list of these kinds of numbers is here http://oeis.org/A005188 . There are only 88 of them.

G.H. Hardy, a renowned number theorist, says there is nothing particularity special about these numbers (i.e. not worth study beyond amusement). Every base will have a different but finite number of them.

 

[James Magan]

+
M

"G.H. Hardy, a renowned number theorist, says there is nothing particularity special about these numbers
(i.e. not worth study beyond amusement). Every base will have a different but finite number of them."​

Thank you for the reference and the information, to which I have found something very similar in the Wikipedia article on "Narcissistic Numbers", as follows:

In "A Mathematician's Apology", G. H. Hardy wrote:
There are just four numbers, after unity, which are the sums of the cubes of their digits:

.
These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician

But the principal point of my inquiry goes beyond "153-narcissism". I do agree that the 153 case would not be very interesting if all the number did was to revert to itself by summing the cube of its digits. But unless I have misunderstood something, your reply does not address the far more surprising property that one third of all numbers also ultimately revert to 153 by using the same process.

Can you or someone tell me what number theorists say about the fact that when you add the digits of any multiple of 3 together, the result is always divisible by 3 (I am sure that I am not the first to discover it!!!)?
I think that this would be heading on the right track since multiples of 3 show the same type of versatility in producing consistent results when they are viewed a) as ordinary numbers in base 10 and b) as strings of decimal digits that can be added together.

I am still open to the possibility of a final judgement that "there is nothing in this that appeals to the mathematician": but it seems to me that the case that there is not something of real interest going on here still has to be proven.

 

[JeffM]

+
M

"G.H. Hardy, a renowned number theorist, says there is nothing particularity special about these numbers
(i.e. not worth study beyond amusement). Every base will have a different but finite number of them."​

Thank you for the reference and the information, to which I have found something very similar in the Wikipedia article on "Narcissistic Numbers", as follows:

In "A Mathematician's Apology", G. H. Hardy wrote:
There are just four numbers, after unity, which are the sums of the cubes of their digits:

.
These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician

But the principal point of my inquiry goes beyond "153-narcissism". I do agree that the 153 case would not be very interesting if all the number did was to revert to itself by summing the cube of its digits. But unless I have misunderstood something, your reply does not address the far more surprising property that one third of all numbers also ultimately revert to 153 by using the same process.

Can you or someone tell me what number theorists say about the fact that when you add the digits of any multiple of 3 together, the result is always divisible by 3 (I am sure that I am not the first to discover it!!!)?
I think that this would be heading on the right track since multiples of 3 show the same type of versatility in producing consistent results when they are viewed a) as ordinary numbers in base 10 and b) as strings of decimal digits that can be added together.

I am still open to the possibility of a final judgement that "there is nothing in this that appeals to the mathematician": but it seems to me that the case that there is not something of real interest going on here still has to be proven.

James

The onus is not on anyone to prove that a fact is not interesting. The onus is on someone to prove that a fact is interesting because it illuminates or connects to other facts. What Hardy means is that this kind of relationship exists because humans have chosen a positional notation based on ten for representing numbers. Certain facts follow from that choice, but would not follow if we chose a different base. Consequently, they are not general facts about numbers and so are not found interesting by number theorists.

 

[James Magan]

James

The onus is not on anyone to prove that a fact is not interesting. The onus is on someone to prove that a fact is interesting because it illuminates or connects to other facts. What Hardy means is that this kind of relationship exists because humans have chosen a positional notation based on ten for representing numbers. Certain facts follow from that choice, but would not follow if we chose a different base. Consequently, they are not general facts about numbers and so are not found interesting by number theorists.

Thank you, Jeff. I appreciate that in a rational enquiry about the significance of any set of data it is counter-productive to try to extort the truth from the facts. One has to be patient and accept that one may or may not come to a (more) complete understanding of the truth as the enquiry progresses.

What I would want to ask about Hardy's position as you state it is: why have humans chosen base ten as their preferred way of ordering numerical data? In various civilizations it was not always so. I believe (though someone may correct me) that the Babylonians worked in base 60 - because 60 is so rich in divisors (1,2,3,4,5,6,10,12,15,20,30): and it is to them that we owe the 360 (6 x 60) degrees of the circle. But I would not say that the Babylonians chose base 60: rather that they used it because they discovered that it had a certain versatility that helped them to grapple with and organize the real conditions of their lives ... i.e. they discovered a genuine correspondence between base 60 and the nature of things in the real world. (At this point I feel it is important to say that personally as a non-mathematician I have got no more than a very faint idea what it means to think or work in base 60!)

Unless and until the progress of human history proves me wrong, I feel that it is more accurate to say that human beings have discovered that a base ten numerical system is actually the one that in many ways is best suited to fulfilling their potential in the world (perhaps one only has to count the number of the digits at the end of one's two hands to see a small part of the reason why!) So the question of why the number 3 behaves in a certain way in base 10, or why the numbers 3 and 153 have a remarkable symbiosis in base 10, are questions that have real implications for us and I believe that they are worthy of as thorough an investigation as we are capable of.

If I may revert from philosophy, like a dog with a bone, to my original question about the mathematical status of the process by which every number divisible by 3 can be reduced to 153 (and leaving the fact that 153 is a "narcissistic number" behind, though not forgotten), I would like just to illustrate this process by a couple more examples that I happen to have to hand (having already used the example of 3 in my first post), so that anybody reading this is not obliged to work out what can be quite a long series of steps for themselves:

A. Here is 144 reverting to 153 (apologies that the automatic formatting has offset some of my spacing)

1³ = 1
4³ = 64
4³ = 64
___129 ...
1³ = 1
2³ = 8
9³ = 729
___ 738 ...
7³ = 343
3³ = 27
8³ = 512
____882 ...
8³ = 512
8³ = 512
2³ = 8
___1032 ...
1³ = 1
0³ = 0
3³ = 27
2³ = 8
___36 ...
3³ = 27
6³ = 216
___243 ...
2³ = 8
4³ = 64
3³ = 27
____99 ...
9³ = 729
9³ = 729
___1458 ...
1³ = 1
4³ = 64
5³ = 125
8³ = 512
____702 ...
7³ = 343
0³ = 0
2³ = 8
___351 ...
3³ = 27
5³ = 125
1³ = 1
____153

B. And this is 276 being reduced to 153
2³ = 8
7³ = 343
6³ = 216
____639 ...
6³ = 216
3³ = 27
9³ = 729
____972 ...
9³ = 729
7³ = 343
2³ = 8
____1080 ...
1³ = 1
0³ = 0
8³ = 512
0³ = 0
____513 ...
5³ = 125
1³ = 1
3³ = 27
____153

Any further thoughts regarding this phenomenon would be gratefully received; particularly anything about the validity (or otherwise) of the flipping that takes place in the process between a) numbers in base 10 (i.e. everything that is displaced from the left margin in the above examples) and b) groups of disassociated decimal digits* (i.e. all the numbers on the left margin). The hypothesis that I continue to propose as being at least plausible is that there may be some deep (and not merely casually coincidental) connection between these two different number types, which may be well illustrated by this process, and which it would be well worth understanding more fully. If you wanted me to characterize the two types, I would say that the first is heirarchical - i.e. the place of the digit in the sequence determines its magnitude and therefore also in a way its importance - and the second is egalitarian: if this characterization has any validity, then intuitively it seems to me that there should be some form of legitimate interplay between the two.

* Although "dissasociated", it seems worth noting that each group when simply added together (before being cubed) will produce a total that is divisible by 3: e.g. 5+1+3 = 9.

______________________________________
PS. I would like to thank the right people for a wonderfully organised website, and if I am trying your patience just shut me down: I will go and squeak somewhere else.

 

[Dale10101]

okie dokie

Dale10101 I didn't read your whole post, but your first formula caught my eye.

It is supposed to be \(\displaystyle \ e \ = \sum\limits_{n = 0}^\infty {\frac{1}{{n!}}}

\)


Source: http://en.wikipedia.org/wiki/E_(mathematical_constant)

Your first formula has a sum of \(\displaystyle \ e - 1.\)

Thanks, fortunately I was just making a point and not trying to advance a calculation. I will edit.

The 153 thing is pretty interesting, but so are some of the comment following it.

 

[Dale10101]

James Magan

Google: No results found for "extort the truth from the facts

I hope you write poetry.

I also appreciate learning about the existence of "Narcissistic Numbers" – “Well, well, aren’t you a gaudy number … strutting in self appreciation and all … my, my?” lol.

Your post has given me the opportunity to think new thoughts and see something about mathematics that had not occurred to me before, so, thanks.

Pondering the meaning of the pattern that you are studying and the comments by M.s Daon2 and JeffM I am led to an analogy, slippery dogs though they be. Suppose the contents of a large book written in a foreign language are laid out in character sequence into a square frame. The book could be read if you knew that language although the breaks at the end of rows would be arbitrary and split words or spaces.

Now looking at this puzzle the question arises, what is being said, what intelligence is being transmitted. To answer that question one begins to look for patterns. But the thing is there are two types of patterns, those having to do with the original contents of the book and those having to do with how you laid the book out.

One might discover word units based upon repeated patterns of characters and spaces, one might confirm this by devising consistent rules for handling row breaks. One might deduce that the intelligence flows left to right, or right to left by row, or up or down by column. These are deductions based upon patterns created by the actual information being conveyed by the book.

One the other hand, now thinking in terms of English, there would be patterns that had nothing to do with the books contents. One would find by happenstance, a column here and there with e’s aligned 3,4,5, perhaps 8 deep, a word or phrase running diagonal, or column wise, in effect, a sort of happenstance cross-word aspect. The larger the book the more “oddities” that might be found.

How would one know the difference between these two types of patterns and not spend a lifetime trying to fathom, essentially a mirage, that had nothing to do with the intelligence being conveyed by the book’s author?. Well, for one thing, lay the book out in a rectangle instead of a square, the patterns dealing with the books intelligence would stay (you could still read the sentences, the rules for the line breaks would be preserved etc) but the “notational patterns”, would all change and new happenstance patterns would appear.

In math, it doesn’t matter whether you are expressing yourself notationally in base 2, 10, or 60, the Pythagorean formula still holds, the ratio of a circles circumference to its diameter doesn’t change, larger integers will be the product of primes and so on. On the other hand if you take 153 to mean a length of 153 units of your choice and translate it to base two for example, all of the coincidences based upon adding cubes of numbers based upon position rather then weight value disappear … but of course other oddities will appear and be just as alluring.

Yes, you might say, but we are not talking about simple coincidences like numbers stacked upon one another in a column or patterns of numbers that appear in a column, we are talking about the results of a formula that after several operations deliver a consistent result!

True enough, but mathematicians ponder types of formulas and their properties with the same interest as they ponder the properties of types of numbers. From that point of view they might say that your formula only impresses you because it is relatively obvious. They might suggest that there are millions of ways that the digits of a number can be separated, processed (cubed in your case) and recombined (summed in your case), and in some cases, for a small number of key numbers (153, 370,371 etc) a consistent pattern will appear. They might say such cases are guaranteed to occur as a matter of probability no matter what notational base you are working in … but, THEY WON’T TRANSLATE from one notational system to another because you are talking about happenstance patterns that are unattached to the nature of “numbers” and “relations” which DO translate between bases.

Well, anyway, that is how I interpret what those previous posts meant by notational patterns that have only notational significance, Happy hunting.

 

[James Magan]

+
M

Dale, I am sorry, I knew you were the last "poster" on 27 January, but I did not realise you posted twice, and had not seen the second one until a few moments ago when I was preparing to send a new post of my own. I think the best thing to do is to post my text now anyway, as I have been thinking about it for a while, then I will read yours of 27 Jan more carefully and give it some thought. So ...

Some further thoughts on 153.

1. The list of Armstrong / Pluperfect / Narcissistic numbers found in the link sent by daon2 (http://oeis.org/A005188 : many thanks again for it) seems to me to add to 153’s “list of qualifications”. I note the following:
a. The first ten numbers in the list are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 153. So in this way of ordering numbers according to one specific property, 153 so to speak “completes” the basic decimal sequence 1 to 9 by “taking the place of 10”!
b. No pluperfect numbers result from a process in which their summed digits are multiplied to the power of two. This seems to add some further emphasis to the “special relationship” between 3 and 153, situated directly after those numbers that are merely “multiplied to the power of 1”.
c. It seems to me that 1 to 9 might be considered as pluperfect numbers only in a limited sense, since there is no multiplication involved in the power of 1, and there is no adding of digits involved in producing the pluperfect result (e.g. 9¹ = 9). If that is a fair comment, then in this sense 153 is the first, the primordial, pluperfect number.

2. My most important conclusion is not a mathematical one, but for me it provides a resolution to the question “Is this serious math?” In a way, this conclusion satisfies my quest for the moment. I do not expect it will satisfy mathematicians, and insofar as they are mathematicians it ought not to; but I will return to that at the end.
I explained in a previous string that my particular interest in the number 153 is that it is the only precise number of three digits or more in the Gospels, which are the core of the Christian Bible. The Gospels were written by four men who we know under the names of Matthew, Mark, Luke and John (which may well have been their actual names); but it is an ordinary part of Christian belief that these men were inspired in a very special way in what they wrote by the Holy Spirit, or Divine Wisdom: so much so that it is also correct for a Christian to say that God, the Holy Spirit, or Divine Wisdom is the Author of the Gospels, and of the rest of the Bible.
So it is my belief (and not mine only!) that the occurrence of the number 153 as the only precise number of three digits or more in the Gospels is ultimately attributable to the designs of Divine Wisdom. Do I think that there was some serious reason for its inclusion in the text? Certainly, and it will very probably be a mystery that will exercise me for the rest of my life. But does that mean that the seriousness of the 153 phenomenon must be insisted on at all costs? This is where I have woken up to a new perspective. In another part of the Bible, Wisdom -- a reflection of the Divine Wisdom Who is ultimately the Author of the Gospels -- speaks of herself (Wisdom is always feminine) in these terms:
“From everlasting, I was firmly set, from the beginning, before the earth came into being. The deep was not, when I was born, nor were the springs with their abounding waters. Before the mountains were settled, before the hills, I came to birth ... When He fixed the heavens firm, I was there, when He drew a circle on the surface of the deep ... I was beside the Master Craftsman, delighting Him day after day, ever at play in His presence, at play everywhere on His earth ...” (Proverbs 8:23ff)
So in the Judaeo-Christian tradition this Wisdom, the wisdom that was instrumental in the creation of the universe, is playful as well as being serious; and therefore 153, as a notable numerical expression of this same wisdom, is understandably both serious and playful in what it is and what it does.

Mathematicians reading this may not share any of the beliefs that are expressed or implied above. But we may still share in common a certain interest and puzzlement in the behaviour of the number 153. For what it is worth, I offer my personal conviction that this number and its properties are profoundly associated with the creative forces at work in the universe; and I have a feeling that any mathematician who wanted to make large profits (in the widest sense) for humanity would do well to keep the 153-phenomenon in mind: it might contain a very abundant harvest of applications – so much so that you would not even have to go looking for them, because they would come looking for you first!
But every harvest comes at its own proper time, and I have no special insights about this with regard to 153. I only know that the harvest will not be complete until we arrive at Eternity and Infinity.

 

[James Magan]

Google: No results found for "extort the truth from the facts

I hope you write poetry.

I also appreciate learning about the existence of "Narcissistic Numbers" – “Well, well, aren’t you a gaudy number … strutting in self appreciation and all … my, my?” lol.

Your post has given me the opportunity to think new thoughts and see something about mathematics that had not occurred to me before, so, thanks.

Pondering the meaning of the pattern that you are studying and the comments by M.s Daon2 and JeffM I am led to an analogy, slippery dogs though they be. Suppose the contents of a large book written in a foreign language are laid out in character sequence into a square frame. The book could be read if you knew that language although the breaks at the end of rows would be arbitrary and split words or spaces.

Now looking at this puzzle the question arises, what is being said, what intelligence is being transmitted. To answer that question one begins to look for patterns. But the thing is there are two types of patterns, those having to do with the original contents of the book and those having to do with how you laid the book out.

One might discover word units based upon repeated patterns of characters and spaces, one might confirm this by devising consistent rules for handling row breaks. One might deduce that the intelligence flows left to right, or right to left by row, or up or down by column. These are deductions based upon patterns created by the actual information being conveyed by the book.

One the other hand, now thinking in terms of English, there would be patterns that had nothing to do with the books contents. One would find by happenstance, a column here and there with e’s aligned 3,4,5, perhaps 8 deep, a word or phrase running diagonal, or column wise, in effect, a sort of happenstance cross-word aspect. The larger the book the more “oddities” that might be found.

How would one know the difference between these two types of patterns and not spend a lifetime trying to fathom, essentially a mirage, that had nothing to do with the intelligence being conveyed by the book’s author?. Well, for one thing, lay the book out in a rectangle instead of a square, the patterns dealing with the books intelligence would stay (you could still read the sentences, the rules for the line breaks would be preserved etc) but the “notational patterns”, would all change and new happenstance patterns would appear.

In math, it doesn’t matter whether you are expressing yourself notationally in base 2, 10, or 60, the Pythagorean formula still holds, the ratio of a circles circumference to its diameter doesn’t change, larger integers will be the product of primes and so on. On the other hand if you take 153 to mean a length of 153 units of your choice and translate it to base two for example, all of the coincidences based upon adding cubes of numbers based upon position rather then weight value disappear … but of course other oddities will appear and be just as alluring.

Yes, you might say, but we are not talking about simple coincidences like numbers stacked upon one another in a column or patterns of numbers that appear in a column, we are talking about the results of a formula that after several operations deliver a consistent result!

True enough, but mathematicians ponder types of formulas and their properties with the same interest as they ponder the properties of types of numbers. From that point of view they might say that your formula only impresses you because it is relatively obvious. They might suggest that there are millions of ways that the digits of a number can be separated, processed (cubed in your case) and recombined (summed in your case), and in some cases, for a small number of key numbers (153, 370,371 etc) a consistent pattern will appear. They might say such cases are guaranteed to occur as a matter of probability no matter what notational base you are working in … but, THEY WON’T TRANSLATE from one notational system to another because you are talking about happenstance patterns that are unattached to the nature of “numbers” and “relations” which DO translate between bases.

Well, anyway, that is how I interpret what those previous posts meant by notational patterns that have only notational significance, Happy hunting.

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[No results found for "extort the truth from the facts”. Maybe because it is not possible to do it ... ? Extort from ex-torquere, to twist out: but if you twist the truth, it is no longer the truth ...]

[Terminology:
If something that appears to be narcissism turns out to have positive results relating things beyond itself, then it is not simple narcissism but at some level transcends it. So the term “pluperfect number” seems preferable for 153: and it emerges that all other narcissistic numbers also relate at least to some other numbers by acting as thier final “anchor”, so pluperfect seems a better description generally.]

Thank you for your excellent and very clear post which describes the "grammar" of the number-base system.

If I had read it during the course of last week, I probably would have been discouraged (and certainly distracted) from writing my post of last Saturday: so you will have to decide whether the oversight on my part was a good thing or not. As a result of what you say and of reflection on the whole of the discussion so far, all of which has helped my understanding of the subject hugely, I would like to make one clarification / qualification concerning my previous post; namely, where I say that I believe the properties of 153 “are profoundly associated with the creative forces at work in the universe”, there is an implication that the 153-phenomenon is somehow involved in the physical processes of the universe, and that practical applications may to flow from it. I do not know that: perhaps it could be true, but its significance in the Bible relates principally to a spiritual harvest. I think that I was to some extent subconciously letting that implication dangle like a carrot just in case (like any pupil struggling with his homework) I could entice a mathematician to do the donkey work of finding out, or showing me, why 153 behaves this way in base 10 – because I was sure that any mathematician could do it with a much better method and ten times quicker than I could. All along I have been trying to get people to supply the answer to this specific question, but I can see that for a mathematician it is not alluring because from a mathematical point of view it may be no more, as you say, than a mirage.

But from my point of view there is a carrot: namely that beyond being possibly a mathematical mirage, 153 will always remain at some level a mystery, and it is undoubtedly rich in its potential for symbolism. So after a number of false starts and at a mule’s pace (I always knew deep down that it was the right role for me) I am begining to work out the patterns of what happens to a continuous sequence of integers in base 10 (say 1-1000) when you apply the process of cubing digits / summing totals /cubing digits / etc.

Various things emerge quite quickly:
- None of the four “narcissistic” numbers to the power of 3 (i.e. 153, 370, 371, 407) is truly narcissistic, because although they revert to themselves, other numbers also return to them as their final “base” in the process. This becomes obvious if one thinks about it. 407, for example, has the same “active” digits as 47, 74, 470, 704 and 740: so therefore at least those numbers will reduce to 407 in just one stage of the process, and there may be others which also reduce to it by one or more stages.
- So far the results for 153 are entirely consistent: it is the final product of every third number.
- It seems that 371 may be nearly as consistent as 153, inasmuch as most numbers prior to a multiple of 3 in the integer sequence (e.g. 2,5,8,11,14,17,20, etc.) that I have looked at so far reduce to 371. But this sequence of 371 results is interrupted by all of the 407 anagrams mentioned above (47, 74, 407, 470, 704, 740), since all of these are also immediately prior to a multiple of 3.
- At least some numbers reduce to 370 (7, 19... maybe other prime numbers?)
- Some numbers reduce to 1 (e.g. 1, 100, 1000: there may be others).
- Some numbers reduce to a repeating circle of results (e.g. 4 reduces to 133 ... 55 ... 250 ... 133, etc).
- All numbers can be treated anagramatically: namely, once the sequence of the process is known for, say, 28, there is no need to process 82 (or 208, or 280, or 802 or 820) since the steps and the result will be the same (it is only necessary to process about 219 numbers between 1 & 1000 to know the results for all between 1 & 1000).
- It seems that when the same process is applied to numbers by using the power of 2 (squaring) rather than the power of 3 (cubing), the results may often be a rather large and untidy looking repeating loop. I have not tried enough examples yet, but if this is so, it is almost certainly because no pluperfect numbers are produced according to the power of 2: the presence of a pluperfect number, or several, would act as an anchor and bring the process to a final halt at least in the case of some numbers.

Thank you again for making the framework, and the limitations, of this operation clear to me. One of my false starts in replying was to make an appeal to the co-naturality of the base 10 system with our human way of conceptualising numbers (and therefore also the co-naturality of the 153-process with ourselves, even if it does not translate into other bases). Maybe there is something in this thought, but I do not think now is the time for it.

 

[mmm4444bot]

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[video=youtube;f5Mdbcbys_s]http://www.youtube.com/watch?v=f5Mdbcbys_s[/video]

 

[James Magan]

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M

The versions may be compound fragments of truth and untruth, in which case they are partial truths (and the parts are always potentially separable) but no truth qualifies for a definite article - "the truth" - unless it is either the WHOLE truth (which is much too big for any of us to manage, or comprehend), or an unalloyed part of that whole (and there is an infinite number of such parts which, by definition, cannot contradict one another: although in our limited perspective they may sometimes appear to do so).

(I could not hear the dialogue on the youtube post probably because I work with a computer that was made in the stone age.)