New whole numbers classification

[Jean-Yves BOULAY]

Hello,

I think this may interest you. I propose a new mathematical definition making no distinction between the set of prime numbers and the numbers 0 and 1.

Here is an overview of my attached article (or linked below):

“According to a new mathematical definition, whole numbers are divided into two sets, one of which is the merger of the sequence of prime numbers and numbers zero and one. Three other definitions, deduced from this first, subdivide the set of whole numbers into four classes of numbers with own and unique arithmetic properties. The geometric distribution of these different types of whole numbers, in various closed matrices, is organized into exact value ratios to 3/2 or 1/1.

Definition of an ultimate number:

Considering the set of whole numbers, these are organized into two sets: ultimate numbers and non-ultimate numbers.

Ultimate numbers definition:

An ultimate number not admits any non-trivial divisor (whole number) being less than it.

 Non-ultimate numbers definition:

 A non-ultimate number admits at least one non-trivial divisor (whole number) being less than it.

 Other definitions:

 Let n be a whole number (belonging to ℕ), this one is ultimate if no divisor (whole number) lower than its value and other than 1 divides it.

 Let n be a natural whole number (belonging to ℕ), this one is non-ultimate if at least one divisor (whole number) lower than its value and other than 1 divides it.”

 Please give your full attention to this. For example, the extension of Sophie Germain's concept of number applied to these new classes of numbers generates other singular phenomena.

Original full text

The_ultimate_numbers_JY_Boulay_2020.pdf

Edited April 22, 2020 by Jean-Yves BOULAY

[taeto]

Maybe the terminology does not work so well when transported from French into English. 

The expression "the 40 primordial numbers" is puzzling. First you might think of the numbers 0,1,...,39. But we see 48 and 81 presented as examples.

And "composite number" is already used for every non-prime number larger than 1. It is unfortunate to attach the same term to two different and closely related concepts.

On the positive side, to find something interesting about the density of natural numbers that have no square factors might be worthwhile. I am not aware of much that is known. 

[Jean-Yves BOULAY]

Taeto, you should read the article to understand: 48 and 81 are not examples, they are part of the first 10 numbers of each class. For the classes of numbers, I distinguish the raised numbers from the other composites. By this process, the different types of numbers oppose in ratio 3/2 for the 10 first. The 10 primordial are the 10 first numbers of the 4 classes which I propose.

 

This is not speculation but a mathematical definition! otherwise prove the opposite:

0, 1 and all primes (ultimate numbers) not admit any non-trivial divisor (whole number) being less than them.

Other numbers (non-ultimate numbers) admit at least one non-trivial divisor (whole number) being less than them.

Please replace this post in mathematics.

Below are listed, to illustration of definition, some of the first ultimate or non-ultimate numbers defined above, especially particular numbers zero (0) and one (1).

 - 0 is ultimate: although it admits an infinite number of divisors superior to it, since it is the first whole number, the number 0 does not admit any divisor being inferior to it.

- 1 is ultimate: since the division by 0 has no defined result, the number 1 does not admit any divisor (whole number) being less than it.

- 2 is ultimate: since the division by 0 has no defined result, the number 2 does not admit any divisor* being less than it.

- 4 is non-ultimate: the number 4 admits the number 2 (number being less than it) as divisor *.

- 6 is non-ultimate: the number 6 admits numbers 2 and 3 (numbers being less than it) as divisors *.

- 7 is ultimate: since the division by 0 has no defined result, the number 7 does not admit any divisor* being less than it. The non-trivial divisors 2, 3, 4, 5 and 6 cannot divide it into whole numbers.

- 12 is non-ultimate: the number 6 admits numbers 2, 3, 4 and 6 (numbers being less than it) as divisors*.

Thus, by these previous definitions, the set of whole numbers is organized into these two entities:

 - the set of ultimate numbers, which is the fusion of the prime numbers sequence with the numbers 0 and 1.

 - the set of non-ultimate numbers identifying to the non-prime numbers sequence, deduced from the numbers 0 and 1.

 * non-trivial divisor.

 

Edited April 22, 2020 by Jean-Yves BOULAY

[taeto]

On 4/22/2020 at 10:47 PM, Jean-Yves BOULAY said:

Taeto, you should read the article to understand: 48 and 81 are not examples, they are part of the first 10 numbers of each class. For the classes of numbers, I distinguish the raised numbers from the other composites. By this process, the different types of numbers oppose in ratio 3/2 for the 10 first. The 10 primordial are the 10 first numbers of the 4 classes which I propose.

That is what I mean by a language problem. Your table contains "the primordial numbers", and 48 and 81 are two of the numbers in the table. Clearly then they are indeed examples of such numbers. 

You also say here that there are two kinds of composite numbers, some that are raised (prime powers) and others that are composite (not prime powers). Instead of talking about composite numbers that are not composite because they are raised, why not call them "unraised" or "true composite" numbers. 

Why the first 10 numbers of each kind? How about the first 12? Or 100? Are you aware of Skewes's number in the theory of prime numbers: 

https://en.wikipedia.org/wiki/Skewes's_number ?

Edited April 23, 2020 by Strange
Fixed link

[Sensei]

5 hours ago, Jean-Yves BOULAY said:

- 4 is non-ultimate: the number 4 admits the number 2 (number being less than it) as divisor *.

- 6 is non-ultimate: the number 6 admits numbers 2 and 3 (numbers being less than it) as divisors *.

In other words, they are product of two prime numbers. 

They are called semiprimes or biprimes. 

https://en.m.wikipedia.org/wiki/Semiprime

6 is discrete semiprime. 

 

[taeto]

9 hours ago, Sensei said:

In other words, they are product of two prime numbers. 

They are called semiprimes or biprimes. 

Sensei, you are right of course. And this assumes the old-fashioned classification. In this new-fashioned classification, the biprime 4 is in the raised class, and the biprime 6 is in the (pure) composite class. 

[Jean-Yves BOULAY]

Sensei and taeto, thank you for this debate. I'm not interested in semiprimes. For example number 30, which I qualify in non-ultimate, but also in pure composite, is with 3 divisors (2, 3 and 5). Number 48, 10^th mixed composite, with 5 divisors whose 4 identical (2,2,2,2 and 3) and called as mixed because for example this is (2×2×2) ×(2×3).

Also, take a good look at my article in detail, I am only interested at the beginning of the lists of numbers and with numbers of entities of multiples of 5 of which mainly 10, 25 and 100. Where a ratio 3/2 can appear (and it appears really a lot in my dozens of matrices presented).

In OEIS site, the ultimate numbers sequence (A158611) is called “0, 1 and the primes”. This is not a precise name! And if you look on the site, there is no clear definition applying simultaneously to primes, to 0 and to 1.

My ultimates concept is without any ambiguity (and includes 0 and 1):

An ultimate number not admits any non-trivial divisor (whole number) being less than it.

Let n be a whole number (belonging to ℕ*), this one is ultimate if no divisor (whole number) lower than its value and other than 1 divides it.

[taeto]

1 hour ago, Jean-Yves BOULAY said:

Also, take a good look at my article in detail, I am only interested at the beginning of the lists of numbers and with numbers of entities of multiples of 5 of which mainly 10, 25 and 100. Where a ratio 3/2 can appear (and it appears really a lot in my dozens of matrices presented).

I would not mind looking at your paper. But please tell me more about which media and audience you intend it for. Not for a dedicated number theory journal, I presume? You might write a monograph?! 

1 hour ago, Jean-Yves BOULAY said:

In OEIS site, the ultimate numbers sequence (A158611) is called “0, 1 and the primes”. This is not a precise name! And if you look on the site, there is no clear definition applying simultaneously to primes, to 0 and to 1.

My ultimates concept is without any ambiguity (and includes 0 and 1):

This comment is confusing. I got convinced from reading your initial post that the sequence of ultimate numbers is exactly the same as the sequence of the primes with 0 and 1 appended at the beginning. How can that description be ambiguous? 

If you are only saying that "0, 1 and the primes" is not "a name", then maybe you have a point. But would you expect them to write "this is the sequence of ultimate numbers, where an ultimate number is defined as a prime or one of 0 and 1"? It doesn't seem natural.

[Strange]

3 hours ago, Jean-Yves BOULAY said:

In OEIS site, the ultimate numbers sequence (A158611) is called “0, 1 and the primes”. This is not a precise name! And if you look on the site, there is no clear definition applying simultaneously to primes, to 0 and to 1.

The definition "0, 1 and the primes" seems pretty clear to me. But it also says "nonnegative noncomposite numbers" which also seems a pretty good definition.

 

3 hours ago, Jean-Yves BOULAY said:

Let n be a whole number (belonging to ℕ*), this one is ultimate if no divisor (whole number) lower than its value and other than 1 divides it.

So this is the primes before the convention was adopted that 1 (and presumably 0) should not be included among the primes?

[Jean-Yves BOULAY]

A very strong entanglement links all these sets of numbers which oppose in multiple ways in ratios of value 3/2 (or reversibly of ratios 2/3). For example, the first 6 ultimates (0-1-2-3-5-7) are simultaneously opposed to the 4 non-ultimates (4-6-8-9) among the first 10 natural numbers, to the 4 raiseds of the first 10 non-ultimates (4-8-9-16) and to the 4 ultimates beyond the first 10 whole numbers (11-13-17-19).

52 minutes ago, Strange said:

So this is the primes before the convention was adopted that 1 (and presumably 0) should not be included among the primes?

Yes as M. Jackson said: this is it!

58 minutes ago, Strange said:

The definition "0, 1 and the primes" seems pretty clear to me. But it also says "nonnegative noncomposite numbers" which also seems a pretty good definition.

It's a bit complicated, it's an enumeration of several criteria contrary to my definition in one way.

[Strange]

35 minutes ago, Jean-Yves BOULAY said:

Yes as M. Jackson said: this is it!

It's a bit complicated, it's an enumeration of several criteria contrary to my definition in one way.

I can't see how it is complicated. It seems perfectly clear to me. (But I have understood almost nothing that you have said.)

36 minutes ago, Jean-Yves BOULAY said:

A very strong entanglement links all these sets of numbers which oppose in multiple ways in ratios of value 3/2 (or reversibly of ratios 2/3). For example, the first 6 ultimates (0-1-2-3-5-7) are simultaneously opposed to the 4 non-ultimates (4-6-8-9) among the first 10 natural numbers, to the 4 raiseds of the first 10 non-ultimates (4-8-9-16) and to the 4 ultimates beyond the first 10 whole numbers (11-13-17-19).

Where does 2/3 come into that? 

What does it mean for a set to "oppose" another set?

 

[Jean-Yves BOULAY]

2 hours ago, Strange said:

Where does 2/3 come into that? 

What does it mean for a set to "oppose" another set?

This has analogies with the organization of matter especially within quarks which are with electric charges of thirds of electrons

[Strange]

2 hours ago, Jean-Yves BOULAY said:

This has analogies with the organization of matter especially within quarks which are with electric charges of thirds of electrons

You didn't answer the questions.

You didn't even attempt to answer the questions. You just dragged in some irrelevant nonsense.

Is that because you can't answer the questions?

5 hours ago, Jean-Yves BOULAY said:

A very strong entanglement links all these sets of numbers which oppose in multiple ways in ratios of value 3/2 (or reversibly of ratios 2/3). For example, the first 6 ultimates (0-1-2-3-5-7) are simultaneously opposed to the 4 non-ultimates (4-6-8-9) among the first 10 natural numbers, to the 4 raiseds of the first 10 non-ultimates (4-8-9-16) and to the 4 ultimates beyond the first 10 whole numbers (11-13-17-19).

Please just answer the following questions 

1. Where does 2/3 come into the relationship between those numbers you have listed?

It is hard to understand your description with all its non-standard terminology, but it sounds like you are saying that there is some relationship between prime numbers ("ultimates" as you call them) and non-prime numbers ("non-ultimates") that involves 2/3. But no primes (other than 2 and 3) result in an integer when multiplied by 2/3 or 3/2. And no non-primes result in primes when multiplied by 2/3 or 3/2.

Maybe you mean the ratio between the number of "ultimates" and the number of "non-ultimates". Well, that is obviously false as well (see below for an example).

But maybe you mean something else. So please explain where 2/3 comes into it.

2. What does it mean for a set to "oppose" another set?

On 4/22/2020 at 11:47 PM, taeto said:

Why the first 10 numbers of each kind? How about the first 12? Or 100?

In the first 100 integers, there are 27 of these "extended primes" (so called "ultimates"). And therefore 73 "non-ultimates".

73 and 27 are not in the ratio 3:2.

So, hypothesis falsified, I think.

[taeto]

10 hours ago, Jean-Yves BOULAY said:

This has analogies with the organization of matter especially within quarks which are with electric charges of thirds of electrons

For those interested in "coincidences" of values of physical constants and their history, there is a well-written (don't let the typesetting fool you) paper by Victor J. Stenger:

https://web.archive.org/web/20120716192004/http://www.colorado.edu/philosophy/vstenger/Cosmo/FineTune.pdf

Jean-Yves, 

I have read your article, and the amount of work that you put into it is obviously impressive. So far as I can see it presents only the consequences of your concept of ultimate numbers up until the number 99, is that not essentially correct? Can you make any predictions, or have you even made observations already, concerning the consequences up to as high as 224 (so that you have square \(15\times 15\) matrices available)?

I find confusing sounding quotes such as (...) the definition of so-called prime numbers did not allow the numbers zero (0) and one (1) to be included in this set
of primes. Thus, the set of whole numbers was scattered in four entities: prime numbers, non-prime numbers, but also ambiguous numbers zero and one at exotic arithmetic characteristics.

You are aware that there is a precise definition of primes, according to which every integer less than 2 is a non-prime, including 0 and 1? So why explain the opposite in your article?

And why so-called prime numbers? I do not know how it works in French, but you should realize that this sounds insulting. You should remove such phrases from your paper if you want to be taken seriously. 

Edited April 24, 2020 by taeto

[studiot]

45 minutes ago, taeto said:

And why so-called prime numbers? I do not know how it works in French, but you should realize that this sounds insulting. You should remove such phrases from your paper if you want to be taken seriously. 

Yeah good advice, we don't know how much is lost or bungled in translation. +1

[Strange]

1 hour ago, taeto said:

And why so-called prime numbers? I do not know how it works in French, but you should realize that this sounds insulting.

It can be insulting, but it can also just be factual. (But the advice to avoid it, in case, is probably good.)

For example, "Athos, Porthos and Aramis, the so-called Three Musketeers" would be OK. But "Boris Johnson, the so-called Prime Minister" not so much.

[Jean-Yves BOULAY]

10 hours ago, Strange said:

In the first 100 integers, there are 27 of these "extended primes" (so called "ultimates"). And therefore 73 "non-ultimates".

73 and 27 are not in the ratio 3:2.

So, hypothesis falsified, I think.

No comments look this:

10 hours ago, Strange said:

You didn't answer the questions.

You didn't even attempt to answer the questions. You just dragged in some irrelevant nonsense.

Is that because you can't answer the questions?

The problem is that I show you the moon but you just look at my finger! Again I ask you to read my entire article (with more than 70 tables where the 3/2 ratio appears) then you will have answers. I will not publish it in full here! For example, there are developments with Fibonacci sequences, Pascal's triangles, Sophie Germain's numbers. Just one more excerpt.

 

[Strange]

12 minutes ago, Jean-Yves BOULAY said:

No comments look this:

I have no idea what that means.

13 minutes ago, Jean-Yves BOULAY said:

I will not publish it in full here!

Which journal will it be published in? Why are you wasting time on a science forum?

[Jean-Yves BOULAY]

2 hours ago, taeto said:

 

You are aware that there is a precise definition of primes, according to which every integer less than 2 is a non-prime, including 0 and 1? So why explain the opposite in your article?

And why so-called prime numbers? I do not know how it works in French, but you should realize that this sounds insulting. You should remove such phrases from your paper if you want to be taken seriously. 

My definition is from another angle by insisting on the inferiority of the diisors:

- 0 is ultimate: although it admits an infinite number of divisors superior to it, since it is the first whole number, the number 0 does not admit any divisor being inferior to it.

- 1 is ultimate: since the division by 0 has no defined result, the number 1 does not admit any divisor (whole number) being less than it.

- 2 is ultimate: since the division by 0 has no defined result, the number 2 does not admit any divisor* being less than it.

- 4 is non-ultimate: the number 4 admits the number 2 (number being less than it) as divisor *.

*non-trivial divisor. 

 

[Strange]

Just now, Jean-Yves BOULAY said:

My definition is from another angle by insisting on the inferiority of the diisors:

- 0 is ultimate: although it admits an infinite number of divisors superior to it, since it is the first whole number, the number 0 does not admit any divisor being inferior to it.

- 1 is ultimate: since the division by 0 has no defined result, the number 1 does not admit any divisor (whole number) being less than it.

- 2 is ultimate: since the division by 0 has no defined result, the number 2 does not admit any divisor* being less than it.

- 4 is non-ultimate: the number 4 admits the number 2 (number being less than it) as divisor *.

Your insistence on "inferior divisors" appears to be redundant.

Can you name any integer that has divisors larger than itself?

[Jean-Yves BOULAY]

When I said: 0, 1 and all primes not admit any non-trivial divisor (whole number) being less than them. Why are You so afraid of this? So also answer this question: is it right or not? If this is true then the whole numbers divides well into two groups: ultimate and non-ultimate numbers.

8 minutes ago, Strange said:

Your insistence on "inferior divisors" appears to be redundant.

Can you name any integer that has divisors larger than itself?

Yes of course: number 0 !

[Strange]

6 minutes ago, Jean-Yves BOULAY said:

When I said: 0, 1 and all primes not admit any non-trivial divisor (whole number) being less than them. Why are You so afraid of this? So also answer this question: is it right or not? If this is true then the whole numbers divides well into two groups: ultimate and non-ultimate numbers.

Why do you think I am afraid of a redundant statement?

Of course it is right (just redundant). The whole numbers divide well into two groups: prime and non-prime numbers. (Where you are including 0 and 1 in the primes; I don't think deserves a new name).

6 minutes ago, Jean-Yves BOULAY said:

Yes of course: number 0 !

Christ. It's like some child's joke.

[studiot]

10 minutes ago, Jean-Yves BOULAY said:

 

 

11 minutes ago, Jean-Yves BOULAY said:

  2 hours ago, taeto said:

 

You are aware that there is a precise definition of primes, according to which every integer less than 2 is a non-prime, including 0 and 1? So why explain the opposite in your article?

And why so-called prime numbers? I do not know how it works in French, but you should realize that this sounds insulting. You should remove such phrases from your paper if you want to be taken seriously. 

My definition is from another angle by insisting on the inferiority of the diisors:

- 0 is ultimate: although it admits an infinite number of divisors superior to it, since it is the first whole number, the number 0 does not admit any divisor being inferior to it.

- 1 is ultimate: since the division by 0 has no defined result, the number 1 does not admit any divisor (whole number) being less than it.

- 2 is ultimate: since the division by 0 has no defined result, the number 2 does not admit any divisor* being less than it.

- 4 is non-ultimate: the number 4 admits the number 2 (number being less than it) as divisor *.

*non-trivial divisor. 

It is extremely bad form and very unhelpful to readers to answer quotes in this fashion.

ie by placing your reply within the quote attributed to another member  makes the reply from that member appear different from what it really is, both in content and date/time.
(I have changed your word colour to red to show this)

Above the second quote is the first attempt to quote your words, using the forum quote function.
It is completely blank.

Please separate your words from those of other members.

This example quote from Strange shows how to do it properly.

15 minutes ago, Strange said:

16 minutes ago, Jean-Yves BOULAY said:

My definition is from another angle by insisting on the inferiority of the diisors:

- 0 is ultimate: although it admits an infinite number of divisors superior to it, since it is the first whole number, the number 0 does not admit any divisor being inferior to it.

- 1 is ultimate: since the division by 0 has no defined result, the number 1 does not admit any divisor (whole number) being less than it.

- 2 is ultimate: since the division by 0 has no defined result, the number 2 does not admit any divisor* being less than it.

- 4 is non-ultimate: the number 4 admits the number 2 (number being less than it) as divisor *.

Your insistence on "inferior divisors" appears to be redundant.

 

[Strange]

Just now, studiot said:

It is extremely bad form and very unhelpful to readers to answer quotes in this fashion.

I have a hypothesis about this, but it would be off topic.

[studiot]

13 minutes ago, Jean-Yves BOULAY said:

If this is true then the whole numbers divides well into two groups: ultimate and non-ultimate numbers.

Yes, but it is also true that many ( actually an infinity) other classifications are possible.

So how is yours useful?

By the way 'group' is another very well defined term in Mathematics, how and why is your definition different?

I too am also waiting to learn how one set can 'oppose' another.

I do wonder if you are referring to the conventional 'complemetary set' since you are partitioning your universal set into two parts.

 

Edited April 24, 2020 by studiot