Author Topic: Modern Mathematics as Religion  (Read 8377 times)

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Silenus

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Re: Modern Mathematics as Religion
« Reply #15 on: January 16, 2021, 11:36:26 am »
Holden,
  Have you noticed any positive changes with taking the B12 supplement?  I have recently begun to supplement Niacin (Vitamin B-3) and NAC (N-Acetyl-Cysteine). Both have shown positive benefits with energy, mental clarity, memory, learning, and reducing depression, especially NAC.

https://healthyfocus.org/niacin-detox/

https://www.healthline.com/nutrition/nac-benefits#TOC_TITLE_HDR_5

Take care.

"And the strict master Death bids them dance."

Holden

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Re: Modern Mathematics as Religion
« Reply #16 on: February 22, 2021, 04:05:00 am »
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
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Holden

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Holden

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1729*y^2 + 1 = x^2
« Reply #19 on: February 24, 2021, 09:24:24 am »
From Holden's link:   What is the smallest positive integer y such that 1729*y^2+1 is a perfect square?

Havens’s problem is an example of Pell’s equation, which is an equation of the form x^2 − N*y^2 = 1 where N is a whole number that is not a square. Havens is asking when 1729*y^2 + 1 = x^2 has an integer solution for both x and y, so you begin by rearranging the equation so the 1 is by itself. And in both Havens’s question and Pell’s equation, we don’t want any old solutions but pairs of numbers x and y that satisfy the equation where both x and y are integers.


I had created a couple programs which use Pell's equation and continued fractions.  I wrote the code using what I learned from CD Olds 1963 book.  I had never studied them in school.  No, not even in university of New Brunswick, NJ.   :(

mwh@coyote:[~]:
$ ----> pell 1729

sqrt(1729) = [41, [1, 1, 2, 1, 1, 2, 1, 2, 1, 8, 1, 1, 27, 5, 6, 5, 27, 1, 1, 8, 1, 2, 1, 2, 1, 1, 2, 1, 1, 82]]


cf: convergent
___________________
41:  41/1
1:  42/1
1:  83/2
2:  208/5
1:  291/7
1:  499/12
2:  1289/31
1:  1788/43
2:  4865/117
1:  6653/160
8:  58089/1397
1:  64742/1557
1:  122831/2954
27:  3381179/81315
5:  17028726/409529
6:  105553535/2538489
5:  544796401/13101974
27:  14815056362/356291787
1:  15359852763/369393761
1:  30174909125/725685548
8:  256759125763/6174878145
1:  286934034888/6900563693
2:  830627195539/19976005531
1:  1117561230427/26876569224
2:  3065749656393/73729143979
1:  4183310886820/100605713203
1:  7249060543213/174334857182
2:  18681431973246/449275427567
1:  25930492516459/623610284749
1:  44611924489705/1072885712316
82:  3684108300672269/88600238694661


(44611924489705)^2 - (1729)(1072885712316)^2 = 13770086920831946257 - (1729)(6921492334693245456) = 13770086920831946257 - 13770086920831946256 = 1

Hence, x = 44611924489705, y = 1072885712316 is a particular solution of x^2 - 1729*y^2 = 1

« Last Edit: February 24, 2021, 08:10:22 pm by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Nation of One

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example of continued fraction
« Reply #20 on: February 24, 2021, 09:36:13 am »
I had mentioned these in the "Depression Tabulator" thread, of all places.   

Quote
I am sometimes amazed at the collection of homegrown command line programs I have collected in my ~/bin folder.  Even when something can be done by an already created computer algebra system like Sage or SymPy, I take great delight in building a stand-alone command line program that can handle a special solitary task, like listing the continued fraction representation of a number.

mwh@coyote2:[~]:
$ ----> cf_rat 13 25

13/25 =
          1
 0 + -----------------
            1
     1 + ------------
               1
          1 + -------

               12

13/25 = [0; 1, 1, 12]


cf: convergent
___________________
0:  0/1
1:  1/1
1:  1/2
12:  13/25

The above is output from one of the programs I created while on a little tangent last spring while going through an old c.1963 book by CD Olds, Continued Fractions.  I went through the little book, engaging with the exercises, and creating programs for various scenarios.

In Sage it would look like the following, the first without the magic of "%display ascii_art":

sage: continued_fraction(13/25)
[0; 1, 1, 12]
sage: %display ascii_art
sage: continued_fraction(13/25)
      1
--------------
         1
 1 + ---------
           1
      1 + ----
           12

I will take a look at my notes to see if I might be able to remember ... maybe I might find a way to dislodge my head from out of my ass. 

Thank you, Holden ... maybe there is a way to revive interest in mathematics other than returning to the exercises where I lost interest.   Maybe I really have to be interested.   I approach my own notes as a stranger to he who wrote them!

It might make more sense if you use parenthesis:    1/(1 + 1/(1 + 1/12)) = 13/25
« Last Edit: February 24, 2021, 08:27:25 pm by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

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Nation of One

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rewrite the book
« Reply #21 on: February 24, 2021, 06:31:42 pm »
The New Mathematical Library (NML) was begun in 1961 by the School Mathematics Study Group to make available to high school students short expository books on various topics not usually covered in the high school syllabus. In a decade the NML matured into a steadily growing series of some twenty titles of interest not only to the originally intended audience, but to college students and teachers at all levels.

Note from Preface of c.1963 "Continued Fractions" :

It goes without saying that one should not “read” a mathematics book. It is better to get out pencil and paper and rewrite the book.  A student of mathematics should wrestle with every step of a proof;   if he does not understand it in the first round, he should plan to return to it later and tackle it once again until it is mastered.

Aha ... this is still so very true.   Maybe the spell I have been under has been broken.   

« Last Edit: February 24, 2021, 08:21:33 pm by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Holden

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Herr Hauser as Sui Generis Regulative Ideal
« Reply #22 on: February 26, 2021, 04:06:04 pm »
This is the thing-you can take up most of the problems in publications like Popular Mechanics and you would comprehend them.You can pick any economics book(which are little more than applied maths) and you would be able to make sense of them. Even if you run into a math concept with which you are not quite familiar you would be able to learn it because you possess very sound fundamentals.


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Nation of One

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beyond conventional genre boundaries
« Reply #23 on: February 26, 2021, 11:48:52 pm »
Sui Generis ?
class by itself ?

Sui generis is a Latin phrase that means "of its/his/her/their own kind, in a class by itself", therefore "unique".

 beyond conventional genre boundaries


Thank you, Holden.  That chart flatters me.  Some of the qualities which make me not an ideal employee or husband or "patriarchal head of household" are those qualities which flatter me.

The "situation" is  both paradoxical and ironic.   As for the sex impulse, I must refer to myself as an Onanist.  Such an image may not be very flattering, and Nietzsche had been mortified at the thought of his doctor telling their mutual musician friend (Richard Wagner) that he [Nietzsche] was a chronic masturbator. 

That I have to wrestle to keep the fundamentals forever front and center is nothing to be ashamed of.  I simply wish to be as honest as possible, and to leave some notes to the future.

I am considering contacting some home-schooling organization which still focuses on those Dolciani texts.  Maybe they would be very interested in incorporating the programming exercises and code (and expanded Solution Keys).  They might help teachers in the future, or just serve as a curious footnote to the otherwise unfavorable records left about me by "official historians," that is, police records and other cold bureaucratic nonsense.

Our day-to-day interactions and our behaviors may be judged by society, and it may be true that we judge ourselves the harshest.   It is kind of you, Holden, to offer me such respect in a world which is really set up to mock someone like me, to drag me through the mud.   Fortunately, I was touched by the stories i read by Dostoevsky, and so I was prepared for the deceptive and cruel traits of attractive women and their alpha and beta male cohorts.

You were on point with your assessment of "Omega Man Cometh" ! ;)

I wonder who each of us really is.  Are we the enthusiastic student of mathematics in one phase of our existence, or are we the shell of a Humpty Dumpty who had a great fall?   Are we equal to our Fate?    If we are equal to our Fate, then is one to hide away to prevent being transformed beyond recognition?

Quote
Sadly, it is most likely more common than we could imagine - a mathematician dying of malnutrition, a genius regressing into idiocy via alcohol poisoning, a beautiful woman blowing her brains out with a gun from heartache and confusion ...

Let us not judge ourselves (or others) too harshly.   Who can ever know the transformations taking place?   Are we to believe the narratives others tell us about our very own existence?   Are we to believe even our own narratives about ourselves?

It is good for us to allow ourselves to entertain Forbidden Thoughts, to even think against our own egos.
« Last Edit: February 27, 2021, 12:13:33 pm by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

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Holden

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Re: Modern Mathematics as Religion
« Reply #24 on: February 27, 2021, 12:19:27 pm »
I am considering contacting some home-schooling organization which still focuses on those Dolciani texts.  Maybe they would be very interested in incorporating the programming exercises and code (and expanded Solution Keys).-Herr Hauser


I hope and pray that this comes to pass.I am sure there would be men interested in mathematics who would greatly appreciate what you are doing.
I have been thinking if a formula could be discovered which would help one to find prime numbers easily.
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
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Dolciani Math / primes
« Reply #25 on: February 27, 2021, 07:07:49 pm »
1.
Quote from: Holden
Quote from: Sticks & Stones
I am considering contacting some home-schooling organization which still focuses on those Dolciani texts.  Maybe they would be very interested in incorporating the programming exercises and code (and expanded Solution Keys).
I hope and pray that this comes to pass.I am sure there would be men interested in mathematics who would greatly appreciate what you are doing.
notes:

https://www.physicsforums.com/threads/wtb-dolciani-math-books.490207/

https://forums.welltrainedmind.com/topic/85830-dolciani-math/

http://dolcianihalloran.squarespace.com/meet-mary/;jsessionid=6B61C7DA6F057FA3D23D1BA24A9C8FE8.v5-web018

Quote from:  Dr. Donald Albers, Publications Director, Mathematical Association of America
Mary Dolciani was a powerful exemplar in mathematics education. She clearly could have gone on to a successful career as a research mathematician, but her great success as a teacher did not leave much room for research. She chose to work primarily with students, especially those entering mathematics teaching.

https://bookstore.ams.org/dol

http://www.hunter.cuny.edu/dolciani

https://math.stackexchange.com/questions/55353/what-book-can-bridge-high-school-math-and-the-more-advanced-topics

I left a note at Stack Exchange:

Quote from: Mike Hentrich

I know this thread is nearly six years old, but maybe someone will stumble across it as I have. Your question motivated me to sign up to Mathematics Stack Exchange just to post what I hope to be a short message. Without telling my life story, I will just say that I graduated with a Computer Science degree in 2002, did well in math classes such as Multivariable Calculus (an elective), Linear Algebra, Mathematical Reasoning, Numerical Analysis, etc, but, for whatever reasons, I never got a job as a programmer, and have actually developed what I call an "unemployable personality".

What I want to suggest is in response to your following question:

"Is there a good “bridge” between high school math and the more advanced topics? I love math and would like to know more of it. However, whenever I try to pick up a book on what I consider to be "advanced" mathematical topics, I often have a hard time understanding some of the terms and symbols right from the beginning."

More specifically: "How can I get myself to the point where I can move on in my mathematical education when what I perceive to be "high school math" is too easy, but the more "advanced" subjects are incomprehensible?"

I can suggest where to look for the bridge you seek. I will point you towards a hidden secret mind treasure to be found in a place where others might be too proud to discover.

Sometime in 2015 I dove back into Linear Algebra and Integral Calculus. I had collected some Physics textbooks (Matter and Interactions which makes use of VPython), and I was psyching myself up, imagining I wanted to become a "Computational Physicist" (in my own imagination, of course).

I will get to the point. By 2017 I found myself frustrated and wish I were more familiar with writing proofs. i longed to reboot my brain, to approach pure mathematics rather than just continue to calculate, compute, and apply algorithms.

I have returned to various incarnations of my high school math books, tracking them down on Amazon, ebay, alibris, and abebooks.

We used the Dolciani series. They are refreshingly challenging, rigorous, and formal. I think that anyone facing this problem with gaps in their knowledge or just a yearning to start their [pure] mathematical education over from a different perspective (besides as a means for gainful employment) might want to invest the time into exploring these text books. You might even want to splurge on the solution keys, especially if you intend to tackle all of the "B" and "C" exercises.

The editions after 1980 even have some computer exercises in BASIC that you might get a kick out of (you can run PC-BASIC or just write in Python or C++ or whatever language you wish).

Trust me, I thought the books would be "beneath me" or "baby math". No, they are challenging in the rigor and formality of the presentation. Many of the exercises will prove to be more challenging than what one might encounter in a community college course sharing the same name. I humbly submit that my brain needs a hard reboot.

The bottom line, from more advanced to most fundamental:

Modern Analytic Geometry (1983) Wooton, Bechenbach, Fleming

Modern Introductory Analysis (1964/1986) Dolciani, Bechenbach, Jurgensen, Donnelly, Wooton

Introductory Analysis (1988/1991) Dolciani, Sorgenfrey, Graham, Myers

Algebra and Trigonometry: Structure and Method Book 2 (1986 edition has least expensive solution key, I think) Dolciani, Sorgenfrey, Brown, Kane

(or you might prefer Modern School Mathematics 1968 edition of Alg2/Trig, but it is harder to track down)

Then there is Geometry ... either Modern Geometry: Structure and Method (1965) Dolciani, Jurgensen, Donnelly

or Geometry (1988 OR 1993 OR 2000/2011) Jurgensen, Jurgensen, Brown

Algebra: Sturcture and Method Book 1 (1987) - overlaps with book 2 ch 1 - 9 ... You might want to skip it, or, if you are some kind of purist who wants to make a fresh start with the Beginner's Mind, it can't hurt to have a copy on the shelf unless you are easily embarrassed. If it helps, don't think of the above texts as high school books. See them for what they truly are: the foundations of real analysis and abstract algebra. You might even become enchanted by the novelty of this formal and rigorous approach.

I do not think there is any shame in filling in the gaps in our knowledge. And we don't have to justify or explain our interests to anyone.

"To be conscious that you are ignorant is a great step to knowledge." ~ Benjamin Desraeli

May we all find the time to study what truly interests us.

2.
Quote from: Holden
I have been thinking if a formula could be discovered which would help one to find prime numbers easily.

I used to do this by hand from a jail cell just to give the demons in my head something to keep them occupied.  There is no simple formula.  You have to check divisibility, starting with 2, then 3, then 5, and so on.

I used long division, not a formula.

Of course, one can formulate the process of checking divisibility with a simple computer program.
Quote
// C program for prime number using function

#include<stdio.h>
 
int check_prime(int);
 
main()
{
   int n, result;
 
   printf("Enter an integer to check whether it is prime or not.\n");
   scanf("%d",&n);
 
   result = check_prime(n);
 
   if ( result == 1 )
      printf("%d is prime.\n", n);
   else
      printf("%d is not prime.\n", n);
 
   return 0;
}
 
int check_prime(int a)
{
   int c;
 
   for ( c = 2 ; c <= a - 1 ; c++ )
   {
      if ( a%c == 0 )
    return 0;
   }                    // Having reached this point, there are no factors
   if ( c == a )
      return 1;
}

// for ( c = 2 ; c <= (int)sqrt(n) ; c++ )
// Only checking from 2 to square root of number is sufficient.
// You could also improve this algorithm by incrementing by 2
// so as to check only odd numbers (after 2)
// You also would not check integers which have prime factors
// That is, you only check primes, not even numbers, not 9 or 15 or ...


The "soul" of this program is the function check_prime(int):

int check_prime(int a)
{
   int c;
 
   for ( c = 2 ; c <= a - 1 ; c++ )  // could use: for ( c = 2 ; c <= (int)sqrt(n) ; c++ )
   {
      if ( a%c == 0 )  // % means "mod" ... See if integer 'a' is divisible by 'c'.
    return 0;        // It returns the remainder after 'a' is divided by 'c'
   }
   if ( c == a )
      return 1;
}

_________________________________________________
So, if doing by hand, say, checking n = 667:

In your mind, with pencil and paper, check_prime(667):

for c = 2 to c = 667 - 1 = 666 (You only have to check up to INT(sqrt(667)) = 25)
667%2 --->  667 mod 2 = 1
667%3 ---> 667 mod 3 = 1
667%5 ---> 667 mod 5 = 2
667%7 ---> 667 mod 7 = 2
667%11 ---> 667 mod 11 = 7
667%13 ---> 667 mod 13 = 4
667%17 ---> 667 mod 17 = 4
667%19 ---> 667 mod 19 = 2
667%23 ---> 667 mod 23 = 0

667/23 = 29

667 = 23 * 29

667 is not prime

no formula, just a process

I would use little peices of crumpled up paper to randomly choose a 9 digit number, and then I would use long division.  How to know when to stop?   You only have to check up to the integer part of the square root of the integer. 

It sounds as though, you, Holden, are drawn to Number Theory, which I also find very interesting --- and it doesn't even have to be useful!   
« Last Edit: February 27, 2021, 10:04:08 pm by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Prime Factorization
« Reply #26 on: February 27, 2021, 10:16:28 pm »
Holden,
I understand that you wish to find some kind of formula to check if an integer is a prime number, but the nature of primes (and factorization of integers) makes this highly unlikely.   An integer that is prime will only have itself as a factor.

Do you see how finding the square root of a number by hand (the long way) would help find the integer part of the square root?  This in turn would help in that you would know when the number is prime once testing all primes up to the integer part of the square root of this number.

The "sra" programs, when compiled and run, were created not so much to find the square root, which it does fine, but, paradoxically, to TEACH a human animal how to do this with paper and pencil.

This program, named hfac, lists the prime factors of a given integer:

/* These are the simplified steps to find all prime factors:

1) While n is divisible by 2, print 2 and divide n by 2.

2) After step 1, n must be odd. Now start a loop from i = 3 to square root of n.
While i divides n, print i and divide n by i, increment i by 2 and continue.

3) If n is a prime number and is greater than 2, then n will not become 1 by above two steps. So print n if it is greater than 2.

*/

#include <stdio.h>
#include <math.h>
#include <stdlib.h>

void primeFactors(int n);

int main(int argc, char *argv[])
{
    int i, h;

    for(i=1; i<argc; i++) {
        h = atoi( argv[ i ] ) ;  // underlined i else this editor reads
        primeFactors(h);               // BBcode "italics"
        printf("\n");
    }
    return 0;
}

void primeFactors(int n)
{
    int i;
    printf("%d:  ", n);

    while (n%2 == 0)
    {
        printf("%d ", 2);
        n = n/2;
    }

    for (i = 3; i <= sqrt(n); i = i+2)
    {
        while (n%i == 0)
        {
            printf("%d ", i);
            n = n/i;
        }
    }
    if (n > 2)
        printf ("%d ", n);
}

/*  How does this work?

The steps 1 and 2 take care of composite numbers and step 3 takes care of prime numbers. To prove that the complete algorithm works, we need to prove that steps 1 and 2 actually take care of composite numbers. It is clear that step 1 takes care of even numbers. And after step 1, all remaining prime factor must be odd (difference of two prime factors must be at least 2), this explains why i is incremented by 2.
Now the main part is, the loop runs until square root of n. To prove that this optimization works, let us consider the following property of composite numbers.
Every composite number has at least one prime factor less than or equal to square root of itself.

This property can be proved using counter statement. Let a and b be two factors of n such that a*b = n. If both are greater than √n, then a * b > √n * √n, which contradicts the expression “a * b = n”.

In step 2 of the above algorithm, we run a loop and do following in loop
a) Find the least prime factor i (must be less than √n,)
b) Remove all occurrences i from n by repeatedly dividing n by i.
c) Repeat steps a and b for divided n and i = i + 2. The steps a and b are repeated till n becomes either 1 or a prime number.

Thanks to Vishwas Garg for suggesting the above algorithm.
*/

EXAMPLE:

mwh@coyote:[~]:
$ ----> hfac 987654321
987654321:  3 3 17 17 379721
« Last Edit: February 27, 2021, 10:32:05 pm by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Re: Modern Mathematics as Religion
« Reply #27 on: February 28, 2021, 02:28:33 pm »
I like the way you have written the message about the Dolciani books. Please do keep me posted about this matter,if it is okay with you.

Thanks for the detailed response.As regards Prime numbers,I was wondering if you have looked into the following paper:

https://www.claymath.org/sites/default/files/ezeta.pdf

The fact is, due to various reasons which I have described in the other messages, I have not been able to put in the hours that you and people you might find in Stack Exchange and Physics Forum.

I am, very slowly, trying to alter this sorry state of affairs.

What I wish for you is this: you find the people who have the "technical know-how" to appreciate your work and they also appreciate the "spirit" which has driven you to this kind of work.

I think ,if I could reach a certain level of mathematical expertise, I would be able to appreciate what you are trying to do ,better than most, because our Weltanschauung is so similar( no surprises there - you are the one who has helped me to form it more than perhaps anyone else).

I think I would especially be able to appreciate it better compared to,someone like,say, Musk or Gates. Now, I think if Musk or Gates were to go through your papers "today", they would be able to comprehend them better than I could. But they might not appreciate the spirit in which the work has been done.

They have the technical know-how to understand it but the appropriate Weltanschauung.I posses the Weltanschauung,but at present,  the  technical know-how "today",more's the pity.

I think,my life might end like van Gogh's in the following way, just when I might begin to comprehend really serious mathematics, my life circumstances, would compel me shuffle off this mortal coil.


« Last Edit: February 28, 2021, 02:31:56 pm by Holden »
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DIVISIBILITY
« Reply #28 on: March 02, 2021, 01:36:17 pm »
Quote from: Holden
As regards Prime numbers,I was wondering if you have looked into the following paper:

https://www.claymath.org/sites/default/files/ezeta.pdf

No, I have not looked at that paper.  I skimmed through a bit of it and see that it is certainly not something you would glean any insight from.  I certainly would not wrack my brains over it.   You would have to study many years just to understand the notation.   That paper is not a bridge to deeper understanding.

Returning again and again to the fundamentals is far more important (for us mere mortals) than stumbling across major breakthroughs which only the over-educated are privy to deciphering.  That is, you will get more satisfaction developing authentic understanding of the fundamental structure of the integers, and the role prime numbers play in Integer Factorization.

This is why a bridge between high school mathematics and advanced mathematics is so crucial, and I believe that this bridge has to be crossed, even for those who have "passed" or done well in advanced mathematics courses.    You see, you will always have the fundamental structure.

Don't let the priesthood-like academic hierarchy intimidate you into pretending to understand things that you have not honestly wrestled with.  I promise you that there are plenty of less advanced concepts and  methods which would afford you more immediate satisfaction.

Topics such as divisibility (of integers) will give you immediate insight into the nature of prime numbers.   Remember back in 2014?

Quote
Do you remember when I tried to lay off the booze back in October 2014, and I took out a weird math book by native German Gerardo Joffe from the library next to my apartment?  Teaching Old Dogs New Tricks : See Section IV: DIVISIBILITY.


IV: DIVISIBILITY


[2]  Of course, all even numbers and only even numbers are divisible by 2


[3]  A number is divisible by 3 if its sum of its digits is divisible by 3, 6, or 9

Try 46,215      4+6+2+1+5 = 18 ---> 1+8 = 9   Therefor 46215 mod 3 = 0  YES


[4]  A number is divisible by 4 if the number formed by its last 2 digits is divisible by 4.

{00, 04, 08, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96}


[5]  Divisible by 5 if it ends in 5 or 0


[6]  For a number to be divisible by 6, since its factors are 2 and 3, it has to be even, and the sum of its digits must be 3, 6, or 9

Try 352,812   even   3+5+2+8+1+2 = 21 ---> 3    YES



[7]  {07, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98}

To determine the divisibility of a number by 7, separate the last two digits from that number, double the number formed by the remaining digits. And add the result to the last 2 digits.  If the result of this operation is a number greater than 2 digits, repeat the operation: separate the last 2 digits of the number, double the number formed by the remaining digits, and add it to the last 2 digits.  Repeat, if necessary, until the result is reduced to 2 digits.  If that 2 digit number is divisible by 7, then the original number is divisible by 7.  If the 2 digit number is not divisible by 7, then the original number is not divisible by 7.

Try 9978   99 | 78      (2*99) + 78 = 198 + 78 = 276   Repeat:  2 | 76      4+76 = 80
80 is not divisible by 7, so 9978 is not divisible by 7

Try 92813   928 | 13   (2 * 928 = 1856) + 13 = 1869   Repeat      18 | 69     
36+69 = 105      Repeat      1 | 05      2+5 = 7
Therefor, 92813 is divisible by 7



[8] In any number of 3 digits or more, the third digit from the right (hundreds) is either even or odd.

If that number is even, for the whole number to be divisible by 8, the last 2 digits have to be divisible by 8.  If that number is odd, the last 2 digits must be divisible by 4 but may not be divisible by 8, for the whole number to be divisible by 8.

Note that 2 digit numbers divisible by 8 is a subset of the set of 2 digit numbers divisible by 4:

{00, 08, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96}

Try 32428   4 is even: 28 is not divisible by 8, so neither is 32428

Try 42548   5 is odd: 48 is divisible by 4, but it is also divisible by 8, so 42548 is not divisible by 8

Try 23632   6 is even: 32 is divisible by 8, so 23632 is indeed divisible by 8



[9]  In order for a number to be divisible by 9, the sum of its digits must be 9   AMAZING!

11307      1+1+3+0+7 = 12    NOPE
111202      1+1+1+2+0+2 =  7   NOPE
226224   2+2+6+2+2+4 = 18   YES



[10]  The number has to end in 0.  Trivial, I know.


[11]  In order for a number to be divisible by 11, the difference of its sums of its alternate digits must be either 0, 11, or a multiple of 11.

2547      12 – 6 = 6   NOPE
36985      17 – 14 = 3   NOPE
505802   10 – 10 = 0   YES
93819      26 – 4 = 22     YES



[12]  prime factors of 12: 2 * 2 * 3

For a number to be divisible by 12, the number formed by the last two digits has to be divisible by 4, and the sum of the digits must be 3, 6, or 9.

Try 13917:  Forget it.  It's odd.  Obviously an odd number can't be divisible by 4.

Try 28972   72 is divisible by 4      2+8+9+1+2 = 28 ---> 10 ---> 1   NOPE

Try 18408   8 is divisible by 4      1+8+4+0+8 = 21 ---> 3   YES



We will go beyond what we have learned, understanding that all numbers, except primes, are made up of prime factors, such as 2, 3, 5, 7, 11, etc., just as 6 was made up of 2 and 3, and 12 made up of 2 and 2 and 3 OR 4 and 3.

Take, for instance 14: prime factors 2 and 7.  We know, in order to be divisible by 2, the number has to be even.  To be divisible by 7, it has to come out with a 2-digit number divisible by 7.

Try 8792   even      87 | 92      (2 * 87 = 174) + 92 = 266      2 | 66      70
70 is divisible by 70, so, YES,  8792 is divisible by 14

The same reasoning holds for 15 and other numbers.  It has to pass the divisibility test for divisibility by 3 (sum of digits must be 3, 6, or 9) and the number must end in 0 or 5.

29610      2+9+6+1+0 = 18 ---> 9   YES
1855      1+8+5+5 = 19 ---> 10 ---> 1  NOPE


Maybe I will take some notes on primes … that's what interests me (besides NOTHING, the void).


Note that the Fundamental Principle of Arithmetic is that “every positive integer can be expressed in one and only way as a product of prime numbers".

I find that principle to be cool.
« Last Edit: March 02, 2021, 05:20:56 pm by Sticks and Stones »
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The Prime Number Theorem
« Reply #29 on: March 02, 2021, 04:34:20 pm »
As a Hentrician Footnote to the paper mentioned by Holden::: On the Number of Prime Numbers less than a Given Quantity :

The Prime Number Theorem: approximating π(x)

The Prime Number Theorem: The number of primes not exceeding x is asymptotic to x / [ ln( x )].

ln(x) ---> Natural Logarithm of x

Sage has a function called prime_pi(n) which takes one argument, an integer n.  It returns the number of primes less than n.

See quick reference for Sage:  Elementary Number Theory

It was actually my interest in Number Theory in 2015 which got me so interested in SageMath (and SymPy) in the first place.

also:  Number Theory commands

http://mathsci.kaist.ac.kr/cms/wp-content/uploads/2017/11/NumberTheory_Sage.pdf

Here is an example of a C++ solution to counting the number of primes less than given positive integer:


#include <iostream>
#include <cmath>

// using namespace std; /* bad practice, best to use full :: notation */
class Solution {
public:
    int count_Primes(int n) {
        int ctr = 0;
        for (int i = 2; i < n; i++) {
            if (is_Prime(i)) {
                   ctr++;
            }
        }
        return ctr;
    }

bool is_Prime(int n) {
        int n_ = int(sqrt(n));
        for (int i = 2; i <= n_; i++) {
            if (0 == n % i) {
                return false;
            }
        }
        return true;
    }
};

int main() {
    Solution *solution = new Solution();
    std::cout << "Enter positive integer: ";
    int n;
    std::cin >> n;
    std::cout << "Number of prime numbers less than " << n << " is " <<  solution->count_Primes(n) << std::endl;
    return 0;
}

https://www.w3resource.com/cpp-exercises/math/cpp-math-exercise-24.php

Here is an example of a solution in C which displays the primes before displaying the count:

    //To find the prime numbers up to any number
    #include<stdio.h>
     
    int isPrime(int n)
    {
        int i,val=1;         
       for(i=2; i<n; i++)
       {
          if(n%i == 0)
          {
             val = 0;
             break;
          }
       }
       return val;
    }
     
    int main()
    {
       int j,n,count=0;
       printf("Enter a number: ");
       scanf("%d",&n);
       printf("Prime numbers upto %d are: ",n);
       for(j=2; j<=n; j++)
       {
           if(isPrime(j))
           {
               printf("%d ",j);
               count++;
           }
       }
       printf("\nTotal primes found = %d",count);
    }


Here is an online C compiler

Just copy everything from  #include<stdio.h> down to the last curly brace } of main().   Then paste into compiler editor and RUN.

I think that your interest in prime numbers and Number Theory in general will serve as some kind of motivation for you to learn just enough C, C++, or even Python within [SymPy/Sage] to assist you in your explorations.
« Last Edit: March 03, 2021, 02:41:49 am by Sticks and Stones »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~