Ivan's island
  • Home
  • Blog
  • CATEGORIES
  • Rules

The math of sorrow

3/21/2023

0 Comments

 
If you had the misfortune of first having colleagues and only then friends, there would exist a danger of another misfortune: to think that friendship is also commutative.
0 Comments

Infinitely many ways to prove the infinitude of primes, 2

3/19/2023

0 Comments

 
Let us start with the observation that for m > 5 the n-th m-gonal pyramidal number is of the form (1/6)*n*(n + 1)*((m – 2)*n – m + 5) (details here).

​The positive m-gonal pyramidal number P and the m-gonal pyramidal number whose index is (6*P + 1) are coprime since the latter is of the form

(1/6)*(6*P + 1)*(6*P + 2)*((m – 2)*(6*P +1) – m + 5) = … =
= (3*P + 1)*(6*P + 1)*(2*P*(m – 2) + 1).

Therefore, for any m > 5 any sequence whose first term a(1) is a positive m-gonal pyramidal number and whose general term is of the form 
a(n) = (3*k + 1)*(6*k + 1)*(2*k*(m – 2) + 1), 
where k = Product_{i=1..n-1} a(i), is a sequence of pairwise coprime m-gonal pyramidal numbers. 

The above implies, by the Fundamental theorem of arithmetic, that there are infinitely many ways to prove the infinitude of primes.

Example:
By taking as a seed the first hexagonal pyramidal number 1 (see OEIS A002412), we can construct the following sequence of pairwise coprime hexagonal pyramidal numbers: 
1, 252, 2310152797, 28410981127871160285705816883937448685, ...
0 Comments

Civilization = Optimization

3/4/2023

0 Comments

 
Instead of feeding thousands with 5 loaves and 2 fishes, today you can feed billions with several lemons from the virtual trees of Facebook and YouTube.
0 Comments

Infinitely many easy ways to prove the infinitude of primes

2/26/2023

0 Comments

 
Elaborating on my previous text called "One easy way to prove the infinitude of primes", I found the following

Theorem
Let m = 2l, for any l > 0. There are infinitely many sequences of pairwise coprime m-gonal numbers, whose first term a(1) is any positive m-gonal number and whose general term a(n) is of the form a(n) = (k +1)((l – 1)k + 1)), where
k = Product_{i=1..n-1} a(i).


Corollary
The fact that any such sequence is infinite implies (by the Fundamental theorem of arithmetic) the infinitude of primes.


Example
Let us take for example the Tetradecagonal numbers (where m = 14 and l = 7) and take the second 14-gonal number 14 as the first term of the new sequence (NS). The general term of NS is of the form a(n) = (k +1)(6k + 1)),
where k = Product_{i=1..n-1} a(i) 
and
NS = {14, 645, 244658821, 14642610579551886703145221, ...}.
0 Comments

One easy way to prove the infinitude of primes

2/23/2023

0 Comments

 
Improvising on the ideas of Wacław Sierpiński*, I found that infinitely many sequences of pairwise coprime octagonal numbers can be constructed, their first term a(1) being equal to any positive octagonal number and their general term being of the form a(n) = (k+1)(3k+1), where k = Product_{i=1..n-1} a(i).

Any such sequence is infinite, which implies (by the Fundamental theorem of arithmetic) the infinitude of primes.

Example
One such sequence is
8, 225, 9727201, 919691230011613567201,...

___________________________________________________________________
* 
 see Problems 42 and 43 here, which involve triangular and tetrahedral numbers
0 Comments

Enrico Fermi, lover of paradoxes

2/22/2023

0 Comments

 
Instead of asking why there are so few intelligent beings on Earth, Enrico Fermi asked why there are so few of them in the Universe.

Вместо да пита защо са толкова малко разумните същества на Земята, Енрико Ферми питал защо са толкова малко във Вселената.
0 Comments

Happy Ending Problem (with wine glasses)

1/30/2023

0 Comments

 
The Happy Ending Problem (named so by Paul Erdős as it led to the marriage of his friends Esther Klein and George Szekeres) was originally stated by Esther Klein in the following manner:

There are five points on a flat surface, no two of which are coinciding and no three of which are on a straight line. Prove that four of these points are vertices of a convex quadrilateral.

This is the latest and simplest proof (outline) of mine, based on the fact that five points in general position always form a pentagon. There are only two cases to consider.

Case 1. If the pentagon is convex we are done (as four of its vertices form a convex quadrilateral).​

Case 2. If the pentagon is concave let us rotate it until one of its edges becomes parallel to the X axis, so that we use the pentagon as a wine glass and pour wine through the diagonal that is outside the pentagon, if this is possible. There exist only four types of glass, which we call Flat Bottom, Flat Top, Flat Middle and No Wine, and for every one of them we can easily construct a convex quadrilateral (see below).
Picture
0 Comments

Happy Ending Problem, 4+1=5

1/28/2023

0 Comments

 
The Happy Ending Problem (named so by Paul Erdős as it led to the marriage of his friends Esther Klein and George Szekeres) was originally stated by Esther Klein in the following manner:
There are five points on a flat surface, no two of which are coinciding and no three of which are on a straight line. Prove that four of these points are vertices of a convex quadrilateral.

This simple theorem has a simple proof but, as the unheard of concept of the so called convex hull is involved, the proof could be unintelligible to non-mathematicians, high school students inclusive. So, here is the latest proof (outline) of mine, in which you can meet the familiar points and lines only. Let us remember that four of those five points form a quadrilateral and let us call the other point extra point. There only two cases to consider.

Case 1. If the quadrilateral is convex we are done.

Case 2. If the quadrilateral is concave let us number its vertices in such a way that 2 is the vertex at the reflex angle of the quadrilateral and 1 is the only vertex, such that the line between 1 and 2 dissects the body of the quadrilateral (we call this line Green Line).
Picture
Picture 1
The Green Line divides the plane into two parts, which we call Left and Right. Let us analize only the Left one (having in mind that the same thinking can be applied to the Right one).

Let us draw the line between points 2 and 3 (3 is the vertex in the Right half of the plane) and call it Red Line. Thus, the Left half of the plane is divided into 7 parts. No matter in which of those 7 parts the extra point may be, we can construct a convex quadrilateral following the suggestions on Picture 1, where x is the name of the extra point.
0 Comments

HEP for Beginners, 2

1/14/2023

0 Comments

 
The Happy Ending Problem (named so by Paul Erdős as it led to the marriage of his friends Esther Klein and George Szekeres) was originally stated by Esther Klein in the following manner:
There are five points on a flat surface, no two of which are coinciding and no three of which are on a straight line. Prove that four of these points are vertices of a convex quadrilateral. 

​This simple theorem has a simple proof but, as the complex concept called "convex hull" is involved, the proof could be unintelligible to non-mathematicians, high school students inclusive. So, here is the latest proof of mine, in which you can meet the familiar points and lines only.

Let us start with the observation that any three of those five points form a triangle and let us call the other two points extra points. Where can the extra points be? There are five cases to consider.

Case 1. There exists at least one extra point in an OK part of the plane (see Fig. 1 below). The convex quadrilateral is easy to construct.
Picture
Fig. 1
Case 2. There exist two extra points in two ? parts of the plane. (see Fig. 2 below). The convex quadrilateral is easy to construct.
Picture
Fig. 2
Case 3. There exist two extra points in one ? part of the plane (see Fig. 3 below). The line connecting those points may intersect zero or two sides of the triangle. Connect those extra points to an unintersected side of the triangle and there it is, our convex quadrilateral.
Picture
Fig. 3
Case 4. There exists two extra points inside the triangle. There is no need to draw here, the reasoning is the same as in Case 3. The only difference in Case 4 is that the line between the extra points always intersects two sides of the triangle.

Case 5. There exist one extra point inside the triangle and one in ? part of the plane (see Fig. 4 below). The line connecting those points will always intersect two sides of the trianble. Start your "walk" from the extra point outside the triangle to the one inside it. Take the points of the side you cross first, connect them to the extra points and there you have it, our convex quadrilateral.

Picture
Fig. 4
We have just exhausted all possible arrangements of a triangle and two points in a plane and proved that a convex quadrilateral can be built in each and every one of them. Q.E.D. 
0 Comments

10 percent of the brain, the rest goes down the drain

1/8/2023

0 Comments

 
They say humans use only 10 percent of their brain capacity. Wrong! It is HUMANITY that uses only 10 percent of human brain capacity.

Казват че човек използва само 10% от мозъчния си капацитет. Грешка! ЧОВЕЧЕСТВОТО използва само 10% от мозъчния капацитет на човека.
0 Comments

Don't interrupt the wise one

1/7/2023

0 Comments

 
Wise men's thinking and talking is regularly interrupted, either by handclapping or by swearing. That is bad, wise men do not want to be applauded (let alone cursed), but to be heard and (possibly) understood.

"Ask for anything and I'll gladly oblige," said Alexander the Great to Diogenes. "Move to the right, you are blocking my sun," answered Diogenes, who was basking in the sun. Alexander left and said to his companions, "Had I not been Alexander, I should have liked to be Diogenes.
" That was the only time in history when a wise man was heard and understood without any interruption.

There is something about this story that bothers me, it seems too contrived. The story of Archimedes, the circles and the sword that interrupted his thought and life sounds much more realistic.
0 Comments

Wisdom

12/7/2022

0 Comments

 
Wisdom is just like our Universe. It can arise out of nothing and can disappear without any trace whatsoever.

This is exactly what happened to the Aristotelian wisdom that heavy bodies fall faster than light ones and that women have fewer teeth than men. It was born out of nowhere, without any connection to reality. Later, it went into the nothingness, thanks to the mental efforts of Galileo and that unknown man who first decided to count his wife's teeth.
0 Comments

Euclid's big mistake

11/26/2022

0 Comments

 
When the pharaoh asked whether geometry could not be made easier, Euclid replied: "There is no royal road to geometry". He was badly mistaken. Not only is there a royal road to geometry (and mathematics as a whole), but there is also a road for ordinary noblemen. Marquess de l'Hôpital decided to pay annual fee to Johann Bernoulli in order to drive on it.
0 Comments

Picasso's great mistake

11/20/2022

0 Comments

 
"Everything you can imagine is real," Picasso used to say. He had never heard of the imaginary numbers, poor fellow.

"Всичко въображаемо е реално", казвал Пикасо. Не бил чувал за имагинерните числа, горкият.
0 Comments

Time is money ...

11/17/2022

0 Comments

 
... is plastic surgeon's motto.
0 Comments

Earthlings on the Moon

11/15/2022

0 Comments

 
Armstrong and Aldrin left a plaque on the Moon, which reads as follows: "We came in peace for all mankind." Then they stuck the American flag next to it. A century earlier, as if especially for this occasion, Kozma Prutkov warned people that when they see the elephant's cage labeled "Buffalo", they must not believe their eyes.
0 Comments

Henrik Ibsen's big mistake

11/10/2022

0 Comments

 
A picture is worth a thousand words, Henrik Ibsen thought. He was badly mistaken. If Nike's swoosh could say it all, their slogan "Just do it." would be superfluous.
0 Comments

Color blindness

10/18/2022

0 Comments

 
Sometimes even colorful people see the world in black and white.

Artist X said about artist Z that he was a white crow, which was why he was never accepted by the people of black&white thinking. Doesn't that sound stupid? Dividing people into people with colorful and people with black&white thinking is a manifestation of binary, i.e. black&white, thinking.
0 Comments

Men and women

9/28/2022

0 Comments

 
Ambiguity is not necessarily a bad thing, sometimes you profit from it. When a woman asks you if the dress makes her look fat, it's wise to be vague about the answer. This is what a writer with a lot of life experience wrote.

My life experience makes me think otherwise: in the above situation, the clear and honest answer is preferable. You answer NO and you are always right:
a) When a woman is skinny, a dress cannot make her fat.
b) When she is fat, the fault is not in the dress.
0 Comments

Civilization

9/7/2022

0 Comments

 
We are so enlightened we can see how the Universe will die of cold (Heat death). We are so blind we can't see how we make the Earth sweat (Global warming).
0 Comments

Weapons and warriors

8/31/2022

0 Comments

 
"I hated war, but I loved the warrior spirit. I hated the sword, but loved the samurai." is what Phil Knight wrote in his autobiography. I smell something fishy there. Without the samurai the samurai sword is a work of art. Without his swords the samurai is nothing.
0 Comments

Best anti-ad ever

8/26/2022

0 Comments

 
Armani can't buy me love
0 Comments

Normalcy

8/21/2022

0 Comments

 
Real people are like real numbers: almost all of them are supposedly normal, but if you want to point which ones are, you'll face great difficulties.

Реалните хора са като реалните числа: уж почти всички са нормални, но aко се наложи да посочиш кои, ще срещнеш големи трудности.
0 Comments

Politics

8/11/2022

0 Comments

 
Let A be the set of your friends and B be the union of C (the set of your friends' friends) and D (the set of your enemies' enemies). Politics is an effective nonmathematical instrument that can easily disprove your naive faith that B is a subset of A.
0 Comments

One flew over the Newton's nest

7/27/2022

0 Comments

 
Giants dislike people standing on their shoulders, as it is them who enjoy standing on the shoulders of others.
0 Comments
<<Previous

    RSS Feed

    This website uses marketing and tracking technologies. Opting out of this will opt you out of all cookies, except for those needed to run the website. Note that some products may not work as well without tracking cookies.

    Opt Out of Cookies

    Categories

    All
    Alan Turing
    Aphorisms
    Art
    Asymmetries
    Bacillus Bulgaricus
    Economics
    Environment
    History
    Hr
    InEnglish
    Intelligence
    Language
    Mathematics
    Music
    Paradoxes
    Politics
    Psychology
    Reading&writing
    Seriouslessness

    Archives

    March 2023
    February 2023
    January 2023
    December 2022
    November 2022
    October 2022
    September 2022
    August 2022
    July 2022
    June 2022
    May 2022
    April 2022
    March 2022
    February 2022
    January 2022
    December 2021
    November 2021
    October 2021
    September 2021
    August 2021
    July 2021
    June 2021
    May 2021
    April 2021
    March 2021
    February 2021
    January 2021
    December 2020
    November 2020
    October 2020
    September 2020
    August 2020
    July 2020
    June 2020
    May 2020
    April 2020
    March 2020
    February 2020
    January 2020
    December 2019
    November 2019
    October 2019
    September 2019
    August 2019
    July 2019
    June 2019
    May 2019
    April 2019
    March 2019
    February 2019
    January 2019
    December 2018
    November 2018
    October 2018
    September 2018
    August 2018
    July 2018
    June 2018
    May 2018
    April 2018
    March 2018
    February 2018
    January 2018
    December 2017
    November 2017
    October 2017
    September 2017
    August 2017
    July 2017
    June 2017
    May 2017
    April 2017
    March 2017
    February 2017
    January 2017
    December 2016
    November 2016
    October 2016
    September 2016
    August 2016
    July 2016
    June 2016
    May 2016
    April 2016
    March 2016
    February 2016
    January 2016
    December 2015
    November 2015
    October 2015
    September 2015
    August 2015
    July 2015
    June 2015
    May 2015
    April 2015
    March 2015
    February 2015
    January 2015
    December 2014
    November 2014
    October 2014
    September 2014
    August 2014
    July 2014
    June 2014
    May 2014
    April 2014
    March 2014
    February 2014
    January 2014
    December 2013
    November 2013
    October 2013
    September 2013
    August 2013
    July 2013
    June 2013
    May 2013
    April 2013
    March 2013
    February 2013
    January 2013
    August 2012

    See also

    My contributions to OEIS

Powered by Create your own unique website with customizable templates.