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.
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Let us start with the observation that for m > 5 the nth mgonal pyramidal number is of the form (1/6)*n*(n + 1)*((m – 2)*n – m + 5) (details here).
The positive mgonal pyramidal number P and the mgonal 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 mgonal 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..n1} a(i), is a sequence of pairwise coprime mgonal 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, ... 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.
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 mgonal numbers, whose first term a(1) is any positive mgonal number and whose general term a(n) is of the form a(n) = (k +1)((l – 1)k + 1)), where k = Product_{i=1..n1} 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 14gonal 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..n1} a(i) and NS = {14, 645, 244658821, 14642610579551886703145221, ...}. 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..n1} 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 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. 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). 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 nonmathematicians, 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). 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. 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 nonmathematicians, 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. Case 2. There exist two extra points in two ? parts of the plane. (see Fig. 2 below). The convex quadrilateral is easy to construct. 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. 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. 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.
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. 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. 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. 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.
"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. 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.
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.
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. 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. 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).
"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.
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.
Giants dislike people standing on their shoulders, as it is them who enjoy standing on the shoulders of others.

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