Let k be a non-negative integer, n be a positive integer and
be the sum of the k-th powers of the positive integer divisors of n. Let
be the sum of the k-th powers of the odd positive integer divisors of n.
Is it true that for every k and n the latter sigma divides the former sigma? In other words, is it true that for every k and n
I believe it is true.
Nassim Taleb invited few friends to drink coffee and discuss antifragility. Fat Tony tasted the coffee and asked Taleb about the coffeemaker he uses. Instead of information related to the coffemaker’s brand or the technology used he was given the following lecture:
Every time you use a coffeemaker for your morning cappuccino, you are benefiting from the fragility of the coffeemaking entrepreneur who failed. He failed in order to help put the superior merchandise on your kitchen counter.
“Man you got it totally wrong. Based on the taste of your coffee I can tell you have a coffeemaker by a coffeemaking entrepreneur who failed,” Fat Tony said.
Nine is the only natural number n such that the sum of the digits of the first n natural numbers and the sum of the digits of the next n natural numbers are equal.
It was Yuval Noah Harari who started this train of thought (there) and I’ll finish it differently here.
God helps those who help themselves and does not help those who do not help themselves. Antibiotics are more powerful than God as they help:
a) those who help themselves, i.e. those people X who take antibiotics when in need, and
b) those who do not help themselves, i.e. those people Y who do not need to take antibiotics, i.e. those Y that have not been infected by X because X had been cured by the antibiotics before they met Y.
With my very eyes I saw how this Prometheus lighted his fire and can honestly tell you, gentlemen, it was nothing special. Any slacker or goatherd could do it …..
Apocryphal Tales, Karel Čapek
G. H. Hardy once said that any fool could have guessed Goldbach's conjecture. He was wrong since there had lived so many fools before Golldbach's time but none of them had guessed the conjecture (that is why Goldbach's name is attached to it).
One might think that any fool could have made G. H. Hardy's remark. But the reasoning in the first paragraph can be applied to the last sentence. Therefore, though G. H. Hardy's remark sounds foolish it is not true that any fool could have made it.
There is a vast area of foolishness reserved for the clever people. Blessed are the fools, for they are unable to explore this "special" area.
Let n be a natural number, D be the set of the divisors of n and sigma(n) be the sum of the elements of D. A number is a Zumkeller number if and only if D can be divided into two disjoint subsets D1 and D2 such that both sums of the elements of D1 and D2 equal sigma(n)/2.
Prove (or disprove) that out of every four consecutive Zumkeller numbers there exists at least one number k such that sigma(k)/2 is also a Zumkeller number.
For the present-day fool "to be, or not to be" is not a question but a question of taste.
За съвременния глупак "да бъдеш, или да не бъдеш" не е въпрос, а въпрос на вкус.
To ask questions like "Where good ideas come from?" is not a good idea. Those with the good ideas do not ask themselves where their ideas come from but what to do with them.
Да питаш откъде идват добрите идеи не е добра идея. Хората с идеите не се питат откъде идват идеите им, а какво могат да направят с тях.
Thank God that Just do it is the philosophy of a company producing sneakers, not nuclear power plants.
Благодаря ти, Боже, че Just do it е философията на производител на маратонки, а не на атомни електроцентрали.
Barbarians, ancient and contemporary, look alike in their belief they worship the best possible God and inhabit the best possible world.
Древните и съвременните варвари си приличат във вярата си че се кланят на най-добрия от всички възможни богове и живеят в най-добрия от всички възможни светове.
Everybody wants to make a difference. That is why everything is getting smaller.
There exist natural numbers N, whose names* share at least one letter with the name of every other natural number. Find some N.
* in American English, where 250 is "two hundred fifty", not "two hundred and fifty"
There is an interesting integer sequence recursively defined in the following manner:
a(0) = 0, a(1) = 1, a(n) = 34*a(n-1) - a(n-2) + 2.
Let sigma(n) be the sum of the divisors of n, k be a nonnegative integer and
m = 2*k + 1.
Prove that for every m, sigma(a(m)) < 2*a(m).
Euclid was able to see the infinitude of primes but was blind to the existence of the number zero. Gregor Mendel left his fingerprint on genetics but did not leave a genetic fingerprint.
Евклид можа да види безкрайността на простите числа, но остана сляп за съществуването на числото нула. Грегор Мендел можа да остави отпечатък в генетиката, но не можа да остави генетичен отпечатък.
Alice and Bob are friends. Alice is an optimist and always sees similarities. Bob is a pessimist and always sees differences. Yesterday they learned that tobacco is more addictive than Facebook.
"It's obvious. There are so many differences between tobacco and Facebook," said Bob.
"You are wrong. They are similar, as they both shorten one's life," countered Alice.
"On the one hand, you are right. But there's a material difference. Tobacco shortens one's life by shortening its end, i.e. life's worst part. Facebook shortens one's life by shortening its middle, i.e. life's best part. That is why it's easier to give up on Facebook," closed the discussion Bob.
Ever the optimist, Amos Tversky used to say that pessimists are unhappy twice: when they predict bad things will happen and when the bad things actually happen. Some pessimist once replied that the pessimists are always happy: when bad things happen, since they are proven right, and when bad things do not happen, since they avoid the bad things.
It seems that Tversky, the optimist, was too pessimistic about pessimism, and the pessimist was too optimistic about it. Who was the pessimist and who was the optimist then?
Albert Szent-Györgyi, a Nobel prize winner in physiology, said that discovery consists of seeing what everybody has seen, and thinking what nobody has thought. This was independently discovered by another Nobelist - Erwin Schrödinger - who said that the task is...not so much to see what no one has yet seen; but to think what nobody has yet thought, about that which everybody sees.
A true discovery is hard to make. Even if you are a Nobelist. Even if what is discovered is only an aphorism.
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:
Given any five points on a flat surface that are in general position, i.e. no two of them coinciding and no three of them in a straight line, prove that four of these points will always form a convex quadrilateral.
Let us start with the observation that a line can intersect three sides of a triangle at most (see below).
Therefore, a side (or its extension) of a triangle can intersect three sides of another triangle at most. If we impose the additional condition that no three vertices of the two triangles are in a straight line, then any side (or its extension) of any of the two triangles can intersect two sides of the other triangle at most. Therefore, the set of points where the sides (or their extensions) of a triangle intersect the sides of another triangle can consist of six points at most. Therefore, for every two triangles T1 and T2, there always exist a pair of sides s1∊T1 and s2∊T2, such that neither s1 nor its extension intersects s2 (and vice versa). Let us call such a pair ‘useful’ (see the black pair on the picture below).
As any three points in general position form a triangle, any five points in general position form two triangles sharing a vertex. By connecting a useful pair of sides (see below) we can always form a convex quadrilateral. Q.E.D.
No rule applies to those who write it. That is why fair and democratic elections put into office unfair and undemocratic people. That is why in the real world the Barber paradox is a no-brainer.
What did Wittgenstein mean when he said, “If the lion could speak, we could not understand him”? Think about the following conversation:
A: What is your zodiac sign?
A: There is no such sign.
B: I am very fine, thank you.
A: What the …bleep… are you talking about?
B: You are full of anger. I am sure this is the meat in your diet talking.
Sartre said that hell is other people. Wittgenstein probably thought that lion means other people.