Letter Bill 0..33: Fill & Download for Free

Oh, this is going to be fun.First, a point of terminology: there are no “counterexamples to mathematical theories”, because a theory isn’t a single statement that can be refuted with a counterexample. Nor are there counterexamples to mathematical theorems, because theorems are true.There do exist interesting, non-trivial and sometimes complicated counterexamples to mathematical statements which, of course, aren’t theorems. Many of those statements have stood as conjectures for a while, until someone found a counterexample.Some of the following counterexamples are profound, some mundane. Some are easy to verify, some aren’t. Some are in Number Theory, some in the Theory of Computation, some in Logic and some in Topology (and maybe a few other fields, we’ll see).Claim: Every number of the form [math]2^{2^n}+1[/math] is prime. (False!)This was a genuine conjecture made by Pierre de Fermat, who verified that [math]2^2+1=5[/math], [math]2^4+1=17[/math], [math]2^8+1=257[/math] and [math]2^{16}+1=65537[/math] are indeed primes.The counterexample comes precisely at the next number, [math]2^{32}+1=4294967297=641 \times 6700417[/math]. This may seem trivial by today’s standards, but when Euler found this factorization it was quite a clever feat (he did this by recognizing that [math]641=2^9+2^7+1[/math] must divide that number).In fact, Fermat was about as wrong about this as can be: as far as we can tell, past [math]n\geq 5[/math] none of the numbers [math]2^{2^n}+1[/math] is prime. We don’t know this for sure, though. We know that [math]2^{2^{32}}+1[/math] is composite (as are all smaller numbers of this form), but nobody knows for sure if [math]2^{2^{33}}+1[/math] is or isn’t prime.Claim (Tait’s Conjecture): Every planar, 3-connected cubic graph has a Hamiltonian cycle. (False!)This was proposed in 1884 by P. G. Tait as a way to prove the famous 4-Color Theorem: Tait showed (quite easily) that the 4-color theorem follows from this statement. It took more than 60 years for Bill Tutte to find this counterexample:This graph is obviously planar, it is obviously cubic (every vertex has exactly three neighbors), it is slightly less obviously 3-connected (it cannot be separated by removing one or two vertices), and it’s not Hamiltonian (there’s no way to visit all the vertices in one continuous path with no loops). This last part isn’t at all obvious but it’s also not super difficult to prove, either. It’s finding this graph that’s the challenge.Claim: If you add up some [math]k[/math]th powers to get a [math]k[/math]th power, you must have added at least [math]k[/math] numbers. (False!)This conjecture was made by none other than Leonhard Euler, as a certain extension of Fermat’s Last Theorem.Euler observed that it’s easy to add up two squares to get a square, as in [math]3^2+4^2=5^2[/math], but you can’t add two cubes to get a cube: [math]a^3+b^3=c^3[/math] has no solution in positive integers (This is the case [math]n=3[/math] of Fermat’s Last Theorem, which Euler was the first to actually prove.) You can, however, add up three cubes to get a cube, for example [math]3^3+4^3+5^3=6^3[/math]. Well, thought Euler, there’s a pattern here: you can add up [math]k[/math] [math]k[/math]th powers to get a [math]k[/math]th power, but you can’t get away with fewer than [math]k[/math] of them.This isn’t true, but it took almost 200 years to find a counterexample.The first counterexample was found in 1966 by Lander and Parkin, using a computer search. They discovered that[math]27^5+84^5+110^5+133^5=144^5[/math].You can find their original paper here. In 1966, this was quite a feat. Nowadays, you can find this example in just a few minutes of coding and running your code on a standard laptop, or even a smartphone (Try it!).This shows that Euler was wrong when [math]k=5[/math]. He already knew that he was right for [math]k=3[/math]. What about [math]k=4[/math]?That took considerably more ingenuity than a mere computer search. In 1986, 20-year-old Noam Elkies used the theory of elliptic curves (plus some computer work) to find infinitely many counterexamples to Euler’s conjecture for fourth powers. The smallest one is[math]2682440^4+15365639^4+18796760^4=20615673^4[/math].Finally, for [math]k \geq 6[/math], the situation is still unknown. It is possible, though in my mind unlikely, that Euler’s conjecture holds for 6th powers and beyond.We aren’t quite done with Euler, though.Claim: There are no Orthogonal Squares of even order which is not divisible by 4. (False!)Look at the following array of cards.A♠ K♥ Q♦ J♣Q♣ J♦ A♥ K♠J♥ Q♠ K♣ A♦K♦ A♣ J♠ Q♥See how clever this is: every row contains each rank (J, Q, K, A) exactly once, as does each column. Also, every row contains each suit (♠,♥,♦,♣) exactly once, as does each column. But moreover, the two arrangements of rank and suits are so mixed up that no card appears twice in the entire array: the Jack of Hearts appears just once, as does the Queen of Spades, and so on for all possible combinations.Without this last “orthogonality” condition, what you’d have here is just two superimposed Sudoku puzzles of order 4. But the requirement to have none of the pairs appear twice makes it a very hard puzzle, hard enough to interest the great Euler. He used latin and greek letters to build those arrays, earning them then name Graeco-Latin square. Here’s such an array of order 3.Euler was able to construct such arrays for every odd order and for every order divisible by 4. It’s easy to see that there’s no such array for order 2 (exercise!), and Euler was unable to build one of order 6, so he proposed that the orders 2, 6, 10, 14, … are impossible. Those are the even numbers that aren’t divisible by 4.Euler was wrong about that, too, and once again it took more than 150 years for this conjecture to be refuted (at this point I should probably mention that Euler was one of the most profound and creative mathematicians of all time, and those conjectural snafus were quite uncharacteristic of him.)Euler was right about order 6, though it was only in 1901 that this was proven. But in 1959, Raj Chandra Bose and Sharadchandra Shankar Shrikhande constructed an orthogonal array of order 22. E. T. Parker found an example of order 10, and later the three of them proved that such arrays exist for every [math]n[/math] except the gaps at [math]n=2[/math] and [math]n=6[/math].Here’s an orthogonal array of order 10: every 2-digit combination appears exactly once, and every row and every column contain every digit in both the first and second places.Claim: There are no Turing degrees between 0 and 0′. (False)!The theory of Turing degrees is a truly beautiful idea. It studies the computational aspects of sets of numbers, and reveals an amazingly intricate structure from a few simple definitions. Here are the minimum basics we need.A set of natural numbers is called computable if you can write a computer program that checks for membership in the set. Since you can write a computer program that checks if a number is prime, the set of primes is computable. Since you can write a computer program that checks if a given number is even and is the sum of two primes, the set of even numbers which are the sum of two primes is computable (We don’t know if this set contains all even numbers greater than 2, which is Goldbach’s conjecture; but the set is nevertheless computable). The family of all computable sets is called 0.Not every set of natural numbers is computable, because there are far too many sets and far too few possible computer programs to detect them. As a concrete example of a non-computable set, consider the set of numbers which encode (in some way) computer programs which halt. No computer program can correctly determine if a number is or isn’t in the set .Now, suppose you are given a magical gadget which can, actually, tell you if a program halts or not. Using this gadget as “black box” (or, in more common parlance, an “oracle”), you can now do better and write computer programs that decide membership in more sets than just the computable ones. Those sets are said to be “computable using a halting oracle”, and the family of those sets is denoted 0′. It also carries the captivating name “The Turing jump of 0”.Emil Post asked the question: are there sets that are harder to verify than 0 but easier to verify than 0′? A bit more formally, is there a set which is not computable but is computable with a halting oracle, and which is itself not powerful enough to serve as a halting oracle? It seems a bit hard to imagine that such intermediate-level problems exist.But they do. Richard Friedberg and Albert Muchnik, independently, developed the idea known as the Priority Method and constructed such peculiar sets. This opened a whole new field of study, exploring the rich structure of Turing degrees and their partial order.Claim: There are no models of ZFC which violate the Continuum Hypothesis. (False!)Georg Cantor, the creator of the Theory of Sets, wondered if there’s an uncountable set of real numbers which cannot be put in correspondence with the real numbers. In other words, is there any cardinality between [math]\aleph_0[/math] and [math]2^{\aleph_0}[/math]?This very famous conjecture became known as the Continuum Hypothesis (CH). In 1938, Kurt Gödel proved that this hypothesis is consistent with ZFC, which is the same thing as saying that there’s a model of ZFC – A “universe of sets” – in which CH is true. Some people believed this was an indication that CH is, indeed, true, which means that there are no models of ZFC in which CH is false.In fact, people generally had no idea how to build models of ZFC other than the “natural” ones like [math]L[/math] which is the model built by Gödel. But Paul Cohen succeeded in doing just that, inventing the fundamental technique of Forcing in 1963. For this he was awarded the Fields medal.People don’t usually think of a model of ZFC+[math]\neg[/math]CH is a “counterexample”, but formally it can be regarded as one: it is a counterexample to the claim that ZFC implies CH, namely that there are no models of ZFC where CH is false. There are, in fact, such models, so Cohen built what might be described as a counterexample.Claim: An uncountable group must have an uncountable proper subgroup. (False!)It’s very easy to construct uncountable groups: the circle, the torus, the group of [math]n\times n[/math] real matrices of determinant 1, and so on. There are tons of such groups, and if you look at them, you’ll see that they all insist on containing uncountable subgroups. Is it possible to create an uncountable group without uncountable proper subgroups?It’s possible, but very far from easy. The first counterexample, dubbed a “Kuroš monster”, was found by Saharon Shelah and published here. It’s beyond my abilities to even sketch the construction.Claim: There are no books dedicated exclusively to counterexamples. (False!)Topology has so many counterexamples, there’s a whole book about them, and this book is so famous, there’s a Wikipedia article about it. Following its success, other books with similar titles popped up in other fields of mathematics.Some of the standard counterexamples in topology are the Cantor set, the long line and the Hawaiian earring. A slightly more exotic counterexample is the “Sticky foot” space constructed by Bing as an example of a countable, connected Hausdorff space. It’s easy to suppose that a connected Hausdorff space must be uncountable, because all the connected spaces everyone knows are uncountable, and clearly the finite ones won’t do. But such weird spaces do exist, and the Sticky Foot space is one of them.Claim: This list is complete. (False!)This claim is nonsense. There are tons of other counterexamples in math, each of which serves as a counterexample to that claim.

In 2007, I got laid off from my high paying biotech job and backpacked through Central America. I was 32 at the time. I took all the money I had and bought Apple stock on margin. It was a good year to buy Apple stock as this was the year the iPhone came out. I moved to LA with plans to fulfill my dream of starting a company. I'd always been in interested in tech and the Internet so I started on a solid year of wantrepreneurship.In 2008, the market came unhinged, the economy experienced the worst recession since the Great Depression and I sold my stock at the bottom, taking huge loses. My start up business hadn't panned out. I'd made very little progress on my idea, and eventually partnered with a friend to raise money for a Facebook apps company. Neither of us knew a thing about Facebook apps.What little personal savings I had left was drying up as we started talking to investors. I didn't have a job and was hoping to raise money to pay the bills and build a company. Then the financial crisis struck. No investors would talk to us. My friend had a job, I was pretty much out of money. By some miracle, unemployment benefits got extended, and I was eligible from having been laid off a year and half earlier. I don't understand why.I was now completely broke except for my biweekly unemployment checks which barely covered rent and food. I was 33 year old, hadn't worked in a year and half and was ostensibly changing careers/industries yet had no experience in the new industry. I was looking for a job as the unemployment rate was rising at rate faster than it had in nearly seventy years. It was a historically bad time to be looking for a job, especially in an industry where I had no experience.At one point, I couldn't pay my cellphone bill. I had no idea, but Verizon has a payment plan when you're back due.I spent months applying to jobs and writing cover letters. I got coffee and met with anyone and everyone I could. This eventually led to an interview and offer from MySpace (at the time bigger than Facebook and a hot company). I was going to start the following week, then the offer letter got delayed one week. Then another week, then a third. Then MySpace put on a hiring freeze.I couldn't pay my credit cards, and creditors were calling non stop. At first I tried to answer the calls, then I just stopped. There wasn't anything I could do. The news on the economy got worse and worse.I kept applying to jobs and tried to keep a positive outlook. That was all I could do. My unemployment benefits were about to end. I didn't know how I was going to pay rent.A couple weeks before they expired, a friend of friend heard I understood social media and asked me to come in and present to her team. I drafted a strategy for her and presented for a couple hours. I charged $200, a nominal amount but it was a massive boost me for emotionally.A week or so later, my neighbor on a whim passed on my resume for an open position onto a company where he freelanced. A month later I was head of sales at a top digital Hollywood production company (really the only one of its kind). I felt like I'd won the lottery.This is just the short version. There were substantially more challenges and adversities I encountered than I've listed or have space to.What I learned:You can only fix one problem at a time. Don't go out and try to fix your life. Focus and solve only the biggest problem you have. Once you've fixed that, move onto the next one. So you're not bald, broke, unemployed and friendless. You're just unemployed. Go get a job and you've fixed that problem. Then move onto the next.Don't create new problems for yourself. It doesn't sound to me like you'd have any problem working in an office job. Are you crippled, maimed or cognitively impaired? No, then you can handle a job. I have to use my left hand for my mouse because of tennis elbow. I don't think twice about it anymore. Don't fortune tell problems or scenarios that don't exist. As Mark Twain said, "I have known a great many troubles, most of which never happened."Don't define yourself by your problems. No ones life is perfect, far from it. Think of who you'd be if you didn't have your problems. Probably the exact same person. So rediscover who you are and just start over from that point. When I was growing up playing hockey and we were getting whopped by another team, our coach would huddle us together between periods, tell the score was 0-0 and that our job was to win the next period, not the game. The game didn't seem winnable at this point, but the opposing teams were always caught of guard in the next period by a team they'd mostly counted out. I don't know that we ever lost a period after those talks, and we came back to win a surprising number of games from a big deficit.Create a routine. You don't have a job. You're job now is to get a job. Wake up early, have a morning routine and go somewhere (coffee house etc) where you will work at getting a job for eight hours a day. Now that doesn't mean writing resumes for that long. It can and should include networking, reading, learning, informational interviews, improving job skills, etc. But create a schedule for yourself. You should make exercise and the outdoors part of the schedule. Research shows that exercise helps alleviate depressions and the NY Times just published an article on another study that shows How Walking in Nature Changes the Brain.Reframe. When I was applying for jobs after being essentially unemployed for 18 months, I listed the 6 months of travel/backpacking I did as "travel sabbatical" on my resume. A director at one of the companies I interviewed at asked about my travels with envy. During my year of wantrepreneurship, I didn't successfully launch the company I envisioned, but I did learn valuable skills such as blogging, social media, SEO and others. In my interviews, I talked about these skills and my successes, not the fact I couldn't ultimately raise money and folded the enterprise. Use your time being unemployed, as a chance to learn new job skills, volunteer, pick up a hobby and/or travel. If you don't think you have the money, a subscription to Lynda runs $25/month. There's more there to learn than you could possibly cover in a lifetime.Get inspired. You're far from the first or only one in your predicament. Countless others have overcome difficulties far worse than yours. It can help immensely to watch and read their stories. James Altucher writes a great blog about personal development and shares his own ups and downs in his career. Nick Vujicic was born without arms and legs. That's right, no arms or legs. He was bullied so badly that he contemplated suicide. He's now a hugely successful motivational speaker. Watch his Ted Talk:Embrace uncertainty. This comes from Deepak Chopra's "Seven Spiritual Laws of Success" which I'd recommend reading and rereading. It's really only part of Deepak's message, but one that ties very deeply to encountering difficulties. Acting from a place of fear or depression won't do anything to help you overcome your situation.See the abundance. When times are hard, you see what's missing and what you don't have. The world feels scarce, because you're focused on the scarcity. But there's another side to it. Think of what's abundant in the world. Make a list. You don't have a job, but that can change very quickly. The U.S. economy is adding 200,000 new jobs a month! Why can't one be yours?Become grateful. Practice gratitude. Read "The Magic." You'll start to see that how even at what may seem like the lowest point in your, you have countless things to be grateful for. When you see that and recognize that everything will change.Follow my adventures on Instagram, Twitter and read more on My Blog.My most popular Quora answers:What is the most effective yet efficient way to get rich?Why do charismatic people easily get what they want in life?What is the quickest way to get people to trust you?How can I overcome the fear of failure?

My 10 year old daughter got a 271818818728% on her test. Normally, her teacher never gives extra credit. I decapitated her, beat her to death, and fed her to the lions. I know this is very lenient, but I am a kind and loving mother. Can you help me discipline her further?Today, a girl breathed 100 feet away from me. Later, I saw her borrow a pencil from another guy. How can I get my revenge on her for cheating on me?What’s that one song where there’s instruments and vocals?I am 8 years old. Recently, my IQ was tested at 272642739823828645218999387383920201928373832001019199288. How do I get smarter so that I can possibly have a small chance of pursuing my future career as a roadkill collector?My children are 196 and 243. When do I allow them to get out of diapers?I joined 0.0000000000000000000000000000000000000000000001 seconds ago. Why do I not have 28326436746836486 followers and upvotes?What is 0+1?I sent a letter to Harry Potter asking to marry him. Why hasn’t he responded? It’s been a minute already and I find it insulting that he didn’t jump out of the book.I rested my finger against a surface that was a bit too cold for my liking. How can I get treatment for mitochondrial neurogastrointestinal encephalopathy syndrome?I work out 272838734392929 hours a day, and I only eat lettuce. Am I at a high risk of morbid obesity?If God isn’t real, explain oxygen.I am already 0.000001 seconds old, and I still don’t even have $100000000. How do I stop being lazy so that I can have at least that much by age 0.000002 seconds?Aren’t Ketchup and Mustard cute names for twins?Is it okay if I killed 484959594 people because they have different races that my own? Some people are calling me racist, but I think that’s a little mean of them.If you give me the answers I want to all of the above questions, then you can have $37384647388337737382292921188! All I need is your credit card info.UPDATE 12/16/2019: Here’s more!16. I am 9,000 years old. My girlfriend is five. Is that okay?17. How do I report my parents for child abuse? They said that soon, I might be getting old enough to put on my own clothes, but I’m only 27!18. Should I allow my 6th grader to have one friend that I pick by myself?19. Why won’t my science teacher accept that the earth is 6,000 years old? Link here: http://imanidiot.org20. My Teecher Is So Rood! He Sed Tat My Esay neds a Litle Wurk. Its Tree Wurds Long And Cunsissts of “I Hat Teecherz”. He Also Sed Tat I Dont Hav Dislecksi Beecuz I Got Tessted 4 it 1882828282881 Tymes and they Al Cam Owt Neggitiv. But I Spel Stuf Rong Bcs I Skipp Clas and Tat Konts Bcs I Dont Lik Clas, So I Shudnt Hv 2 Do It. How Do I GetHim Fiered?21. I exist. Why don’t I have a Nobel Prize?22. I’m too lazy to do my own homework. Wendy has $20.75. Erasers cost $3 each and notebooks cost $5 each. Solve for parts a and b.a. If Wendy buys 1 notebook and 4 erasers, how much did she spend?b. How much money does she have left?23. Why?24. I am allergic to peanuts. I saw someone from 1829281911821891172662281 feet away eat chocolate. Since chocolate and peanuts both have the letter E in them, this person is violating my rights. How do I force them to pay all my medical bills?25. I beat up someone because I wanted to see what would happen. Why did he get mad at me?26. Why doesn’t my 7 year old have a boyfriend yet?27. Why doesn’t my 11 year old make $100000000 a month?28. My computer isn’t working because I threw it out the window. Why won’t the Apple store give me a free new one?29. How do I explain to my 10 and 12 year old kids that 1 minute a year is more than enough time for educational video games?30. What country is China in?31. There’s this really annoying kid in my class. He got to skip school just because he has cancer, meningitis, and 28383837 other illnesses, while I had to go to school even when I sneezed. How can I let him know that his behavior is immature?32. My friend is a little unreasonable and thinks the earth is round. How do I explain that he’s wrong?33. I’m 35 years old. My dad is 5′2 and my mom is 1′8. I’m 3′6. Will I grow up to be 8′11?34. Is it okay to kill my 1 second old because it didn’t win 10 Olympic medals?35. How do I ask a question on Quora?UPDATE 2: 12/24/2019 (yay Christmas Eve) Here’s more:36. I’m 14 years old, and I have 1838282 iPhones. All my clothes are from Gucci, and I get 8299229929191883838299999918 dollars a week for doing no chores. My parents don’t have any rules for me. Are they too strict?37. Does (insert popular Quoran) sneeze?38. I got banned from Quora for making 789292817262818182 spare accounts to bully and troll people. How do I explain that this is a perfectly reasonable excuse?39. I’m 91. Am I old enough to start talking to boys, or should I wait until I’m 92?40. Why didn’t you guys give me your credit card info?