Comments by "L.W. Paradis" (@l.w.paradis2108) on "Veritasium" channel.

  1. This problem is so cool because of its relationship with chaos theory. It gives you a simple way to explain sensitive dependence on initial conditions that even a middle school student could understand and appreciate.
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  2.  @mortkebab2849  So . . . Isn't it a matter of being unlucky, for everyone who encountered your comments on this thread? Especially since we are here watching videos on science and mathematics, which does require a modicum of effort, say, by comparison to watching cute babies and animals? Isn't everyone's life just the tiniest bit less happy, all because we read your posts? Sheer bad luck. Trivial in itself, but consider the aggregate. Especially these days.
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  3.  @Meilan39  Exactly! The sciences are empirical, mathematics is purely deductive. Once a mathematical theorem is proved, it can never be refuted based on future "evidence."
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  4.  @mortisnoctu  No, the equation does NOT prove the theorem. The theorem allows us to apply the equation to every right triangle.
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  5. Huh? You mean in order to prove it? What do you mean? The conjecture itself has nothing to do with the reals or the rationals, much less imaginary numbers. It is a conjecture about the set of all integers > 0. Mathematicians always seek to prove conjectures. Once proved, they usually provide insight into many practical areas of mathematics and applied mathematics. This one obviously has a relationship with chaos theory, since it gives us a simple way of studying sensitive dependence on initial conditions, that you could show to a middle school student. Really, really cool.
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  6.  @mortisnoctu  All mathematical proofs are about infinite sets. Even the Pythagorean Theorem is about infinitely many right triangles. That's why deduction is necessary; examples cannot prove any theorem.
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  7.  @grimjawx1650  More motivation: after 30 years, your body is almost the same as if you had not smoked (though it will never be exactly the same). I had my 30 years on January 1, 2021. :)
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  8.  @JureDoon  You're talking about applied mathematics. I'm only considering mathematics itself, apart from whether it has any applications at all. You're right that when a particular application of mathematics fails to describe reality, it's not reality that's "wrong." What I'm saying is neither is the mathematics.
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  9. 2:33 sensitive dependence on initial conditions (compare starting with 26 versus 27, etc.)
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  10.  @Chungustav  Not a Jewish belief at all. More like a corrupted version of Calvinism.
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  11.  @mercyjohnson6436  Sigh. Blocked.
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  12. Were you prepared to be born? Were you prepared to be born where you would speak English?
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  13. If that fact affects you psychologically, then definitely consider January 1, 2021, the first day of the new decade. Sounds like a good plan anyway . . . :/
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  14.  @mercyjohnson6436  Your Bible doesn't have this? "I returned, and saw under the sun, that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill; but time and chance happeneth to them all."
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  15.  @randomslomo1875  The problem is to prove that this pattern holds for every positive integer, or to refute that it does by producing a counterexample (a positive integer for which we do not end in the 4 2 1 loop after a finite number of steps). The problem is to prove or refute a theorem about the infinite set of positive integers. (BTW, I was educated in one of those countries with tuition-free university. You were expected to study really, really hard, contrary to what the person who was rude to you implied.)
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  16.  @randomslomo1875  You're welcome! This problem is cool simply to point out that it sounds so simple, but no one knows the answer yet, and a computer cannot give you the answer.
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  17.  @BBBrasil  The anthropologist Marshall Sahlins did extensive work on this question. He titled one of his essays provocatively: The Original Affluent Society.
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  18.  @mortisnoctu  How would you know in advance that the sides of every right triangle do have that relationship, or that they have any set relationship, every single time, when there are infinitely many right triangles? Um, it had to be . . . proved. ("LOL," huh?)
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  19.  @OHFORPEATSAKES  I did it like that, too.
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  20. ​@gauloise6442 At what age?? I can believe a single woman between the ages of 18 and 29 is happy. At 49? 69? It is not evident.
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  21.  @JureDoon  Exactly what Godel disagreed with. A priori reality is reality. Empirical reality has no superiority over a priori reality. Mathematics itself is not invented, it is discovered.
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  22.  @theo49476  What Adam Smith described was a free market, not capitalism. Like the difference between a star cluster and a black hole. Some of the things he says about a rentier class controlling enough wealth to control massive numbers of people suggest he would see modern-day billionaires as a return to feudal overlords. He didn't come close to Marx in describing what would actually happen, but some of his insights do show that the capitalists who invoke him today are lying.
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  23.  @mercyjohnson6436  If you believe in revealed religion, then perhaps it was god who chose not to reveal himself to some people. Faith may not be a "choice" like any other.
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  24.  @woswasdenni1914  Why did you post this? Did it make sense to you?
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  25.  @lukea977  You worked on your parents' marriage? Did you start at birth, or sooner?
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  26.  @RavageStudio  Socrates was put to death. Plato lived to 80, Sophocles to 90, and both of them wrote to the end. Is this misconception about age due to advertising/propaganda? If so, wow.
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  27.  @quietobserver5480  I took it to mean that the agricultural revolution occurred 12000 years ago, while the industrial revolution is at most 300 years old, and mass literacy even more recent. There just were more farmers in human history, and so often today you see brilliant people emerge from families that only recently left the land. On the other hand, something happens in urban centers that is unusual, with sudden, immense spurts of creativity, and cities (though not like contemporary cities) certainly predate industrialization by millenia. Anyway, that was what I thought he meant. (?)
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  28. You can study number theory before completing calculus, though how far you can go will be somewhat limited. But number theory existed long before calculus, so you're good.
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  29.  @truegamerking  Thank you!
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  30.  @mortisnoctu  You clearly don't know.
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  31. What is being asked is to produce a proof (for all positive integers -- an infinite set), OR a refutation (a counterexample). IF the conjecture is false, a computer could in principle produce a counterexample. IF the conjecture is true, only a person can prove it. No computer can do that. How cool is that?
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  32. Russia is going to win. Oh well.
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  33. Luck is no sudden Category 1 storm. Luck is not having a father who set sail with you when the weather was "a little choppy," and jeered at you for crying. Am I lucky to have imagination? It's not evident.
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  34.  @davemwangi05  You beat me to it.
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  35.  @Xda13ombX  Exactly. Calling capitalism a free market is sheer propaganda and sleight of hand.
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  36. IF it is true, then it is true of all positive integers. Of course that's an infinite set. You don't have to check a number ever again; if the number is a positive integer, then the conjecture holds. Not a limit problem.
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  37.  @dazey856  Right. Proofs are about infinite sets, in this case, the set of all positive integers. Even a huge sample where the conjecture works won't be enough to prove it. There's always infinitely more integers.
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  38. Take a look at Epstein's black book. NONE of these people had any problem with what Epstein did. Some lives don't count. Not to the hyperrich. And this one, Gates, in particular was heavily involved, and lied about it, and keeps lying.
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  39.  @likhithravi2797  Who said anything about money? It's very unlucky to live in a society where people always and only think in terms of money.
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  40.  @MrJest2  Nothing to do with "imprecision." The problem is precisely stated. The issue is to prove or refute that the defined function always gives rise to a string that is finite until reaching the 4 2 1 loop (that it therefore has a finite stopping time).
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  41.  @lskdfoIS  There is no series of numbers generated by this procedure that does not contain even integers, no matter what positive integer you choose as your starting point n. This is obvious, and easy to prove. Until the conjecture is proved, it's obvious you can't say whether the next number tested won't be the one that escapes the 4 2 1 loop. We only know a counterexample has never been found. A proof would settle it, for the infinite set of positive integers. You wouldn't need to test a single one after that. You know they all do.
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  42.  @JustPotBro  You're joking. Right??
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  43. Finding proofs IS the mathematical problem. Every real mathematical problem.
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  44.  @acidbath3226  A computer cannot prove this conjecture, no matter how advanced it is. It can aid the mathematician in finding a proof. People prove theorems.
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  45.  @garglemuh2813  We are unlucky in the comments this great video seems to have attracted.
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  46.  @midimoog  Other mathematical objects are not in the set we are hypothesizing about.
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  47.  @AndrewDavidBoyd  I didn't mean to say it was "enough." I meant no one planned for these sorts of events, or prepared for them. Most of the most important things in life, good and bad, simply happen. Having parents who loved each other, stayed married, and lived until you turned 50 or older, is another example. Having a sister you get along with is known to be a bulwark against clinical depression.
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  48. Great insight.
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  49. Of course. Look up ancient Greece.
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  50. 10:32 start listening there
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  51. It isn't an equation. It's a function.
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  52. Sure. There were so many farmers, and for millennia.
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  53. Right, or prove that every positive integer does.
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  54. The studies that say happiness returns to baseline after a major life-changing success have been shown to be flawed. The sheer relief of not having to worry about money decreases everyone's distress. That creates a new minimum of well-being. Obviously, that does not necessarily increase positive affect, but it rids you of many negative emotions forever. The rest is up to you.
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  55. Nope. All theorems are about infinite sets of mathematical objects.
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  56. So, how identical are women identical twins? When does twinning occur? Are we (reasonably) sure?
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  57. Huh?? Let n=2. 3n, 3n+1, 3n+2 partition the positive integers. An integer having any one of these three forms can be even or odd, depending on whether n is even or odd. 0, 3, 6, 9, . . . , 3n, . . . 1, 4, 7, 10, . . . 3n+1, . . . 2, 5, 8, 11, . . . 3n+2, . . .
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  58.  @MaaikeM4  It makes perfect sense. No number of examples that work will prove that it is true for all positive integers.
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  59. Sure. It is a conjecture about integers. Numbers like 2.7 and 13.3 are not partitioned into even and odd, so you cannot apply the formula.
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  60.  @djddm8760  Read some real biographies of truly great people. I don't mean some rich guy, I mean a Martha Argerich, or a J. M. Coetzee, or a Picasso, or a Bette Davis, or Miles Davis. Find out that they had broke absolutely every "rule" you could imagine, and that a good many years of their lives were, by conventional measures, a mess. Well, I'm inspired, anyway. "Make no small plans" is about the only cliche I like. You haven't noticed that the self-help genre, especially on the subject of making money, is propaganda? It's telling you what the current elite wants you to do. I mean, that's useful to know. :D
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  61. Everyone knows this.
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  62. The problem is to prove it or find a counterexample.
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  63.  @benjamineyoungop5535  The statement that a given mathematical theorem is not provable has to be proved. You can't just say it. That would be called a guess. Every "pattern" can be stated as a theorem, then proved or refuted. Once proved, it is true of an infinite set. This pattern hasn't been proved to be true of ALL positive integers.
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  64.  @benjamineyoungop5535  I guess you didn't watch the video, huh?
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  65. THANK YOU.
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  66. ​ @idahoviking Well, are you going to obey, finally? 😂
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  67. ​ @bluecigars That doesn't sound good, lol.
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  68. ​ @havabrownkittycat7107 Thank you. I love your comment. It's the best one.
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  69. ​@TheMrVersetti Being married to the wrong person is not just miserable, it is soul-crushing and dangerous. The decline of happy marriages is the real tragedy. Ending a horrible marriage is better than forcing yourself to stay.
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  70. @sambalbudanny2235  Sometimes it seems as though people give the advice they themselves most need to hear, huh? 😊
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  71.  @ilikehiking  The diversity is in the societal environment.
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  72. @sambalbudanny2235  Who did what to me? I've never been married! 🤣 The happiest I've ever been were the five years I lived in Paris. Needless to say, Parisians are never fooled by that "it's your own fault-ism" that seems to afflict the Five 👀.
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  73.  @debracisneroshhp2827  Tell that to Gaza.
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  74.  @sambalbudanny2235  Yes. I did mean you. But you meant someone other than me, though you were addressing me. Advice: gaslight better 🤣
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  75.  @steveschilling5966  This was a series of solid questions. A critique of reasoning for logical soundness is neither opinion nor fact, under the usual meaning of those words.
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  76. @nirorit  Yup, we shouldn't rely on what we've studied. Watch videos made with commercial motives in mind. That's the best way to progress forward. I mean, just look around. Have we ever been smarter? Have we ever had it better?
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  77. Positive integers are infinite. Theorems about infinite sets have to be proved. Examples are never enough.
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  78. See 10:33
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  79.  @dules812  Try it out and see what happens. Then you'll figure it out.
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  80. We need a serious conference on experimentation with viruses, bacteria, and other pathogens.
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  81. Ha, ha, ha. Computers can't prove mathematical theorems.
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  82.  @MelonMafia1  Because there is no infinite time. When WE prove a theorem, it is true of well-defined infinite sets of mathematical objects, in this case, the entire set of positive integers. You don't have to test a single one ever again. You know.
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  83. Not only that, but the 3rd and 7th of a 7th chord determine its quality.
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  84. @Star17Platinum17  No it doesn't. The calculator display is finite.
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  85. That's got to be the worst book I ever read, LOL.
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  86.  @tuxedobob2  oh well ;/
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  87. Um, proving or disproving the conjecture is the problem.
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  88.  @RustyShackleford66  Mathematics is about proving theorems that are true of infinite sets. It's actually near-miraculous. Within the realm of mathematics, such theorems are universal truths. If you explain them correctly to any other person, so that they understand every step, that person will either agree, or pinpoint exactly where they don't agree (you invoked an axiom they reject, for example). Is anything else like that?
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  89.  @RustyShackleford66  Mathematics has nothing in common with sensory experience.
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  90. Actually, based on Martin Seligman's research, believing you are in complete control of your destiny could backfire. It's often advantageous to see setbacks as weird bad luck that happened at random, but that you can now analyze dispassionately for information, almost as if it had little to do with you.
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  91. That is the worst math website ever.
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  92. If she loves you for it . . . most of the time.
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  93. If you teach college kids, try this experiment: ask them if they think they could become multimillionaires. All of them do, and they'll be furious if you give them any impersonal descriptions of economic forces that might suggest not. (After all, "attracting abundance" might be true -- kind of a primitive Pascal's wager they've got going.) Now, give them a challenge: take a year of Calculus and get an A on every exam. Hmm. Or, for a whole month, cook everything you eat. You will hear groans. No one has ever leaped to take on the challenge and achieved it. If becoming a multimillionaire is easier, how can it be based solely on hard work?
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  94. Insight into chaos theory could help with the environmental sciences. No. Really.
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  95. Mine was January 1, 1992! Thanks for reminding me. ;)
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  96.  @vader88x  LOL, that was great, thanks.
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  97. I wish I had spent far more time at work. I wish I had had the rewarding work I prepared myself to do, but never got the opportunity. EDIT: Yet. I haven't gotten the opportunity yet. I don't give up easily.
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  98. 7:09 is crucial.
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  99.  @davemwangi05  Don't worry. He'll never read The Jakarta Method, for example. Those of us who have know we are lucky it was written. Journalists get killed for less.
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  100.  @cijeel3975  One percent of Bezo's wealth puts a person into the billionaire class. One one-hundredth of one percent (0.01%) is plenty. With prudence, you and your children can almost certainly have basic necessities covered, for life.
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  101. @Mass Debater Prove, not solve.
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  102. @Mass Debater Prove that every possible string generated by the function ends in a 4 2 1 loop. There are infinitely many, one per positive integer. Watch it until you get to that part.
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  103. 😂😂😂
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  104. You forgot viciousness.
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  105. ​@davidabulafia7145 The more skill and hard work, the more unwanted attention by those who seek to pull you down. Whether they can is mostly a matter of luck. Too bad no one in America reads The Apology of Socrates. It's all there, 2400+ years ago.
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  106. Amazing video.
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  107. This relates to Brownian motion and chaos theory, both of which have applications.
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  108.  @jonathanwhite3507  The video gives you a start.
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  109. 0 isn't a positive integer yo.
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  110. Don't feel bad. Now there is a major called Financial Engineering. No. Not a joke.
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  111. That's the man to ask.
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  112. So someone's Bible doesn't contain Ecclesiastes 9: 11? Was it censored?
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  113. How do you know Brownian motion is not useful?
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  114.  @jameshale5331  Try this search term: "application of Brownian motion in daily life." Or try "importance of Brownian motion." Or don't. You may be too smart to read these articles; you may have far better things to do. Like, maybe you're an artist.
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  115. Cool! They know you. ;)
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  116. No.
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  117. Hard work in the wrong context can harm you, too. You've never been in a office where hard work is so resented that people actually try to trip you up? Or where an incompetent supervisor imagines you are after her job? (A classic pattern in employment discrimination cases.) It is very important to be in a context where your work matters, to "find the right pond." That is always partly luck.
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  118. Chaos theory is essential in studying global warming.
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  119.  @Didnt_ask69  Jawohl
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  120.  @MrJest2  I don't think this is right. Mathematics isn't a "language," it is not imprecise, and its symbolic conventions are distinct from what it is about, namely, the things that are symbolized (mathematical objects and their relations).
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  121.  @MrJest2  The problem is precisely stated, and consists in proving or refuting the given conjecture. Once proved, we know the conjecture holds for the infinite set of positive integers. "Infinity" isn't an abstraction any more than other parts of mathematics are. There is a rigorous theory of infinite sets and transfinite numbers, and every theorem addresses the properties of some infinite set of objects. Even the Pythagorean Theorem is about infinitely many right triangles.
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  122.  @RadarFinsR  Applied mathematics does not require the same architecture of deductive demonstration. What is practically useful in an engineering context has no bearing on the discussion in this video. Applications are secondary. It's not what mathematicians are after.
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  123.  @RadarFinsR  Mathematicians use axioms and rules of inference that are not approximative. They must be precisely defined.
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  124.  @RadarFinsR  I thought people knew what proof was. This is all, like, wow.
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  125.  @TheNeomare  Brilliant observation -- but there are ways to do it, similar to how convergent infinite series are proved. I wonder what it would look like?
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  126.  @TheNeomare  I came back just to see this again. Great insight, make it a separate post. This was gratifying. To see how many people didn't even have the patience to watch the video and think about it was upsetting. A lot of people must be browbeaten in school and turned off of math. Here, when it's in a great video and nothing is at stake, they still can't focus. It must be due to some bad experience of some sort.
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  127. Prove the conjecture or refute it.
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  128. ​@m4l490n It was luck in reference to him, the son. The son was a child, he couldn't control whether the parents he was born to valued books.
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  129. ​@ggh_-ts6pn When in fact viciousness plays a role.
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  130. These are often seen as lucky numbers.
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  131. Well, he has money. And he dropped out of college, so take that! Ha!
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  132. 12:34 AWESOME
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  133. That's why a proof is necessary. That is exactly why.
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  134. Epigenetics: What Lyshenko noticed and completely misunderstood.
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  135. 10:32 could we do better? Maybe, but I wonder.
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  136. Nope.
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  137.  @thalinpans9302  You can translate the conjecture into binary code, no problem.
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  138. Odd positive integers have 1 as their final digit, evens have 0. You can do the rest.
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  139.  @thalinpans9302  Start with 7. In binary code, 111 (seven). Multiply by 11 (three) and add 1 (one). You get 10110 (twenty-two). Divide by 10 (two). You get 1011 (eleven), etc. Same thing.
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  140. Every kid has wondered this.
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  141.  @_nines8270  LOL
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  142.  @_nines8270  root of 9.869 is not half bad . . . yeesh
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  143. 9:19 YOU think Gates will tell you "the real story?" Look, when the pandemic started, we came to the realization that most of our medicine is manufactured in China and India. But when to came to vaccine PATENTS, Gates argued strongly against lifting them, during a pandemic. Then India had a huge outbreak of the new variant. This fawning interview is a disgrace.
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  144. But the simulation he did showed that all of the qualified applicants who were rejected -- a very large number -- were simply unlucky.
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  145. ​@dariusus9870 And I think anyone who (1) obsesses this much about "happiness," and (2) believes "happiness" is within a person's control, is not only not happy, or any other thing I'd aspire to be, but should be given wide berth. Keep your distance from these people. Chances are good they will stab you in the back at the first opportunity, for no other reason than to achieve the most banal and limited definition of "success" you could imagine. (For those who think this is a cynical view, I invite you to read Rick Rubin's book, The Creative Act, and in particular the chapter entitled "Non-Competition.")
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  146. ​ @mikhail.mp4 Eat super-well, exercise, make art, take walks in nature, and sleep tight. Don't worry.
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  147. Even THIS discussion is being sense__ord. I'm not happy about that. 😅
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  148. ​ @dariusus9870  And I think anyone who (1) obsesses this much about "happiness," and (2) believes "happiness" is within a person's control, is not only not happy (or any other thing I'd aspire to be), but should be given wide berth. Keep your distance from these people. Chances are too good that they will stab you in the back at the first opportunity, in service of achieving the most banal definition of "success" you could imagine. (A version of this went poof. I wonder if it is visible resorted.)
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  149. @Александр Кузнецов I'm sorry you don't know how to do computer simulations. Maybe if you work harder at it, or if someone suggested the right book to you, you could replicate the data for yourself and feel pride instead of phony tough-guy scorn.
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  150.  @bwake  Well, the people I replied to have since left the discussion, so I don't know what this was about.
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  151. If you teach college kids, try this experiment: ask them if they think they could become multimillionaires. All of them do, and they'll be furious if you give them any impersonal descriptions of economic forces that might suggest not. (After all, "attracting abundance" might be true -- kind of a primitive Pascal's wager they've got going.) Now, give them a challenge: take a year of Calculus and get an A on every exam. Hmm. Or, for a whole month, cook everything you eat. You will hear groans. No one has ever leaped to take on the challenge and achieved it. If becoming a multimillionaire is easier, how can it be based solely on hard work?
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  152.  :D  Are you multilingual or non-Western? If neither, that's both expected and bloody funny.
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  153.  :D  On second thought, thanks for writing this. I have several excellent comments on this video I had forgotten about, and enjoyed rereading.
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  154. You have to prove things in math. You can't generalize. There are infinitely many positive integers. So 158,743^1087 examples isn't enough.
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  155. That's the hypothesis.
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  156.  @icemancometh1188  This is a function with the positive integers as domain and range. F(x) = 3x + 1, if x is odd, and x ÷ 2, if x is even. The conjecture is that for every positive integer (infinitely many), F(x) generates a string that, after some finite number of steps, always terminates in a 4 2 1 loop.
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  157.  @icemancometh1188  These aren't equations. This is a function. The problem is to prove the conjecture.
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  158.  @icemancometh1188  Or disprove, for that matter.
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  159. @girl That is a very good question.
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  160. @The Chosen one It's a problem no one HAS solved. We don't know that it can't be.
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  161.  @xenphoton5833  That is exactly the point. When you PROVE a theorem, it is true of an infinite set. No, computers are not "smarter" than people. They process in finite time, they cannot test every number.
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  162.  @lskdfoIS  Does everyone know this person is faking?
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  163.  @cute_funny_gamer  Like 12-dimensional Dunning Kruger.
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  164.  @JustPotBro  Cool
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  165.  @ruy_mascarua  0.1 doesn't go anywhere.
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  166.  @shanejohnson587  It's not an equation, it's a function, and its domain is the positive integers.
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  167.  @philippschmitz1787  Nicely said. Math either is not taught, or is taught really badly.
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  168.  @shanejohnson587  It is not an equation but a function, where the domain is the positive integers. And it has two parts. (Apply 3x +1 to all odd outputs, x/2 to all even outputs).
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  169.  @existenceisillusion6528  Looks to me like Philipp can do math and you can . . . Argue about stuff, in the usual trash talkin' style. Oh well.
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  170.  @existenceisillusion6528  No, theorems are not "the result of a proof." There are some good books on the philosophy of mathematics you might want to look at, as you don't seem to fully appreciate the positions you are taking with respect to mathematical objects and their relations. It seems you may be inadvertently saying something other than, or more than, what you mean. I don't know why you cast aspersions at Philipp's education, when his posts were spot on and he clearly can do the math. Where have you studied for any substantial length of time where university is tuition-free? What languages do you speak? And do you equate money with "success?" Every mathematician you see here has less money than the C-suite at Facebook, or any other large corporation. McDonald's, WalMart, Uber, Black Rock, etc. I know who I'd trade places with if I had to. In any event, we call the hypothesis of this video a "conjecture" because it has not been proved. But to say that proving it thereby makes it a theorem is a little strange, as it seems to suggest mathematics is invented, not discovered. If this conjecture is true, a lot of mathematicians would say it was always true, and hence was always a theorem of number theory; we just needed to discover exactly why, and satisfy ourselves that it is. (Godel was in that camp, surprisingly enough.) I don't know if you are disagreeing. I can't tell. Scoring points you are not. Interesting fact: when the great state universities in America were nearly free, to the point where a minimum-wage summer job paid for a full year of school, students read more than they do today, and put in longer hours studying. Why is that? The most common major today by far is Business, and has been for decades. Without enough accounting to pass a CPA, that is not a rigorous degree at all. (It sure does demonstrate docility to employers, though.) That is not the usual course of study that you see Europeans embarking on.
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  171.  @existenceisillusion6528  I don't care. Doesn't matter.
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  172.  @ruy_mascarua  LOL
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  173. Only one prime number: the even prime.
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  174. Because we can't find a counterexample and yet we don't know if it's actually true. No one has proved it.
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  175. Coolest thing ever.
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  176.  @wouldbabyhitlerkillyou4217  You think? Yet not one of them has posted a solid, reasoned critique of any element of it. What are the chances that they are all too busy to toss off a few cogent lines? I guess we are unlucky that they haven't seen fit to share their knowledge with us. P. S. Thank you for swearing. Swearing is information: don't bring up this video at parties, or at work, etc., at least not without being prepared to elicit hostility. Most people know not to, but just in case a few optimists might be tempted, you provided the corrective.
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  177. This is a REALLY nice guy. Gratitude!
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  178. But will it? Needs a proof.
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  179.  @goldnutter412  Just study the easy parts. Then it's easy. That's what I did. 🤣
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  180. It's a function: f(x) = (3x + 1), for all odd positive integers. That's one part of the hypothesis, which allows you to generate these strings.
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  181.  @rebeccacarlson9166  What's the big deal to mention this? It's not a cut. Most people probably don't know that the premise that hunter-gatherer societies were very poor and had no leisure has been seriously questioned. I appreciate facts, especially now.
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  182. Prove it.
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  183. A conjecture that has no known exceptions needs to be proved, because there are infinitely many integers. Proving the conjectured holds for all positive integers is the problem.
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  184. The letter after a "th," if there is one, is a vowel, not necessarily e. Anyway, that's my thought on this.
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  185. Only the lucky ones.
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  186. Not only that, but the 3rd and 7th of a 7th chord determine its quality.
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  187. The frequency of perfect squares x^2 can approach 0 as x approaches infinity, but it never reaches 0. We know there are infinitely many perfect squares: one per integer. (Remember an infinite set can have an infinite proper subset.)
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  188. You have to subtract the $2 from the $27 dollars, not add it, LOL. That comes to the actual amount paid after the refund: 25 = 30 -- 5.
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  189. The answer is a proof (or a refutation) that no one has found yet.
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  190.  @AndrewDavidBoyd  If no one ever spoke English to you, you would never acquire it at all.
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