Mathematician Proves Huge Result on 'Dangerous' Problem (quantamagazine.org) 167
Mathematicians regard the Collatz conjecture as a quagmire and warn each other to stay away. But now Terence Tao has made more progress than anyone in decades. From a report: It's a siren song, they say: Fall under its trance and you may never do meaningful work again. The Collatz conjecture is quite possibly the simplest unsolved problem in mathematics -- which is exactly what makes it so treacherously alluring. "This is a really dangerous problem. People become obsessed with it and it really is impossible," said Jeffrey Lagarias, a mathematician at the University of Michigan and an expert on the Collatz conjecture. Earlier this year one of the top mathematicians in the world dared to confront the problem -- and came away with one of the most significant results on the Collatz conjecture in decades. On September 8, Terence Tao posted a proof showing that -- at the very least -- the Collatz conjecture is "almost" true for "almost" all numbers. While Tao's result is not a full proof of the conjecture, it is a major advance on a problem that doesn't give up its secrets easily. "I wasn't expecting to solve this problem completely," said Tao, a mathematician at the University of California, Los Angeles. "But what I did was more than I expected."
Lothar Collatz likely posed the eponymous conjecture in the 1930s. The problem sounds like a party trick. Pick a number, any number. If it's odd, multiply it by 3 and add 1. If it's even, divide it by 2. Now you have a new number. Apply the same rules to the new number. The conjecture is about what happens as you keep repeating the process. Intuition might suggest that the number you start with affects the number you end up with. Maybe some numbers eventually spiral all the way down to 1. Maybe others go marching off to infinity. But Collatz predicted that's not the case. He conjectured that if you start with a positive whole number and run this process long enough, all starting values will lead to 1. And once you hit 1, the rules of the Collatz conjecture confine you to a loop: 1, 4, 2, 1, 4, 2, 1, on and on forever.
Lothar Collatz likely posed the eponymous conjecture in the 1930s. The problem sounds like a party trick. Pick a number, any number. If it's odd, multiply it by 3 and add 1. If it's even, divide it by 2. Now you have a new number. Apply the same rules to the new number. The conjecture is about what happens as you keep repeating the process. Intuition might suggest that the number you start with affects the number you end up with. Maybe some numbers eventually spiral all the way down to 1. Maybe others go marching off to infinity. But Collatz predicted that's not the case. He conjectured that if you start with a positive whole number and run this process long enough, all starting values will lead to 1. And once you hit 1, the rules of the Collatz conjecture confine you to a loop: 1, 4, 2, 1, 4, 2, 1, on and on forever.
Interesting (Score:2, Insightful)
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Then you can do the same thing, but instead of multiplying it by 3, you CUBE it. Then you can do the same thing, but instead of cubing it, you divide by 6 and subtract 1. Then you can...ad infinium.
Re:Interesting (Score:4, Informative)
Mathematical ice-9 (Score:4, Funny)
Dammit. Now I, one of the worst mathematicians in the country have become obsessed with it.
Why did you have to go and tell me about it ? Why ?
(n*3+1)/2 (Score:2)
The sequence will slowly "walk forward", halving every time it's even, until it reaches a power of 2 - then it goes down to 1 and stays there.
Bet the same thing will happen with n*5+1 or n*X+1 with X being any odd number.
Re:(n*3+1)/2 (Score:5, Insightful)
Oh yeah? Prove it.
Re:(n*3+1)/2 (Score:4, Informative)
Bet the same thing will happen with n*5+1 or n*X+1 with X being any odd number.
Actually, the general mathematical consensus is that it is very likely that if one uses an odd number n greater than 3 (such as the 5 in your example) that there will likely be sequences which increase without bound. Here's the basic idea:
Let's first think about the Collatz example itself. There are two equally likely possibilities given a given x, either we go from x to x/2 (when is even) or we go from x to 3x+1. But in the second case, we immediately will always then divide by 2. So we should think of two cases, where with 50% chance we go to x/2 and with 50% chance we go to (3x+1)/2. Now, this means that given a large x, about half the time it goes to about x/2 and about half the time it goes to about (3/2)x. Now, that means that we should expect on "average" that the growth for a given iteration is about x times the square root of (1/2)(3/2), which is less than x, since 3/4 But what if we replace the 3 with a 5? Well, we can try the same logic, but now we have a growth which looks like the square root of (1/2)(5/2), and that is greater than 1. So we expect that on average to see sequences get very big.
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You and the other mathematicians that reached the consensus are way over-thinking it.
The crux of the issue is that the divide by two case is an entrapment case. If repeated infinitely odd_number*n+1 will eventually land on a power of two. Once that occurs, the whole ball of wax unwinds down to one very quickly. The average square root is a non-sequitor. One way to prove the conjecture would be to show that there is no way to get caught in a higher-order loop before the algorithm can hit a power of 2.
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Yes they'll get very big but eventually they'll still hit a power of two. There will just be (way) more iterations.
Even or Odd (Score:2)
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This is the easy to prove part. Whether the amount of extra time spent even "out-divides the multiplication while odd" is the less easy part.
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Why is the amount of time spent even or odd make any difference? Once a 2^n number is encountered, its going to all unravel down to one. All that must be shown is that 3n+1 will eventually meet 2^n
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How much time it spends as odd or even is a non-sequitor. The only thing the algorithm has to do is eventually land on a number equal to 2^n, where n is between 1 and infinity. 2^n will devolve to 1 in this conjecture for all values of n. The question is if repeated an infinite number of times, would 3n+1 eventually encounter a whole power of 2.
If you can prove that 3n+1 will encounter a power of 2 if repeated less than an infinite number of times, then you have proven the conjecture is true.
Just a search for a multiple of 2? (Score:2)
It seems that this is just a way to start with a prime number and increase its value till you reach a multiple of two. Once you reach a multiple of two, the rules will make it roll down to one, and then the repeating pattern starts. A "proof" for this is to simply show that N*3+1 will produce a number divisible by two?
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I think you've misunderstood, unless I've misunderstood you:
1: 1*3+1 = 4 ... 4/2 = 2 ... 2/2 = 1. .. 10/2 = 5 .. 5*3+1 = 16 .. 16/2 = 8 ... 8/2 = 4 .. leads to 1. ... 2/2 = 1 .. 22/2 = 11 ... 11*3+1 = 34 ... 34/2 = 17 .. 17*3+1 = 52 .. 52/2 = 26 .. 26/2 = 13 .. 13*3+1 = 40 .. 40/2 = 20 .. 20/2 = 10, see 3.
2: 2/2 = 1
3: 3*3+1 = 10
4: 4/2 = 2
5: 5*3+1 = 16, see 3.
6: 6/2 = 3, see 3.
7: 7*3+1 = 22
Number 7 there illustrates it. You get 52, then 26, then 13 .. it's not rolling all the way down to 1 before being mult
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So how to you end up on a power of 2? Two possible ways:
1. Half of the next higher power of two.
2. 3n + 1
So what are the powers of two that minus
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Congratulations!!
You stumbled over the halting problem!
https://en.wikipedia.org/wiki/... [wikipedia.org]
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Trivial, as long as you remember that you're always doing this to an odd number. Multiplying two odd numbers together will always produce another odd number, and adding one to it will produce an even number, so the step after the multiplication and addition is always division. (If you don't understand why the multiplication always produces an odd number, you're probably not ready to work on the conjecture.)
A new quiver for my bow - explination (Score:3)
Here is the very simplified version of the argument. Take a large sample of numbers that have a certain random(ish) weighted distribution based on some criteria. This weighting is such that after you apply the Conjecture to each number in the set the weighting of the resultant set stays about the same. One weighting would be no numbers that are multiples of 3, lots that are 3n+1 and a few that are 3n+2. (You can easily see that the conjecture quickly weeds out numbers that are multiples of 3). Now if I can show that 99% of the numbers in these sets get smaller over time then with a lot more work I could generalize this to say 99% of numbers behave this way.
How to work on a proof (Score:5, Insightful)
Why? (Score:5, Insightful)
It's not the problem itself, it's the tools and knowledge for proving it that's the real prize.
The Collatz Conjecture is a very famous problem. One of the greats of 20th century mathematics, Paul ErdÅ's, said of it, "Mathematics may not be ready for such problems.".
What he was meaning is that we just didn't have the tools and the view in order to be able to deal with such a simple but elusive question about something as basic as the integers.
We're not there yet by any means, but perhaps mathematics is closer to being ready for such a question.
Numberphile (Score:2)
Do I feel a Numberphile video coming on? They've talked about Collatz before [youtube.com].
...laura
Proof (Score:2)
I have a complete proof for this conjecture. Unfortunately, it's too large to fit in this post.
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Re:Seriously (Score:5, Informative)
Yeah, that's great, but this guy is supposed to be doing meaningful work.
He's not allowed to have a hobby? My hobby, this time of year anyway, is Christmas lights. Yours is apparently shitting on other peoples hobbies. His is trying to solve quirky math problems. What do those three hobbies have in common? None of them matter, they give us something to do outside of work, we like to talk about them, and they make us happy.
Re:Seriously (Score:5, Insightful)
You know these guys are spending work time on this
And your probably spending time at work, shitting on some other nerd's hobby.
Re:Seriously (Score:4, Funny)
Re:Seriously (Score:5, Insightful)
Much of mathematics is seemingly pointless. They just keep proving new theorems without worrying about applications. In fact, a Belgian maths professor once said: "Every now and then someone calls me to say they found an application for my work. I always feel that's a bit of a pity".
But the applications do arise. Who cares about factoring semiprime numbers? Totally useless, right? Or non-cartesian coordinate systems? Hilbert spaces? All seemed perfectly useless at first, until cryptography, relativity and quantum mechanics came along.
If mathematicians hadn't already thought way ahead, all that theory would have had to be invented when it finally became clear that somebody needed it, taking lots of extra time. But thanks to mathematicians and their pointless hobby, the theories were ready ahead of time before anyone even knew they would be useful.
Re:Seriously (Score:4, Informative)
This ain't no hobby. Sounds like an obsession.
It is not an obsession. Terence Tao has said that he limits himself to only a few days each year to work on "hopeless" problems like the Collatz Conjecture. It is amazing that he was able to get the result he did with such a small amount of time. But he is an amazing guy. Probably the world's most famous living mathematician, and one of the most productive.
You know these guys are spending work time on this while their grad students teach hapless freshmen how to solve algebraic equations.
Terence Tao is considered a superb instructor.
What is it like to take a class from Terence Tao? [quora.com]
Re:Seriously (Score:5, Interesting)
This ain't no hobby. Sounds like an obsession.
It is not an obsession. Terence Tao has said that he limits himself to only a few days each year to work on "hopeless" problems like the Collatz Conjecture. It is amazing that he was able to get the result he did with such a small amount of time. But he is an amazing guy. Probably the world's most famous living mathematician, and one of the most productive.
You know these guys are spending work time on this while their grad students teach hapless freshmen how to solve algebraic equations.
Terence Tao is considered a superb instructor.
What is it like to take a class from Terence Tao? [quora.com]
The work of Terrence Tao is directly applicable to my work in random number generators and has led to real world measurable reductions in transistor count and the associated power and silicon area savings in entropy extraction circuits.
An apparently completely useless series of papers at the time.
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It's a dangerous problem because it's nerd sniping.
Ob XKCD: https://xkcd.com/356/ [xkcd.com]
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That one is funny!
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Don't believe the danger? As usual, xkcd shows why:
https://www.xkcd.com/710/ [xkcd.com]
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Nice diagram! In short, if you repeat the process long enough, you will eventually fall on a number that is a power of 2. Said like this, it doesn't seem so mysterious after all.
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As with any basic research, we can never know if finding a solution is just a dead end, or in fact leads to new insights and new techniques. For thousands of years mathematicians have played with these sorts of problems, and I don't think it's at all hyperbolic to say that from these often strange little trivialities that a lot of how our world works today stems from crazy brilliant guys trying to solve peculiar puzzles.
Re:Seriously (Score:5, Insightful)
There was a time when calculus was a peculiar puzzle. Before that, the Pythagorean Theorem.
Almost everything mathematicians do are strange puzzles with no obvious use... until some engineer, physicist, chemist, programmer, banker, etc. finds a use for it.
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That's the point of the natural sciences. Proof's and theorems about the natural world inevitably lead to things in the real world when the engineers and people down the chain start to apply what was learned.
Math works the same way these theorems and proofs can lead to improvements in things you can't even predict like improvements in computers or even better biological or pure science theories that then lead to things like Quantum Dot TV's.
Work done in the pure sciences and math during the 20th century has
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I can think of a billion variations on this "conjecture". None of them really matter.
But... if you solve this one you get to be really famous and get invited to mathematician parties.
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Yeah, I guess it beats trying to teach algebra to moronic freshmen. I withdraw my objection.
Re:Seriously (Score:5, Funny)
I figured out the proof long ago, but hid it so that I wouldn't be invited to mathematician parties.
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YAY!! PUN THREAD!!
Because, I didn't like the products?
Re: Seriously (Score:2)
He spends a few days a year looking at it. That's all. What's the problem with that?
Meaningful work (Score:5, Insightful)
Pierre de Fermat wasn't doing "meaningful work" when he noticed that for any prime p and any integer a, a^p - a is a multiple of p. In 1640, this fact was just as "meaningless" as the Collatz conjecture today: a mathematical curiosity.
Leonhard Euler later extended this property to composite numbers via his "totient function".
These facts became useful in the 20th century when they became Diffie-Hellman key exchange and the RSA cryptosystem. That "s" after "http" in the URL you're using to access Slashdot is thanks to these guys not doing "meaningful work".
Re:Seriously (Score:5, Interesting)
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He got a slashdot headline out of it, that's something,
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You're trying too hard.
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"He conjectured that if you start with a positive whole number and run this process long enough, all starting values will lead to 1.." If you get obsessed with stuff like that you need to reexamine your life. Who cares? If you are interested, run a billion random numbers through a computer and see if it leads to 1. If it does, then you are most likely correct, otherwise you are not. The real question is, if you ARE correct what difference does it make?
And so it turned out that 3208 was, in fact, the least interesting number ever.
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And so it turned out that 3208 was, in fact, the least interesting number ever.
Isn't that interesting?
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And so it turned out that 3208 was, in fact, the least interesting number ever.
Isn't that interesting?
A very interesting property indeed! [wikipedia.org] The actual number, much less so. ;)
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A while back, I got interested in the collatz conjecture, so I wrote a few scripts that would cycle through numbers and give information about results. One of the things I found interesting is that certain numbers came up a lot more often than you'd think. One of these was that 9232 would show up quite often as the highest number reached while doing the calculation on numbers below 10,000. For instance, checking the numbers from 1 to 5000, you'll find 9232 was the highest number reached 1225 times. That's 2
Woo hoo! Proof! 8) (Score:2)
current=100000000000000
while current > 1:
print(current)
if (current % 2) == 0:
current = int(current / 2)
else:
current = int(current*3+1)
print(current)
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replace the initial value of current with a series starting at 1 until a few quadrillion and run it. If they all end up leading to 1 then you have "solved" it. Great. Send the guy an email and ask him "what is next?"
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Haha, clearly a wrong proof without even reading it. The algorithm should work the same in any base since it's just multiplying by 3 and adding 1, or dividing by 2. There's no way that ending digit base 10 is relevant, since the base isn't relevant. Funny.
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The whole point of a mathematical proof is that you don't have literally sit there and enter every possible number. That's not a solution, that's just an iterative workhorse that cannot actually prove the conjecture.
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What a fascinating world view.
So what should people who are not you waste their brain power on, can you describe the set of problems you find satisfactory for them to work on ?
Possibly their choice of problems to solve is not the issue here.
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Forget about him.
He just keeps proving his stupidity and functional illiteracy regarding almost anything.
Thus he keeps trolling along the topics.
The way he's trolling though...
Comment above revels he doesn't understand the concept of algorithms or personal advancement through intellectual pursuits.
Another one up above shows he harbors secret existential fears of inadequacy and value to his existence that are eating him from the inside.
The "Why?" he's trolling...
Re: Woo hoo! Proof! 8) (Score:2)
Because solving stuff like this can lead to advances in mathematics that are useful elsewhere. The really tough problems require new approaches.
Getting a "yes" or "no" answer is trivial compared to _how_ you got the answer.
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I think this might be the worst take I've ever seen on /.
Nobody knew what the hell to use lasers for for years. The research was considered frivolous. Now I'm not sure how the world would function without them.
Some of the patents CSIRO holds with respect to WiFi came out of research on black holes. Arguably meaningless work, since black holes have little to no effect on us here, but WiFi is a pretty useful technology.
Useful stuff comes out of useless research all the time, and you can't pick and choose what
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Apparently you have never taken anything beyond basic math. No frustrated mathematics professor (or philosophy grad student for that matter) explained to you that demonstrations on a trivial subset is not proof.
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We know it isn't proof. However it is good enough to give an answer of "what the f does it matter?"
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We know it isn't proof. However it is good enough to give an answer of "what the f does it matter?"
No, it's not. It doesn't even begin to address that question.
You may not care about it. I may not either. Even if this result is irrelevant now, it may become critically important for something in the future. Your code sample has no bearing on that possibility.
I agree with you that your code sample doesn't matter in the slightest.
Re: Woo hoo! Proof! 8) (Score:2)
How is that a proof?
Itâ(TM)s a program / implementation that simply shows that for the supplied number, that it may converge to 1 or not.
It does not say why this is or is not the case ...
A proof would demonstrate for the set of all whole numbers, that the algorithm will always converge on 1 before entering the endless repeating sequence without fail.
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current=100000000000000
while current > 1:
print(current)
if (current % 2) == 0:
current = int(current / 2)
else:
current = int(current*3+1)
print(current)
But that doesn't show that 100000000000001 fits.
A proof has to show that it works for all numbers.
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Seems to be true for the first 2 billion numbers (code runs in like 10 seconds)
Can't check much further, numbers are starting to exceed 64 bits.
Highest 7125885122794452160 for 1410123943
Re:Seriously (Score:5, Insightful)
Often times conjectures aren't directly applicable to anything useful. But new math will result from solving them that may have application elsewhere. As XKCD puts it [xkcd.com] math is the purest form of science, or in other worlds the least directly applicable one. Math is just math after all, its an abstraction of the physical world. You can think of it like an API. On its own it does nothing but it allows you to make a lot of cool things.
Tell me, how useful is it to be able to determine the length of a triangle's hypotenuse? Well without it navigation wouldn't work.
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As XKCD puts it [xkcd.com] math is the purest form of science, or in other worlds the least directly applicable one.
Of course, without historians you wouldn't have any clue who all those old mathematicians were. Humanities win again! \o/
Re:Seriously (Score:4, Funny)
And without English majors you wouldn't have baristas serving coffee to keep the historians awake.
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And without English majors you wouldn't have baristas serving coffee to keep the historians awake.
That's what grad students are for.
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It's the core justification of all the work on string theory. Even if string theory turns out to be wrong, it has produced new mathematical insights. Sometimes even a dead end can prove profitable.
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"Tell me, how useful is it to be able to determine the length of a triangle's hypotenuse"
It is very useful. Without it navigation wouldn't work. Tell me, how useful it is to be able to prove the "conjecture" above? Not useful at all. What if the answer was "Yes" vs "No"? Would it make a difference to anything meaningful? No.
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I'm not a mathematician so I'm not qualified to really say. But neither are you.
Re:Seriously (Score:5, Insightful)
Once you peel back the layers of economic causation, you will see that compelling 'useful' work for any other reason than self-fulfilment is dehumanizing, as you are in a sense being used as a tool and not a person. Ultimately, fulfilment, be it through aesthetic pleasure or otherwise, is what all the "useful work" is trying to provide opportunity for. So asking for economic or scientific justification for it is backwards.
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"With your line of thinking, what purpose does art have? Bodybuilding? Aesthetic pursuits of any kind?"
This isn't art. This is obsessive mental masturbation.
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Historically, we invented civilization so that a bunch of us working together could support a small subset of the population that did things like play with mathematical puzzles, or paint, or write poetry. Why did we do this? For all the reasons you list, but also because it turned out to be so fantastically productive that it required fewer and fewer of us to work in the fields and more and more of us could create art.
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Um no. We didn't invent civilization so some dude can waste his time proving things that don't matter.
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Your trolling is usually much more clever. I guess it's Friday?
I say that with great respect by the way. Reminds me of the golden age of Slashdot, when the trolls actually put in the time to create art. Useless art, of course. You could even call it mental masturbation.
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Number theory has provided a large number of advances, both applied and theoretical, from banking to fundamental physics.
Public key encryption comes from a similar mathematical game.
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Ha ha, yes it did. Modulus math is *literally* going around in circles.
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Correct. I am.
Nothing makes any difference. (Score:5, Insightful)
Absolutely anything that anyone cares about can be presented as pointless. Whatever things YOU care about can be revealed to be absurd in the big picture.
There is no ultimate purpose to existence, so all purposes are arbitrary. A lot of them boil back down to survival....which is just as pointless as anything else especially since it is ultimately impossible. We all die and that's it. So, by pointing out the pointlessness of proving or disproving this conjecture, you aren't actually adding any wisdom to the conversation. Of course it is pointless. Just like everything else.
There is no rule that says that everyone must spend their time doing things that matter (and certainly not things that matter to you, specifically). And, as shocking as this may be, there are some people in the world who are wired differently than you, and so they find things like this interesting even though you don't.
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"Absolutely anything that anyone cares about can be presented as pointless"
Nope. There is useful math and there is Sudoku.
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Ah crap. I responded to an obvious flame bait post, and fed a troll. My bad.
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We know what formal mathematics is. The main question is "if the answer is A, or if it is B, does it matter?". The answer is no. It doesn't matter.
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Just retire already, go fishing or something ... you are just another bitter boomer.
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OK boomer.
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run a billion random numbers through a computer ...
Traditional computers cannot generate random numbers.
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"It's a siren song, they say: Fall under its trance and you may never do meaningful work again."
Why are mathematicians portrayed as insanely obsessed AF with finding the pattern? Doesn't have to be madness, learn to let go of things.
I can totally picture it. Someone breaking down a wall to find a room where a mad mathematician had walled himself in to give himself a 360 degree chalkboard, a skeleton in ragged clothes laying in a corner, one bony arm still reaching up holding a piece of chalk against the board, the walls covered floor to ceiling in proofs and formulas.
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Because "normal" people don't care about *any* math problems. In fact, most of them find it unpleasant to even hear the statement of the problem. So, when they see a person who finds such problems to be interesting, and spends any time *at all* trying to solve them, such people appear to be obsessing.
So, that's why.
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But it is the mathematicians themselves who say they are obsessed. No one else even cares.
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Re:Insanity (Score:4, Insightful)
I don't know. There are times when I've gone down the proverbial rabbit hole over a programming problem. I don't think it's insanity, I think its a trademark of an analytical mind, that it will inevitably seek out challenges, and yes, will obsess over them.
I remember when I first starting programming, I was about 12 or 13 years old. My uncle had loaned me a VIC-20 with a RAM expansion pack. One morning I woke up, and decide I'd right a BASIC interpreter in BASIC. I had very little notion about tokens, string handling, and of course it was a ludicrous project, considering my skill level and the limited hardware I was using even in the mid-1980s. I never completed the project, but I taught myself a helluva lot of a basic principles of coding, lexical analysis, and so forth. The project was a dud, my uncle thought I was wasting my time, but I came out of that pointless project a lot smarter and with a far more intuitive feel for how computers work.
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Because the Journey is more rewarding then the Destination.
My friend coined this quote which I think beautifully and succinctly summarizes it:
Necessity is the mother of invention;
Curiosity is the father.
-- Michaelangel007
As programmers we can get obsessed with performance. i.e. Quick Sort, Radix Sort, etc. Programming is no different then puzzle solving -- just that for us Mathematics is a subset. We also wear many hats: Logic, Philosophy, Engineering, Psychology, Design, etc. hats.
Without an obses