Patterns of Primes
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Terry Tao has been thinking productively about patterns of primes for most of his life:
Terence Tao: Structure and Randomness in the Prime Numbers, UCLA – 47:51
— UCLA
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Yeah, you "just" don't know all the variables. Or what is even relevant.
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Well, since my degree certificate says 'Computer Science' and not 'Meteorology', I'm not going to be doing any weather forecasting anytime soon. So it doesn't matter
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This post is deleted!
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This thread should be moved to a prime number ID
(next one is 55511, according to http://compoasso.free.fr/primelistweb/page/prime/liste_online_en.php)
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Sure, we can predict primes. We might say that their distribution is not random at all, since we know exactly how to produce all of them less than n. But we're talking about more than their distribution. We're talking about their relationship to each other, which I think is not identical to their identity.
That there are patterns in one representation of them does not convince me of their non-randomness. As mentioned up thread, there are other patterns in random data. Instead of normally or exponentially or whateverly distributed, we could call these Ulam-spirally distributed.
The white vertical columns on that map all have a = 30, and the longest columns (b = 7, 107, 359 and 541) have 6 elements. It was recently proved (2004) that prime arithmetic progressions of arbitrary length exist though as far as I know there is still no analytic method for finding those; the longest one currently known (found with a software search) is the 26-element sequence with a = 43142746595714191, b = 5283234035979900 for n in 0..25.
There is no randomness whatsoever in the prime sequence. It is constructed by an algorithm that is quite specific as to the criteria of the numbers included in the set.
What is lacking is predictability. We cannot look at an arbitrary number and prove whether or not it is prime, except by exhaustive application of the algorithm. But that's true of many algorithms and is, in fact a fundamental principle of chaos...that an algorithm is known exactly but there is no way to predict specific membership. The Mandelbrot set is typical: patterns are evident everywhere and yet there is no way to predict whether a given number will be deep or near.
Don't confuse randomness with unpredictability.
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Don't confuse randomness with unpredictability.
Ironically, the ability to generate that exact confusion on demand is the key property of a CSPRNG.
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Ironically, the ability to generate that exact confusion on demand is the key property of a CSPRNG
Exactly. PRNG's as a class are a perfect example of the same thing I was talking about above. Absolutely algorithmic, yet prediction (without access to the internal state) is difficult or impossible.
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And the interesting thing about the output of a good CSPRNG is that it's not only prediction that's impossible: analysis of the output, by any method, will also never yield results different from those obtained from applying the same method to comparably distributed randomness.
This sometimes leads the part of the brain that engineers share with stoners to spend more time pondering the nature and origin of "true" randomness than is healthily sustainable.
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For example (I'm not a fan of predestination) but sometimes I wonder if there is such a thing as random, in this universe. "Hidden variables" are so omnipresent that its possible every random event would appear not so random if we knew all the details.
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But what if the "hidden variables" are not hidden at all? What if they're hiding in plain sight, in the form of everything else that we don't think of as "part of" the subsystem whose behavior we're trying to predict? What if the whole thing is all cross-linked such that the precise behavior of any part ultimately depends on the total behavior of the whole?
Don't Bogart That Joint – 02:57
— D. Chris Johnson
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http://gizmodo.com/mathematicians-discovered-something-super-freaky-about-1764839266
1764839266 is actually not a prime number.
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But what if the "hidden variables" are not hidden at all? What if they're hiding in plain sight, in the form of everything else that we don't think of as "part of" the subsystem whose behavior we're trying to predict? What if the whole thing is all cross-linked such that the precise behavior of any part ultimately depends on the total behavior of the whole?
It doesn't mean much to us even if it is. The nature of iteratively applied functions means we wouldn't be able predict outcomes in the world any better than we could for CSPRNG. Even if everything is crosslinked.
1764839266 is actually not a prime number.
Well, duh! Even!
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Importantly, this observation has nothing to do with the base-10 numbering system, and is something inherent to primes themselves.
You're talking about the patterns observed in the sequence of last digits. How can this possibly be unrelated to using base 10? I guarantee you consecutive 9s will show up a lot less often in base 8...
....
OK, reading the actual article clarifies a lot.
@boomzilla said:In terms of the back-to-back distribution of the other numbers, primes ending in 3 and 7 appeared 30 percent of the time, and consecutive 9s appears about 22 of the time.
The description of this part is way off. The percentages given appear to be the distribution of the last digit of the second prime number, where the preceding one ended in 1.The overall distribution of the last digit is very even:
1: 24,999,437
3: 25,000,135
7: 25,000,401
9: 25,000,027But taking consecutive pairs of primes, the distribution of their last digits is much less even with a distinct bias against repeated numbers:
1, 1: 4,623,042
1, 3: 7,429,438
1, 7: 7,504,612
1, 9: 5,442,3453, 1: 6,010,982
3, 3: 4,442,562
3, 7: 7,043,695
3, 9: 7,502,8967, 1: 6,373,981
7, 3: 6,755,195
7, 7: 4,439,355
7, 9: 7,431,8709, 1: 7,991,431
9, 3: 6,372,941
9, 7: 6,012,739
9, 9: 4,622,916They state that this occurs in multiple bases, though the only other concrete example they give is for base 3, for the first million primes (greater than 3):
1, 1: 215,873
1, 2: 283,957
2, 1: 283,957
2, 2: 216,213The bulk of the paper seems to be, from a rapid skimming, developing an approximation term for this based on some well-known conjectures. They also develop some consequences, such as that when working mod n, a sequence (a, b, c, ...) should have very similar frequency to the sequence (..., n-c, n-b, n-a). So for example in the base 10 data we see that, e.g. 9, 3 and 7, 1 have very similar frequencies (6,372,941 and 6,373,981 respectively).
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Weather involves processes with multiple feedbacks and is therefore inherently chaotic i.e. exquisitely sensitive to small alterations in initial conditions. And since some of the processes that set those initial conditions are chemical, and chemistry is most accurately described by quantum mechanics, and quantum mechanics is non-deterministic: weather is ultimately random.
In practice, you hit the limit of what you can predict due to the non-linearity of the fluid dynamics equations at a far larger scale than at the point where you need quantum mechanics to explain the chaos. Yes, there's QM in the depths of the detail there, but it really doesn't need to be invoked to explain why weather is hard to forecast for any length of time.
I did work with meteorologists on weather forecasting codes for a while. (We were working on mechanisms for scaling up the process for conversion of information at a global level down to a sub-city-scale level so that it could be offered as a paid-for service.)
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Meteorology is where chaos was first studied. But we're back into deterministic vs random. If weather is purely macroscopic but chaotic then it can be deterministic but unpredictable. If large scale events are affected by non-deterministic quantum effects then it's random and unpredictable.
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Weather involves processes with multiple feedbacks and is therefore inherently chaotic i.e. exquisitely sensitive to small alterations in initial conditions. And since some of the processes that set those initial conditions are chemical, and chemistry is most accurately described by quantum mechanics, and quantum mechanics is non-deterministic: weather is ultimately random.
But.... I thought the science was settled and all scientists everywhere knew exactly what was going to happen with the climate?
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It's weird really; we can predict climate change 1,000 years in advance, but we can't say for sure if it'll rain tomorrow.
INB4
: Unless you're in Manchester, in which case, yes, it will rain tomorrow.
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we can predict climate change 1,000 years in advance, but we can't say for sure if it'll rain tomorrow
It's weird, we can predict when the next high tide will be but can't say for sure how far up the beach the next wave will break
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Don't confuse randomness with unpredictability.
But. That's exactly what randomness is, with the caveat that it's due to incomplete knowledge of the causes.
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Yes, there's QM in the depths of the detail there, but it really doesn't need to be invoked to explain why weather is hard to forecast for any length of time.
Has anyone showed any larger than atomic scale observations of quantum randomness? What's the largest apparent random effect?
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It's weird really; we can predict climate change 1,000 years in advance, but we can't say for sure if it'll rain tomorrow.
Faith is an amazing thing!
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you hit the limit of what you can predict due to the non-linearity of the fluid dynamics equations at a far larger scale than at the point where you need quantum mechanics to explain the chaos. Yes, there's QM in the depths of the detail there, but it really doesn't need to be invoked to explain why weather is hard to forecast for any length of time.
Quite so. Invoking it in some justifiable fashion does, however, put to bed any notion that we're dealing with a deterministic system even in principle.
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I thought the science was settled and all scientists everywhere knew exactly what was going to happen with the climate?
You've been smoking @boomzilla's straw men again, haven't you? Drugs are bad. Just say no.
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@PJH said:
I thought the science was settled and all scientists everywhere knew exactly what was going to happen with the climate?
You've been smoking @boomzilla's straw men again, haven't you? Drugs are bad. Just say no.
Faith is an amazing thing!
It fills the emptiness!
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It's weird really; we can predict climate change 1,000 years in advance, but we can't say for sure if it'll rain tomorrow.
Individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events (or "trials") is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, and applies to concepts of chance, probability, and information entropy.
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Has anyone showed any larger than atomic scale observations of quantum randomness?
Does this count?
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Does this count?
I'm not enough of a physicist to say. Does demonstrating wave particle duality demonstrate randomness?
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I'm not either, but I found it interesting that a molecule that large could exhibit wave-like characteristics. It made me wonder where the limit really is, as I wouldn't have expected it for anything larger than a particle.
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But what if the "hidden variables" are not hidden at all? What if they're hiding in plain sight, in the form of everything else that we don't think of as "part of" the subsystem whose behavior we're trying to predict? What if the whole thing is all cross-linked such that the precise behavior of any part ultimately depends on the total behavior of the whole?
You're joking but this theory is called 'superdeterminism' and it's an unpopular resolution to the EPR paradox.
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Yes.
Things small enough to have wave-particle duality are superimposed on themselves during the two-slit experiment, and are therefore fundamentally hidden from the outside world/completely unpredictable in their wave-function collapse.
Unless you believe in superdeterminism/hidden local variables, but those require either nonlocality(probably impossible) or giving up counterfactual definiteness(if we have to give up the ability to reason about things that have not happened, most of science breaks down anyway).
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Has anyone showed any larger than atomic scale observations of quantum randomness?
I think superpositions have been demonstrated up to the scale of viruses. Not so sure about the uncertainly principle; the minimum information uncertainty involved is really rather small in the first place.
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@accalia said:
There are patterns in primes, far more than we would expect in a truly random set.
Ah, the No True Random defense. Your still wrong. That the human brain, pattern detector par excellence, can detect patterns in randomness does not show that said randomness isn't random.
Time to drink another Ouisghian Zodah do trown oru dispare wee carn'd proofe de pattrons we regnize.
Has anyone showed any larger than atomic scale observations of quantum randomness? What's the largest apparent random effect?
Interesting thought from a physics book: How long would it take a pencil with a sharp tip to tip over if it was positioned exactly balanced on its tip? (According to quantum mechanic, there's an uncertainty in the angular momentum around an axis, in this case, we take one of the horzontal ones.)
1764839266 is actually not a prime number.
Did you try all possible bases?
[spoiler]It's always even, no matter the base (as long it's greater than 9) reason: [spoiler]even number of odd figures[/spoiler][/spoiler]
[spoiler] tags don't nest very well...
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I think superpositions have been demonstrated up to the scale of viruses
What's the real world effect of something like that?
Interesting thought from a physics book: How long would it take a pencil with a sharp tip to tip over if it was positioned exactly balanced on its tip? (According to quantum mechanic, there's an uncertainty in the angular momentum around an axis, in this case, we take one of the horzontal ones.)
So, could those minuscule things add up to anything terribly significant? Intuitively, it seems like it should be possible, but then again the world seems pretty deterministic.
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What's the real world effect of something like that?
Someone got to publish a paper in a fancy journal? :D
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What's the real world effect of something like that?
You might not need vaccination?
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You're joking
Why would you assume that? I've never seen a convincing refutation of superdeterminism. In fact it seems tautologically self-evident to me: only what has happened has happened, only what is happening is happening, and only what will happen will happen. The fact that in general we personally have or can even in principle gain only limited knowledge about what will happen (including interior happenings, such as what our "free" will is going to make us do next) doesn't change that.
Freedom, as applied to will, strikes me as an adjective much like "random": it's a reflection of our inability to predict something. Working from that definition, there is no essential conflict between the notion of free will and that of superdeterminism. They're just describing the same thing from two different reference frames.
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@boomzilla said:
Has anyone showed any larger than atomic scale observations of quantum randomness?
I think superpositions have been demonstrated up to the scale of viruses. Not so sure about the uncertainly principle; the minimum information uncertainty involved is really rather small in the first place.
It's pretty easy to design scale amplifiers for subatomic randomness. Geiger counters, scintillation counters, avalanche tubes, cloud chambers... the list goes on.
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Geiger counters, scintillation counters
I'm more one for charge and -1/-1 counters, myself.
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It's weird really; we can predict climate change 1,000 years in advance, but we can't say for sure if it'll rain tomorrow.
Yeah but the predictions will go to shit if some mutant kid who can control emotions is born.
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Hidden variable theories don't explain quantum randomness.
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I'm pretty sure I read something ages ago about how, for many datasets, the first digit is irregularly distributed, with
1more common than2, etc. The same applied to the second, etc. digit to lesser degrees, with distribution more even for less significant digits.
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Hidden variable theories don't explain quantum randomness.
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I'm pretty sure I read something ages ago about how, for many datasets, the first digit is irregularly distributed, with 1 more common than 2, etc.
IIRC, it's mostly related to the fact that you're overall talking about random numbers picked from a random range. That effectively means that you're picking random values that are evenly distributed in the log domain, and that in turn strongly biases the leading digit. It also means that you'd expect to see the effect in other number bases (except for binary and unary) though with different ratios.
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You're thinking of Benford's Law which is most often used to catch people fraudulently altering numeric data, for instance company accounts.
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Why would you assume that?
You linked "don't bogart that joint", is why I assumed that.
In any case, whether or not the universe is superdetermined is irrelevant, there's strong evidence free will is a bedtime story your brain tells itself.
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You linked "don't bogart that joint", is why I assumed that.
Fair enough. My main reason for doing that was that I think this particular corner of philosophy is so full of undecidable pseudo-profundity that most of it may as well be conversations among stoners.
there's strong evidence free will is a bedtime story your brain tells itself.
There's strong evidence that everything we consider to be true about the world is layer upon layer upon layer of exactly that kind of bedtime story. I am unconvinced that there's much that can be reasonably assumed about the so-called fundamental nature of reality beyond "something's going on". Even the very basis of reason - the act of identifying oneself as conceptually separable from the rest of the world - strikes me as likely, at its root, to be one of those convenient fictions.
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whether or not the universe is superdetermined is irrelevant
Only reason I brought up free will there is that the strongest arguments I've read against superdeterminism rest squarely on the assumption that the universe could not conceivably predetermine the outcome of any given experiment in any sense, because the decision about whether to even perform an experiment ultimately rests on the free will of potential experimenters.
It should be fairly obvious by now that I think that's an utterly cart-before-horse view of the way things work.
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I agree! People are so precious about 'free will'.
I think superdeterminism is stupid because it actually doesn't resolve nonlocality vs hidden variables, because it implies one or the other. But what do I know, IANAP.
