| Topic: |
Science > Physics |
| User: |
"" |
| Date: |
30 Aug 2006 10:38:08 PM |
| Object: |
Randomness debate, some ground rules |
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
That is crucial for my more political claim that mathematicians wrongly
ignore randomness that can be found with primes.
But hey, maybe I'm wrong, but what does that mean exactly?
Well, I say primes show no preference for a particular residue modulo a
lesser prime, so the behavior is random because no rules are around to
make it not be random.
If I am wrong, then that statement is what is wrong, and primes DO show
some preference modulo a lesser prime.
To understand what that means consider yet again what I showed with the
first 23 primes after 3:
5 mod 3 = 2, 7 mod 3 = 1, 11 mod 3 = 2, 13 mod 3 = 1, 17 mod 3 = 2,
19 mod 3 = 1, 23 mod 3 = 2, 29 mod 3 = 2, 31 mod 3 = 1, 37 mod 3 = 1,
41 mod 3 = 2, 43 mod 3 = 1, 47 mod 3 = 2, 53 mod 3 = 2, 59 mod 3 = 2,
61 mod 3 = 1, 67 mod 3 = 1, 71 mod 3 = 2, 73 mod 3 = 1, 79 mod 3 = 1,
83 mod 3 = 2, 89 mod 3 = 2, 97 mod 3 = 1
So the sequence is
2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1
and if the sequence is random, you can only say that 1 or 2 is next in
the sequence.
In case that doesn't make sense, randomness is about NOT knowing what
is coming next beyond the 50% probability, like with a coin, can you
predict whether it will be heads or tails?
(If you can you can make a lot of money betting people against your
ability.)
If there are no rules so that either possibility is equally likely,
then either possibility can occur.
That is crucial, if I am wrong, then there are rules that slant whether
you get 1 or 2, one way or the other and rules mean PREDICTION is
possible.
So someone might be able to say that after the 321st prime, you are
actually more likely to get 1 than 2, considering p_321 mod 3.
PREDICTION is key here.
If p mod 3 is NOT random, then knowing rules governing the behavior
could allow you to predict, that say, 2 is more likely to be next.
So remember, and this is crucial in this debate, that if I am wrong,
then there are some rules that could help you figure out which
number--1 or 2--would come next in that sequence.
There can be no other way, logically.
My fear, which is why I'm making yet another post, is that too many of
you have been beaten down by the easy tactic that mathematicians and
math people have of simply being abstruse.
If things get complicated, people zone out or fear that they're just
too stupid to get it.
Well let me be the one who looks stupid here. I'll ask the questions
no matter how stupid I look, and notice that people from math circles
have no problems with the put-downs.
And neither do I.
The politics here are HUGE. If I can convince some of you that math
people routinely lie in this area it could be an immense swing.
If I am wrong, then, oh well, I'll learn something myself about primes.
My credibility is not an issue.
Math people have already marked me as a crackpot across the web.
So now you know the stakes. I say math people are vicious and
ruthless, quite capable of lying on a huge scale, and of course, they
would be mad as hornets and ornery as cornered rattlesnakes for that to
come out, so I expect it to get very brutal.
But make no mistake, if p mod 3 is NOT random, then damn it, somebody
better start talking about how you predict the next number in the
sequence beyond saying there is a 50% chance it is 1, or 2.
Concrete tests are CRUCIAL when dealing with intelligent people who
have a lot to lose, and a history of successfully lying to a lot of
people all over the world.
James Harris
.
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| User: "Afterthen" |
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| Title: Re: Randomness debate, some ground rules |
30 Aug 2006 11:14:12 PM |
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<jstevh@msn.com> wrote in message
news:1156995488.140167.22710@74g2000cwt.googlegroups.com...
You prove it is random, first.
Go do your homework.
find out what "random" means.
Stop asking others to prove your ideas work, do it yourself.
Read some books.
your current approach is trivial, it is not random, it is not PN either.
.
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| User: "" |
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| Title: Re: Randomness debate, some ground rules |
30 Aug 2006 11:09:42 PM |
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wrote:
The politics here are HUGE. If I can convince some of you that math
people routinely lie in this area it could be an immense swing.
James Harris
James, you are all about the politics, you always have been. You want
recognition as someone with great abilities. You want to convince
others of this in the worst way. Going on what, 10 years now and no
luck. :-(
.
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| User: "" |
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| Title: Re: Randomness debate, some ground rules |
30 Aug 2006 11:02:37 PM |
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wrote:
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
That is crucial for my more political claim that mathematicians wrongly
ignore randomness that can be found with primes.
But hey, maybe I'm wrong, but what does that mean exactly?
Well, I say primes show no preference for a particular residue modulo a
lesser prime, so the behavior is random because no rules are around to
make it not be random.
If I am wrong, then that statement is what is wrong, and primes DO show
some preference modulo a lesser prime.
To understand what that means consider yet again what I showed with the
first 23 primes after 3:
5 mod 3 = 2, 7 mod 3 = 1, 11 mod 3 = 2, 13 mod 3 = 1, 17 mod 3 = 2,
19 mod 3 = 1, 23 mod 3 = 2, 29 mod 3 = 2, 31 mod 3 = 1, 37 mod 3 = 1,
41 mod 3 = 2, 43 mod 3 = 1, 47 mod 3 = 2, 53 mod 3 = 2, 59 mod 3 = 2,
61 mod 3 = 1, 67 mod 3 = 1, 71 mod 3 = 2, 73 mod 3 = 1, 79 mod 3 = 1,
83 mod 3 = 2, 89 mod 3 = 2, 97 mod 3 = 1
So the sequence is
2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1
and if the sequence is random, you can only say that 1 or 2 is next in
the sequence.
In case that doesn't make sense, randomness is about NOT knowing what
is coming next beyond the 50% probability, like with a coin, can you
predict whether it will be heads or tails?
(If you can you can make a lot of money betting people against your
ability.)
If there are no rules so that either possibility is equally likely,
then either possibility can occur.
That is crucial, if I am wrong, then there are rules that slant whether
you get 1 or 2, one way or the other and rules mean PREDICTION is
possible.
So someone might be able to say that after the 321st prime, you are
actually more likely to get 1 than 2, considering p_321 mod 3.
PREDICTION is key here.
If p mod 3 is NOT random, then knowing rules governing the behavior
could allow you to predict, that say, 2 is more likely to be next.
So remember, and this is crucial in this debate, that if I am wrong,
then there are some rules that could help you figure out which
number--1 or 2--would come next in that sequence.
There can be no other way, logically.
My fear, which is why I'm making yet another post, is that too many of
you have been beaten down by the easy tactic that mathematicians and
math people have of simply being abstruse.
If things get complicated, people zone out or fear that they're just
too stupid to get it.
Well let me be the one who looks stupid here. I'll ask the questions
no matter how stupid I look, and notice that people from math circles
have no problems with the put-downs.
And neither do I.
The politics here are HUGE. If I can convince some of you that math
people routinely lie in this area it could be an immense swing.
If I am wrong, then, oh well, I'll learn something myself about primes.
My credibility is not an issue.
Math people have already marked me as a crackpot across the web.
So now you know the stakes. I say math people are vicious and
ruthless, quite capable of lying on a huge scale, and of course, they
would be mad as hornets and ornery as cornered rattlesnakes for that to
come out, so I expect it to get very brutal.
But make no mistake, if p mod 3 is NOT random, then damn it, somebody
better start talking about how you predict the next number in the
sequence beyond saying there is a 50% chance it is 1, or 2.
Concrete tests are CRUCIAL when dealing with intelligent people who
have a lot to lose, and a history of successfully lying to a lot of
people all over the world.
James Harris
Why are you humiliating yourself? Just give up. It's not random.
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| User: "Mike Kelly" |
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| Title: Re: Randomness debate, some ground rules |
31 Aug 2006 03:46:33 AM |
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wrote:
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
That is crucial for my more political claim that mathematicians wrongly
ignore randomness that can be found with primes.
But hey, maybe I'm wrong, but what does that mean exactly?
Well, I say primes show no preference for a particular residue modulo a
lesser prime, so the behavior is random because no rules are around to
make it not be random.
If I am wrong, then that statement is what is wrong, and primes DO show
some preference modulo a lesser prime.
To understand what that means consider yet again what I showed with the
first 23 primes after 3:
5 mod 3 = 2, 7 mod 3 = 1, 11 mod 3 = 2, 13 mod 3 = 1, 17 mod 3 = 2,
19 mod 3 = 1, 23 mod 3 = 2, 29 mod 3 = 2, 31 mod 3 = 1, 37 mod 3 = 1,
41 mod 3 = 2, 43 mod 3 = 1, 47 mod 3 = 2, 53 mod 3 = 2, 59 mod 3 = 2,
61 mod 3 = 1, 67 mod 3 = 1, 71 mod 3 = 2, 73 mod 3 = 1, 79 mod 3 = 1,
83 mod 3 = 2, 89 mod 3 = 2, 97 mod 3 = 1
So the sequence is
2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1
and if the sequence is random, you can only say that 1 or 2 is next in
the sequence.
In case that doesn't make sense, randomness is about NOT knowing what
is coming next beyond the 50% probability, like with a coin, can you
predict whether it will be heads or tails?
(If you can you can make a lot of money betting people against your
ability.)
If there are no rules so that either possibility is equally likely,
then either possibility can occur.
That is crucial, if I am wrong, then there are rules that slant whether
you get 1 or 2, one way or the other and rules mean PREDICTION is
possible.
So someone might be able to say that after the 321st prime, you are
actually more likely to get 1 than 2, considering p_321 mod 3.
PREDICTION is key here.
If p mod 3 is NOT random, then knowing rules governing the behavior
could allow you to predict, that say, 2 is more likely to be next.
So remember, and this is crucial in this debate, that if I am wrong,
then there are some rules that could help you figure out which
number--1 or 2--would come next in that sequence.
There can be no other way, logically.
My fear, which is why I'm making yet another post, is that too many of
you have been beaten down by the easy tactic that mathematicians and
math people have of simply being abstruse.
If things get complicated, people zone out or fear that they're just
too stupid to get it.
Well let me be the one who looks stupid here. I'll ask the questions
no matter how stupid I look, and notice that people from math circles
have no problems with the put-downs.
And neither do I.
The politics here are HUGE. If I can convince some of you that math
people routinely lie in this area it could be an immense swing.
If I am wrong, then, oh well, I'll learn something myself about primes.
My credibility is not an issue.
Math people have already marked me as a crackpot across the web.
So now you know the stakes. I say math people are vicious and
ruthless, quite capable of lying on a huge scale, and of course, they
would be mad as hornets and ornery as cornered rattlesnakes for that to
come out, so I expect it to get very brutal.
But make no mistake, if p mod 3 is NOT random, then damn it, somebody
better start talking about how you predict the next number in the
sequence beyond saying there is a 50% chance it is 1, or 2.
Concrete tests are CRUCIAL when dealing with intelligent people who
have a lot to lose, and a history of successfully lying to a lot of
people all over the world.
James, the next number in the sequence is more likely to be whatever
the curent one isn't. So if the current number is a 1, the next one is
more likely a 2. And visa versa. This means the sequnce isn't random in
the way that a sequence of coin tosses is random. Because you can
predict what is coming next with >50% accuracy! I hope this makes sense
to you.
In that other thread Tim Peters did a good analysis to show this is the
case using Chi Squared tests. But I think you don't know much about
these statistical methods. That's OK, but you shouldn't get angry just
because you don't know about this method! It is a good and common way
of checking a statistical hypothesis. Not smoke and mirrors to try and
fool people!
You might like to read
http://www.georgetown.edu/faculty/ballc/webtools/web_chi_tut.html to
get an introduction to this method.
--
mike.
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| User: "Gib Bogle" |
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| Title: Re: Randomness debate, some ground rules |
31 Aug 2006 04:07:47 AM |
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wrote:
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
That is crucial for my more political claim that mathematicians wrongly
ignore randomness that can be found with primes.
But hey, maybe I'm wrong, but what does that mean exactly?
It means that nothing has changed.
.
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| User: "Jim Ferry" |
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| Title: Re: Randomness debate, some ground rules |
31 Aug 2006 12:54:50 AM |
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wrote:
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
That is crucial for my more political claim that mathematicians wrongly
ignore randomness that can be found with primes.
Mathematicians are, in fact, quite familiar with this heuristic notion
of randomness. A famous example of a result derived via such reasoning
is the Hardy-Littlewood conjecture on the density of twin primes. See
http://en.wikipedia.org/wiki/Twin_prime_conjecture
Note, however, that because this notion of randomness is heuristic, the
only results that can be obtained in such a manner are conjectures and
hypotheses rather than lemmas or theorems.
But hey, maybe I'm wrong, but what does that mean exactly?
Heuristically speaking, your statement is correct. What does that mean
exactly? Well, it doesn't exactly mean anything. It's like saying, "A
googolplex is big". Yep, it's pretty big all right. But unless "big"
is defined, it remains an imprecise statement.
For your statement that the (ascending) sequence {p mod 3 : p prime >
3} is random to be right or wrong entails defining precisely what it
means for an infinite sequence to be random. I looked over what you
wrote below, and it does not amount to a precise definition at all.
Several precise definitions of what it means for a sequence to be
random are given in Paul Vitanyi's review paper "Randomness":
http://arxiv.org/PS_cache/math/pdf/0110/0110086.pdf
I haven't been keeping up with the debate, but someone has probably
argued that the notion of a single sequence being random is
non-sensical. Taking a single point from a probability space and
arguing whether it's "random" or not? That certainly sounds
ridiculous. Nevertheless, we, like James, have an intuitive, heuristic
idea that some sequences are patterned, while others are "random".
This is the idea that Kolmogorov and others formalized . . . well, see
Vitanyi's paper.
Glossing over many details and distinctions, we may follow Kolmogorov
and think of an object as "compressible" if it has a description (e.g.,
a program that generates it) much shorter than itself. A random
sequence is one which is not compressible. It is clear that the
sequence {p mod 3 : p prime > 3} is not random in this sense, because
one can write a short program to generate it.
Nearly all infinite sequences are random in this sense, though of
course all sequences we can explicitly specify in some reasonable way
are necessarily non-random.
<rhetorical crescendo snipped>
So now you know the stakes. I say math people are vicious and
ruthless, quite capable of lying on a huge scale, and of course, they
would be mad as hornets and ornery as cornered rattlesnakes for that to
come out, so I expect it to get very brutal.
But make no mistake, if p mod 3 is NOT random, then damn it, somebody
better start talking about how you predict the next number in the
sequence beyond saying there is a 50% chance it is 1, or 2.
You predict the next number by computing it, i.e., by determining the
next prime and taking it mod 3. I assume you'll want to protest that
this against the rules, somehow, but it's not -- which is to say it's
not "against the rules" of the definitions of randomness given in
Vitanyi's review paper.
You may certainly make up your own rules by making your own precise
definition of what it means for an infinite sequence to be random.
However, the lack of ability to make precise statements is your
hallmark. This imprecision is what allows you to produce your
pseudo-proofs of great mathematical results.
Concrete tests are CRUCIAL when dealing with intelligent people who
have a lot to lose, and a history of successfully lying to a lot of
people all over the world.
Calm down. There's no conspiracy. It all makes sense when you define
the terms you use and reason clearly. It must be confusing not to
understand the distinction between actual mathematical arguments and
your imprecise ones. If you could just take a class where you turned
in proofs for homework, and allowed the grader to point out your
mistakes so you could learn from them . . .
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| User: "" |
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| Title: Re: Randomness debate, some ground rules |
01 Sep 2006 07:43:38 PM |
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Jim Ferry wrote:
jstevh@msn.com wrote:
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
That is crucial for my more political claim that mathematicians wrongly
ignore randomness that can be found with primes.
Mathematicians are, in fact, quite familiar with this heuristic notion
of randomness. A famous example of a result derived via such reasoning
is the Hardy-Littlewood conjecture on the density of twin primes. See
http://en.wikipedia.org/wiki/Twin_prime_conjecture
Note, however, that because this notion of randomness is heuristic, the
only results that can be obtained in such a manner are conjectures and
hypotheses rather than lemmas or theorems.
Well, physicists know that non-random behavior is predictable behavior.
There are then RULES that you can figure out which will guide the
behavior, like, you know, mathematical equations and things.
So there would be this maybe complicated mathematical equation that you
could plug in x, like at x=100, and it'd tell you that p mod 3 in that
area is more likely to be 1 than 2.
Or you could go to the 100th prime, and it'd tell you that prime mod 3
is more likely to be 1 than 2.
Scientists know that random means that prediction is about probability.
Non-random behavior is about rules which is what the scientific process
is all about--finding rules i.e. the laws of physics--which guide our
world.
So really there is no debate. If p mod 3 for a prime greater than 3 is
NOT random, then there are equations that would embody the laws of the
primes mod 3, you could say, just like in physics there are rules that
embody electromagnetic behavior.
In contrast, in quantum mechanics, you can find areas where the best
the rules tell you is that there is a 50% probability something will
happen one way versus another, so it is random.
I am surprised at your lack of intelligence in this area.
You sound somewhat muddled in your presentations which is why I felt a
need to remind you of the above.
Are you trained at all as a scientist? If so, I apologize for being
somewhat pedantic in this area, or worse, condescending?
Is any of it new to you?
Do you understand what random means in physics? Or how like in the
twin slit experiment you can have random behavior by electrons, yet in
a wire you can have electrons as a group following the laws of
electromagnetism in non-random behavior?
I'm needling Ferry a bit because he's supposedly very bright in terms
of his IQ.
So yes, he should know everything I said above without me having to go
through the boring effort of reminding him. What is your IQ Ferry?
James Harris
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| User: "Frank J. Lhota" |
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| Title: Re: JSH: Randomness debate, some ground rules |
02 Sep 2006 09:08:50 AM |
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<jstevh@msn.com> wrote in message
news:1157157818.600801.35380@e3g2000cwe.googlegroups.com...
Well, physicists know that non-random behavior is predictable behavior.
By that criteria, James Harris's posts are definitely not random. They
follow a highly predictable course:
- First, James Harris makes some sort of Mathematical discovery. He presents
his new discovery in a post that heralds his results as "new" and
"revolutionary". Because he proudly admits that he cannot be bothered to
research the topic, he is totally unaware of whether his discovery is flawed
or already discovered.
- Naturally, this is followed up by critiques written by people who actually
know what they are talking about. James Harris then does what he does best:
write florid rants about how the whole Mathematics community lie and cheat
when they deny his "obviously true" results. His unconstrained, paranoid
screeds definitely have their entertainment value, as he colorfully
describes how "the hammer of truth" will smite all these faux
Mathematicians, or present his ludicrous fantasy of Math professors being
carted off to jail. (BTW, what is your estimate of the number of people
serving in federal penitentiaries on the "bad Mathematics" rap?)
- Finally, some post gets through to him and James has to admit he is wrong.
Of course, this is *never* accompanied by any apology to those who he had
called liars and cheaters. Usually, he admissions of error are laced with
complaints of how rudely he was treated. The utter hypocracy of these
complaints gives them some entertainment value.
- After milking the now obviously worthless idea for all its worth, James
finally drops the topic. But wait! James has a new discovery! And so the
cycle begins again...
Will James Harris ever recongise this pattern, and change his tactics? No,
all of these points have been made to him before, many times. At some level,
he apparantly prefers to be famous as a crank to not being famous. And
enough posters are amused by his antics to engage him in threads. So it
looks like this pattern will repeat for quite awhile.
.
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| User: "W. Dale Hall" |
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| Title: Re: Randomness debate, some ground rules |
01 Sep 2006 09:14:39 PM |
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******** *********************
******** Welcome to the JSH Lecture Series *********************
******** *********************
Periodically, the JSH Lecture Series presents a lecture
on The Nature Of Things, presented from the refreshing
perspective of an alimentary canal with nonvanishing
first homology. While somewhat unorthodox, these lectures
provide clear insights into the workings of the Ouroborotic
Mind, and its relation to the Cabal of Lazy Mathematicians
Ruling the World and Preventing JSH from Getting the Babes.
Please join us today in welcoming our Guest (and Only) Lecturer,
the maligned JSH. As our Lecturer needs no introduction, we
begin our Lecture immediately. As per usual, no questions
will be entertained.
Enjoy your reading.
*****************************************************************
*****************************************************************
*****************************************************************
jstevh@msn.com wrote:
Jim Ferry wrote:
jstevh@msn.com wrote:
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
That is crucial for my more political claim that mathematicians wrongly
ignore randomness that can be found with primes.
Mathematicians are, in fact, quite familiar with this heuristic notion
of randomness. A famous example of a result derived via such reasoning
is the Hardy-Littlewood conjecture on the density of twin primes. See
http://en.wikipedia.org/wiki/Twin_prime_conjecture
Note, however, that because this notion of randomness is heuristic, the
only results that can be obtained in such a manner are conjectures and
hypotheses rather than lemmas or theorems.
Well, physicists know that non-random behavior is predictable behavior.
There are then RULES that you can figure out which will guide the
behavior, like, you know, mathematical equations and things.
So there would be this maybe complicated mathematical equation that you
could plug in x, like at x=100, and it'd tell you that p mod 3 in that
area is more likely to be 1 than 2.
Let me remind you of what you recently wrote:
Well, I say primes show no preference for a particular residue
modulo a lesser prime, so the behavior is random because no rules
are around to make it not be random.
which I take to mean that the limiting distribution of the residues of
P mod q (for P = all primes > q) is uniform as one considers larger
initial segments of P. If I've been reading correctly, this much has
been agreed to, and proofs pointed at. It is apparently not the piece
of cake you want it to be, but it's been proven nonetheless.
However, later in the same article, in discussing the case q = 3,
you conflate the notion of random as stated above with a different
one:
So the sequence is
2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1
and if the sequence is random, you can only say that 1 or 2 is next
in the sequence.
In case that doesn't make sense, randomness is about NOT knowing what
is coming next beyond the 50% probability, like with a coin, can you
predict whether it will be heads or tails?
(If you can you can make a lot of money betting people against your
ability.)
If there are no rules so that either possibility is equally likely,
then either possibility can occur.
That is crucial, if I am wrong, then there are rules that slant
whether you get 1 or 2, one way or the other and rules mean
PREDICTION is possible.
In this case, the present thread contains several articles that
demonstrate that your notion of unpredictability fails for this
particular case: if I know the residue mod 3 of a particular prime,
then I can predict with something like 57.5% success what the residue
mod 3 of the next prime is. Just in case you didn't notice, 57.5%
is *larger* than 50%. Casinos would love to have a margin of 15% (which
in reality amounts to a proportional margin of more like 30%, since
we're only talking about comparison to an ideal 50%).
Using q = 101, the failure of your heuristic is spectacular: Tim Peters
showed that he could guess the correct residue (mod 101) at something
ranging between 15% and 17%, as contrasted to your measly 0.99%.
Or you could go to the 100th prime, and it'd tell you that prime mod 3
is more likely to be 1 than 2.
Scientists know that random means that prediction is about probability.
Scientists know better than you: they actually conduct experiments.
Non-random behavior is about rules which is what the scientific process
is all about--finding rules i.e. the laws of physics--which guide our
world.
So really there is no debate. If p mod 3 for a prime greater than 3 is
NOT random, then there are equations that would embody the laws of the
primes mod 3, you could say, just like in physics there are rules that
embody electromagnetic behavior.
What you fail to realize is that your shoddy intellectual standards
and your inability to formulate precise questions will forever doom
you to making bonehead assertions. Jim Ferry was pointing out that
in order to say anything *meaningful* about the facts you're accusing
the mathematics community of laziness over, you need to be precise
about what you're saying. In particular, you need to be *precise*
in your definition of randomness.
You've already conflated the asymptotically-flat distribution of
the residues P (mod q) with the notion of predictability; while
the residues themselves approach a flat distribution, it seems
that the sequences of two residues may do nothing of the sort. Of
course, we've only seen anecdotal evidence in the present thread,
since the mod 3 numbers only went over the primes up to 10^7, and
for the mod 101 numbers only went over the primes up to 10^6.
However, even given the restriction of the census to an insignificant
portion of the set of all primes, that work provided more evidence
than all your arguments from the heart of ignorance.
In contrast, in quantum mechanics, you can find areas where the best
the rules tell you is that there is a 50% probability something will
happen one way versus another, so it is random.
I am surprised at your lack of intelligence in this area.
You imagine your miniscule experience to be mastery, don't you?
You sound somewhat muddled in your presentations which is why I felt a
need to remind you of the above.
Muddled? That's a laugh, especially coming from the person who says:
There are then RULES that you can figure out which will
guide the behavior, like, you know, mathematical equations
and things.
Spoken like a Nobel Laureate, I'm certain. Like, you know, someone
who might have, like, dude, um, won some kinda yeah, sorta prize of
Nobel and all, man.
Are you trained at all as a scientist? If so, I apologize for being
somewhat pedantic in this area, or worse, condescending?
The blind man lectures the seeing person on the nature of color.
Is any of it new to you?
Do you understand what random means in physics? Or how like in the
twin slit experiment you can have random behavior by electrons, yet in
a wire you can have electrons as a group following the laws of
electromagnetism in non-random behavior?
I'm sure he has heard the "expectation value of observable obeys
classical laws of motion" story. Who hasn't? I haven't met a third
grader who doesn't know that one.
I'm needling Ferry a bit because he's supposedly very bright in terms
of his IQ.
As contrasted to someone who's supposedly very clumsy in terms of his
lack of command of his native language. Just a thought, duh.
So yes, he should know everything I said above without me having to go
through the boring effort of reminding him. What is your IQ Ferry?
You *do* realize that this wouldn't be nearly as much fun if
you forgot to wear the "don't bump me, i'm full of gas" sign
around your neck.
Or don't you?
James Harris
Dale
.
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| User: "Tim Peters" |
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| Title: Re: JSH: Randomness debate, some ground rules |
02 Sep 2006 03:33:49 AM |
|
|
[added "JSH:" to subject]
[W. Dale Hall, to JSH]
Thank you for an entertaining and incisively argued post! I'm chopping
almost all of it because I'm done bothering with JSH on this topic, but I
want to clear up a few technical points, and raise a few others. "Few"
doesn't necessarily imply "brief" ;-) Readers are encouraged to go back and
read Dale's original.
....
In this case, the present thread contains several articles that
demonstrate that your notion of unpredictability fails for this
particular case: if I know the residue mod 3 of a particular prime,
then I can predict with something like 57.5% success what the residue
mod 3 of the next prime is. Just in case you didn't notice, 57.5%
is *larger* than 50%. Casinos would love to have a margin of 15% (which
in reality amounts to a proportional margin of more like 30%, since
we're only talking about comparison to an ideal 50%).
Using q = 101, the failure of your heuristic is spectacular: Tim Peters
showed that he could guess the correct residue (mod 101) at something
ranging between 15% and 17%, as contrasted to your measly 0.99%.
1% (exactly 1%) is the correct chance rate, as there are 101-1 = 100
non-zero residue classes mod 101. mod(p, 101) = 0 is impossible for prime p
101, so 0 is never seen and my program never guesses it.
Just for fun, here's a run with p=541, where the chance rate of guessing
right is the substantially smaller 1/540 ~= 0.185%.
simple_learning_guesser(10**6, p=541, n=1)
14693 correct in 100000 tries ~= 14.693% at prime 1301099
29872 correct in 200000 tries ~= 14.936% at prime 2751571
44867 correct in 300000 tries ~= 14.956% at prime 4257763
59574 correct in 400000 tries ~= 14.894% at prime 5801729
74190 correct in 500000 tries ~= 14.838% at prime 7370303
88416 correct in 600000 tries ~= 14.736% at prime 8962081
102421 correct in 700000 tries ~= 14.632% at prime 10572361
116397 correct in 800000 tries ~= 14.550% at prime 12197093
130351 correct in 900000 tries ~= 14.483% at prime 13835749
144155 correct in 999899 tries ~= 14.417% at prime 15485867
As sketched below, there's a particular reason for much of this seemingly
spectacular success that's easy to understand, but I hesitate to mention it
since James will surely leap to more bogus conclusions. OTOH, to reach
those, he'll have to learn something first :-)
....
You've already conflated the asymptotically-flat distribution of
the residues P (mod q) with the notion of predictability;
Yup.
while the residues themselves approach a flat distribution, it seems
that the sequences of two residues may do nothing of the sort.
Yup x 2, although see below.
Of course, we've only seen anecdotal evidence in the present thread,
since the mod 3 numbers only went over the primes up to 10^7, and
And we've also seen that the computed chi-square value gets worse as the
upper limit increases. It's already beyond human conception just /how/
unlikely it is for a random sequence to produce a chi-square statistic in
the low thousands with only 3 degrees of freedom. "No chance in hell"
doesn't even scrape it. For the primes up to 10^9:
non_overlapping_prime_mod_k_tuple(k=2, p=3, limit=10**9)
(1, 1) 5657771
(1, 2) 7051254
(2, 1) 7055917
(2, 2) 5658824
chisq 306302.383232 with 3 degrees of freedom
How unlikely is that? Let's cut him a break and drop the 0.383232 part :-)
The probability of seeing a chi-square statistic w/ 3 df >= 306302 is of
course reported as exactly 0 by every tool and reference easily available.
The "true" probability is:
UIG(3/2, 153151) / gamma(3/2) =
UIG(3/2, 153151) * 2 / sqrt(pi)
where UIG is the upper incomplete gamma function. Using the asymptotic
expansion (useful for large x):
UIG(a, x) ~~ x^(a-1)/e^x * [1 + (a-1)/x + (a-1)*(a-2)/x^2 + ...]
that's roughly (just taking the leading term -- (a-1)/x is already too small
to care about):
sqrt(153151)/e^153151 * 2 / sqrt(pi) ~=
1 / 10^66510
Of course a random sequence /should/ give a 3 df chi-square value >= 306302
about once per 10^66510 tries, so maybe we were just lucky here. LOL.
for the mod 101 numbers only went over the primes up to 10^6.
The results posted earlier for that were through the first 10^6 primes, not
the primes <= 10^6. The millionth prime is about 15*10^6. As I noted, but
didn't show, the 3-tuple learning guesser continued to improve through about
the first 7.2 million primes, which is all the primes through about
126*10^6.
Moving on, in a different post I explained (and this is mostly obvious) how
the counts of residue pairs in the mod(p, 3) sequence are determined by the
gaps g_i between successive primes, g_i = p_(i+1) - p_i. Skipping a repeat
of the derivation,
The number of <1, 2> pairs equals the number of g_i in
{4, 10, 16, 22, 28, ..., 6*j+4, ...}
The number of <2, 1> pairs equals the number of g_i in
{2, 8, 14, 20, 26, ..., 6*j+2, ...}
and the /sum/ of the number of <1, 1> and <2, 2> pairs equals the number of
g_i in
{6, 12, 18, 24, 30, ..., 6*j+0, ...}
It's important here to recount how the last one was obtained:
p_(i+1) = p_i (mod 3) if and only if
p_(i+1) - p_i = 0 (mod 3) if and only if
g_i = 0 (mod 3) if and only if, assuming i > 1,
g_i = 0 (mod 6) since all g_i are even when i > 1
That's why the sum of the <1, 1> and <2, 2> counts equals the number of g_i
divisible by 6.
Now what do we know about the distribution of prime gaps? Very little
non-trivial has been proved, but it's been /conjectured/ that the number of
gaps equal to g across the primes <= x is asymptotically equal to
C_g * integral of dt/ln(t)^2 from 2 to x
where
C_g = D_g * 2 * PI2
where PI2 is the twin-primes constant ~= 0.6601618158, and D_g is another
constant depending on g. Note that D_g is the /only/ part of the conjecture
that depends on g.
Skipping more details, these are some of the conjectured values:
D_2 = 1
D_4 = 1
D_6 = 2
If the conjectures are true, note that in the sets listed above it /is/
expected that the counts for g=2 and g=4 are asymptotically equal, and that
the count for g=6 is asymptotically equal to their /sum/. So if 2, 4, and 6
were the only gaps, it's quite possible that the distribution of the 4
residue pairs here /would/ be asymptotically equal.
Accounting for the infinite number of other possible gap values too is more
than I've had time for today ;-), but I wouldn't be notably surprised if it
were possible to prove that (assuming the relevant conjectures are true) the
4 possible residue pairs /are/ asymptotically equidistributed.
But, in practice, while for g=2:
C_2 * integral of dt/ln(t)^2 from 2 to x
is an empurically very good estimate of the number of twin primes <= x, and
also good for g=4, for reasons explained elsewhere it's a poor estimate for
g > 4 across "tractably small" primes (I've referenced a relevant paper of
Brent's several times, so search for that if you want to know a lot more).
For example, although it's conjectured (and even widely believed) that the
g=6 count is asymptotically twice the g=2 count, looking at the primes <
10^9 you probably wouldn't guess that. Even skipping primes < 10^6 to give
g=6 a "fairer chance", in [10^6, 10^9] the g=6 count is 6076242 and the g=2
count 3416337. The ratio g=6/g=2 ~= 1.78 there doesn't even particularly
/suggest/ "2".
And the conjectured asymptotic ratios above only get farther from actual
counts across "small" primes for g > 6. For example, it's not a coincidence
that D_2 = D_4 = 1, as it's in fact conjectured that D_i = 1 for /every/ i
that's a power of 2. Now just count the number of primes less than, say,
10^50, with prime gap = 2^1000. Of course the count is exactly 0 (there's
at least one prime between 10^50 and 2*10^50, so no gap in the range could
exceed 10^50), not related at all to the count of twin primes.
(BTW, Brent develops what appear to be very good estimates for g > 4, which
are asymptotically equal to the ones above, but very different at "small x")
The upshot is this: even if the residue tuples /are/ asymptotically
equidistributed, unlike as for the distribution of the primes alone we don't
get anywhere near the asymptotic behavior across the range of primes small
enough to work with. The ludicrously LOL large chi-square value above is
sufficient proof of that even at the tiny p=3.
Finlly, what about p=541? In that case, by the same reasoning as above,
p_(i+1) = p_i (mod 541) if and only if
g_i = 0 (mod 1082)
While it's conjectured that the number of primes <= x for which g_i = 1082
asymptotically approaches ~1.32277 * x/ln(x)^2, I don't even know of /one/
prime for which 1082 | g_i. According to:
http://www.research.att.com/~njas/sequences/A053303
the largest gap for any prime < 10^15 is 906.
Therefore we can never see two consecutive equal values in the mod(p, 541)
sequence before hitting primes > 10^15. In fact, a huge number of (i, j)
pairs are entirely missing from the pairs actually seen during the program
run shown above. For example, not only was 1 never followed by 1, there
were only 37 (of 540 possible) values 1 /was/ followed by.
Why wasn't, e.g., 1 ever followed by 2? Well,
p_i = 1 (mod 541)
and
p_(i+1) = p_i + g_i = 2 (mod 541)
implies
g_i = 1 (mod 541)
so we'd need a gap of the form 541*j + 1. Since g also needs to be even,
the smallest gap that could work is 541*1 + 1 = 542. Again consulting the
table above, there are no primes with gap that large < 10^13. Of course
similar arguments explain why many other pairs can never be seen before
reaching large primes.
And it only gets worse for p > 541: even /if/ the asymptotics are
equidistributed, it's flatly impossible for these sequences to behave even
vaguely randomly across ranges of primes that are computationally tractable.
.
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| User: "David Bernier" |
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| Title: Re: JSH: Randomness debate, some ground rules |
02 Sep 2006 08:18:30 AM |
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Tim Peters wrote:
[...]
And we've also seen that the computed chi-square value gets worse as the
upper limit increases. It's already beyond human conception just /how/
unlikely it is for a random sequence to produce a chi-square statistic in
the low thousands with only 3 degrees of freedom. "No chance in hell"
doesn't even scrape it. For the primes up to 10^9:
non_overlapping_prime_mod_k_tuple(k=2, p=3, limit=10**9)
(1, 1) 5657771
(1, 2) 7051254
(2, 1) 7055917
(2, 2) 5658824
chisq 306302.383232 with 3 degrees of freedom
How unlikely is that? Let's cut him a break and drop the 0.383232 part :-)
The probability of seeing a chi-square statistic w/ 3 df >= 306302 is of
course reported as exactly 0 by every tool and reference easily available.
The "true" probability is:
UIG(3/2, 153151) / gamma(3/2) =
UIG(3/2, 153151) * 2 / sqrt(pi)
where UIG is the upper incomplete gamma function. Using the asymptotic
expansion (useful for large x):
UIG(a, x) ~~ x^(a-1)/e^x * [1 + (a-1)/x + (a-1)*(a-2)/x^2 + ...]
that's roughly (just taking the leading term -- (a-1)/x is already too small
to care about):
sqrt(153151)/e^153151 * 2 / sqrt(pi) ~=
1 / 10^66510
Of course a random sequence /should/ give a 3 df chi-square value >= 306302
about once per 10^66510 tries, so maybe we were just lucky here. LOL.
[...]
There's no disputation regarding your "once per 10^66510 tries"
estimate. On the other hand, we do know that your data were not
obtained by paper and pencil computations. There are reasonable
grounds to believe that they were obtained from a computer having
one or more hard drives ("drives"). Frequently, these drives
interrupt your computer's CPU though some IRQ line. Unbeknownst
to you, one of your hard drives may in fact be an Improbability
Drive, which causes your CPU to malfunction. While the origin of
these drives is not clear, there could very well be some connection
to Area 51 and/or elite mathematicians.
David Bernier (playing the role of JSH's advocate)
.
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| User: "Jim Ferry" |
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| Title: Re: Randomness debate, some ground rules |
03 Sep 2006 12:22:01 AM |
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wrote:
I am surprised at your lack of intelligence in this area.
You sound somewhat muddled in your presentations which is why I felt a
need to remind you of the above.
Are you trained at all as a scientist? If so, I apologize for being
somewhat pedantic in this area, or worse, condescending?
Is any of it new to you?
Do you understand what random means in physics? Or how like in the
twin slit experiment you can have random behavior by electrons, yet in
a wire you can have electrons as a group following the laws of
electromagnetism in non-random behavior?
Thank you for your personalized response James. And I thought the
Golden Age was past. Maybe it's dawning.
.
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| User: "" |
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| Title: Re: Randomness debate, some ground rules |
01 Sep 2006 08:38:29 PM |
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wrote:
Jim Ferry wrote:
wrote:
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
That is crucial for my more political claim that mathematicians wrongly
ignore randomness that can be found with primes.
Mathematicians are, in fact, quite familiar with this heuristic notion
of randomness. A famous example of a result derived via such reasoning
is the Hardy-Littlewood conjecture on the density of twin primes. See
http://en.wikipedia.org/wiki/Twin_prime_conjecture
Note, however, that because this notion of randomness is heuristic, the
only results that can be obtained in such a manner are conjectures and
hypotheses rather than lemmas or theorems.
Well, physicists know that non-random behavior is predictable behavior.
There are then RULES that you can figure out which will guide the
behavior, like, you know, mathematical equations and things.
So there would be this maybe complicated mathematical equation that you
could plug in x, like at x=100, and it'd tell you that p mod 3 in that
area is more likely to be 1 than 2.
Or you could go to the 100th prime, and it'd tell you that prime mod 3
is more likely to be 1 than 2.
Scientists know that random means that prediction is about probability.
Non-random behavior is about rules which is what the scientific process
is all about--finding rules i.e. the laws of physics--which guide our
world.
So really there is no debate. If p mod 3 for a prime greater than 3 is
NOT random, then there are equations that would embody the laws of the
primes mod 3, you could say, just like in physics there are rules that
embody electromagnetic behavior.
In contrast, in quantum mechanics, you can find areas where the best
the rules tell you is that there is a 50% probability something will
happen one way versus another, so it is random.
Doesn't 01010101010101010101... have a 50% probability?
But you wouldn't say that's random, would you? There's
more to randomness than simply probability. When you
aggregate multiple probabilities you have to look at
distribution.
I am surprised at your lack of intelligence in this area.
Look, forget all that chi stuff and look at your coin
flipping model again. When you flip coins, sometimes
you get 2 heads in a row, sometimes you get 3, often
you only get 1 before your run is interrupted by a tails.
Were you to actually flip the coin a whole bunch of times,
say, 17984 times and counted how many times you got
groups of 1, 2, 3, etc., you'll see something like this:
group # of groups # of digits
1 2277 2277
11 1140 2280
111 561 1683
1111 263 1052
11111 142 710
111111 81 486
1111111 29 203
11111111 16 128
111111111 10 90
1111111111 2 20
11111111111 2 22
Probability tells us that the mean digits/group is the inverse
of the probability. Since the probability of the coin flip is
1/2, then the mean number of consecutive heads ought to be 2.
sum of groups
4523
sum of digits
8951
mean digits/group
1.978996241
How about that? Randomness IS predictable to a certain extent.
Would it matter if we looked at consecutive tails?
group # of groups # of digits
2 2261 2261
22 1122 2244
222 560 1680
2222 304 1216
22222 146 730
222222 66 396
2222222 34 238
22222222 13 104
222222222 7 63
2222222222 5 50
22222222222 1 11
2222222222222 2 24
22222222222222 1 14
sum of groups
4522
sum of digits
9031
mean digits/group
1.997125166
Guess where this is leading? How are your primes mod 3 clustered?
If they were random, the mean group size should be approximately
2, right?
group # of groups # of digits
1 3049 3049
11 1400 2800
111 578 1734
1111 194 776
11111 79 395
111111 22 132
1111111 12 84
11111111 1 8
1111111111 1 10
sum of groups
5336
sum of digits
8988
mean digits/group
1.684407796
group # of groups # of digits
2 3040 3040
22 1426 2852
222 549 1647
2222 198 792
22222 85 425
222222 33 198
2222222 6 42
sum of groups
5337
sum of digits
8996
mean digits/group
1.685591156
Oh dear, those mean group sizes aren't anywhere near what they
should be if they were randomly distributed. You have way too
many groups of size 1. And look, the 2's never got more than 7
in a row. That's way less than would be expected for a random
run of 17984.
Are you still conviced the primes mod 3 are random?
You sound somewhat muddled in your presentations which is why I felt a
need to remind you of the above.
Are you trained at all as a scientist? If so, I apologize for being
somewhat pedantic in this area, or worse, condescending?
Is any of it new to you?
Do you understand what random means in physics? Or how like in the
twin slit experiment you can have random behavior by electrons, yet in
a wire you can have electrons as a group following the laws of
electromagnetism in non-random behavior?
I'm needling Ferry a bit because he's supposedly very bright in terms
of his IQ.
So yes, he should know everything I said above without me having to go
through the boring effort of reminding him. What is your IQ Ferry?
James Harris
.
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| User: "David Moran" |
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| Title: Re: Randomness debate, some ground rules |
02 Sep 2006 10:08:25 AM |
|
|
wrote:
Jim Ferry wrote:
wrote:
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
That is crucial for my more political claim that mathematicians wrongly
ignore randomness that can be found with primes.
Mathematicians are, in fact, quite familiar with this heuristic notion
of randomness. A famous example of a result derived via such reasoning
is the Hardy-Littlewood conjecture on the density of twin primes. See
http://en.wikipedia.org/wiki/Twin_prime_conjecture
Note, however, that because this notion of randomness is heuristic, the
only results that can be obtained in such a manner are conjectures and
hypotheses rather than lemmas or theorems.
Well, physicists know that non-random behavior is predictable behavior.
There are then RULES that you can figure out which will guide the
behavior, like, you know, mathematical equations and things.
So there would be this maybe complicated mathematical equation that you
could plug in x, like at x=100, and it'd tell you that p mod 3 in that
area is more likely to be 1 than 2.
Or you could go to the 100th prime, and it'd tell you that prime mod 3
is more likely to be 1 than 2.
Scientists know that random means that prediction is about probability.
Non-random behavior is about rules which is what the scientific process
is all about--finding rules i.e. the laws of physics--which guide our
world.
So really there is no debate. If p mod 3 for a prime greater than 3 is
NOT random, then there are equations that would embody the laws of the
primes mod 3, you could say, just like in physics there are rules that
embody electromagnetic behavior.
In contrast, in quantum mechanics, you can find areas where the best
the rules tell you is that there is a 50% probability something will
happen one way versus another, so it is random.
I am surprised at your lack of intelligence in this area.
You sound somewhat muddled in your presentations which is why I felt a
need to remind you of the above.
Are you trained at all as a scientist? If so, I apologize for being
somewhat pedantic in this area, or worse, condescending?
Is any of it new to you?
Do you understand what random means in physics? Or how like in the
twin slit experiment you can have random behavior by electrons, yet in
a wire you can have electrons as a group following the laws of
electromagnetism in non-random behavior?
I'm needling Ferry a bit because he's supposedly very bright in terms
of his IQ.
So yes, he should know everything I said above without me having to go
through the boring effort of reminding him. What is your IQ Ferry?
James Harris
Better question: What's your IQ, Harris?
Dave
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| User: "William Hughes" |
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| Title: Re: Randomness debate, some ground rules |
31 Aug 2006 09:32:55 AM |
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wrote:
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
And,as has been pointed out by many people you are wrong.
You will shortly admit this.
I suspect that you will then inform everyone that the concepts
of randomness in use (including your own "no predictability") are
wrong (this will involve some conspiracy). Amazingly, your
comments about the uselessness of current research into prime
pairs will not change.
-William Hughes
.
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| User: "Pavel314" |
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| Title: Retro Randomness |
31 Aug 2006 09:47:09 AM |
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IIIIIVVIIXIXIIIXVIIXIXXXIIIXXIXXXXIXXXVIILXILXIIILXVII.....
The above is the first few primes written in Roman numerals. Are the letters
in the sequence random?
I suppose it would be easy enough for someone with the proper software to
translate thousands of primes to Roman numerals, string the individual
letters into a sequence and run a check for randomness. You'd probably have
to terminate the list at a fairly low prime, like just before 1,000, the
Roman system's highest-valued numeral as the number of M's would become
overwhelming for numbers in the range of 500,000.
Of course, you could expand the Roman system for higher valued numerals. The
current pattern is:
1
5
10
50
100
500
1,000
So we could keep going on the multiples of 5 and 2 sequence, making new
numerals for:
5,000
10,000
50,000
100,000
etc.
Paul
.
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| User: "Richard Henry" |
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| Title: Re: Retro Randomness |
31 Aug 2006 11:37:19 AM |
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Pavel314 wrote:
IIIIIVVIIXIXIIIXVIIXIXXXIIIXXIXXXXIXXXVIILXILXIIILXVII.....
I believe that should be
IIIIIIVVIIXIXIIXVII .. etc
The above is the first few primes written in Roman numerals. Are the letters
in the sequence random?
.
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| User: "Pavel314" |
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| Title: Re: Retro Randomness |
31 Aug 2006 02:32:30 PM |
|
|
"Richard Henry" <pomerado@hotmail.com> wrote in message
news:1157042239.002102.203660@h48g2000cwc.googlegroups.com...
Pavel314 wrote:
I should have spaced them out a bit.
II III V VII XI XIII XVII XIX XXIII XXIX XXXI XXXVII LXI LXIII LXVII.....
2 3 5 7 11 13 17 19 23 29 31 37
wrong! wrong! wrong!
I flipped the X and L for 41, 43 and 47. Should be XLI XLIII XLVII
I believe that should be
Spacing the next series
II III IV VII XI XII XVII .. etc
gives
2 3 4 7 11 12 17
Makes you glad we switched to Arabic numerals, doesn't it?
Paul
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| User: "Eric Gisse" |
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| Title: Re: Randomness debate, some ground rules |
01 Sep 2006 06:55:09 PM |
|
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wrote:
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
[...]
Why don't you take a break from such groundbreaking research to explain
why you insist on crossposting to sci.physics and sci.skeptic all the
time?
Don't worry, I didn't touch the headers - I'm sure that I am not the
only one wondering.
.
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| User: "kunzmilan" |
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| Title: Re: Randomness debate, some ground rules |
03 Sep 2006 07:47:10 AM |
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Eric Gisse wrote:
jstevh@msn.com wrote:
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
We have 2 values. Replace 2 = 1, 1 = 0 and get the binomial
distribution.
The distances between consecutive 0 and 1 in their strings are
described by the negative binomial distribution.
The randomness of any distribution is evaluated by its chisquare value,
thus it can be measured, similarly as randomness of digits in
irrational numbers.
kunzmilan
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| User: "ldb" |
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| Title: Re: Randomness debate, some ground rules |
31 Aug 2006 11:09:26 AM |
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In case that doesn't make sense, randomness is about NOT knowing what
is coming next beyond the 50% probability, like with a coin, can you
predict whether it will be heads or tails?
As a summary of the arguments that you won't read, allow me to
simplify...
If you predict that the next number in the sequence is the opposite of
the current one, you will be right ~60% of the time.
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| User: "David Moran" |
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| Title: Re: Randomness debate, some ground rules |
04 Sep 2006 10:01:31 PM |
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My credibility is not an issue.
Because you have none, except for being a crackpot.
Math people have already marked me as a crackpot across the web.
Hmm, I wonder why you're marked as a crackpot. LEARN SOME MATH!
Dave
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| User: "Mark VandeWettering" |
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| Title: Re: Randomness debate, some ground rules |
30 Aug 2006 11:55:48 PM |
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["Followup-To:" header set to alt.math.recreational.]
On 2006-08-31, <> wrote:
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
There is no actual debate. Such a sequence isn't random. It isn't
pseudorandom. It's hopelessly biased.
That is crucial for my more political claim that mathematicians wrongly
ignore randomness that can be found with primes.
They ignore the claim because it is patentedly and obviously false.
But hey, maybe I'm wrong, but what does that mean exactly?
There is no maybe about it. You're wrong.
Well, I say primes show no preference for a particular residue modulo a
lesser prime, so the behavior is random because no rules are around to
make it not be random.
Yes, well, that's just the kind of precise thinking that people have come
to expect from you.
If I am wrong, then that statement is what is wrong, and primes DO show
some preference modulo a lesser prime.
No, it shows that you haven't a clue what the term "random" denotes.
To understand what that means consider yet again what I showed with the
first 23 primes after 3:
5 mod 3 = 2, 7 mod 3 = 1, 11 mod 3 = 2, 13 mod 3 = 1, 17 mod 3 = 2,
19 mod 3 = 1, 23 mod 3 = 2, 29 mod 3 = 2, 31 mod 3 = 1, 37 mod 3 = 1,
41 mod 3 = 2, 43 mod 3 = 1, 47 mod 3 = 2, 53 mod 3 = 2, 59 mod 3 = 2,
61 mod 3 = 1, 67 mod 3 = 1, 71 mod 3 = 2, 73 mod 3 = 1, 79 mod 3 = 1,
83 mod 3 = 2, 89 mod 3 = 2, 97 mod 3 = 1
So the sequence is
2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1
and if the sequence is random, you can only say that 1 or 2 is next in
the sequence.
If the sequence was uniformly random, you'd be unable to predict the next
number with a success rate different than 50%.
In case that doesn't make sense, randomness is about NOT knowing what
is coming next beyond the 50% probability,
Wow, you are really quite dense. You can know. That's rather the point.
If we look at all the primes > 3 and less than 200,000, and look at adjacent
pairs of entries from your sequence, we get the following distribution.
3652 11
5335 12
5336 21
3658 22
If we just saw a 1 from your sequence, about sixty percent of the time
it will be followed by a 2, and only forty percent by a one. Similarly
if we see a 2, we are more likely to see a 1 than another two. We can
extend this analysis to longer strings like three digits...
1365 111
2287 112
3039 121
2296 122
2287 211
3048 212
2296 221
1362 222
or four...
478 1111
887 1112
1262 1121
1025 1122
1236 1211
1803 1212
1425 1221
871 1222
887 2111
1400 2112
1777 2121
1271 2122
1051 2211
1244 2212
871 2221
491 2222
like with a coin, can you
predict whether it will be heads or tails?
(If you can you can make a lot of money betting people against your
ability.)
If there are no rules so that either possibility is equally likely,
then either possibility can occur.
No, either possibility cannot occur. One will occur. The reason that
a particular number occurs is the result of a well defined sieving
process, about which nearly nothing is random.
That is crucial, if I am wrong, then there are rules that slant whether
you get 1 or 2, one way or the other and rules mean PREDICTION is
possible.
You don't even need to know the rules. Just do the experiment. If you'd
like to give me 1:1 odds that I can't predict the next number in the sequence,
I've got a stack of quarters all ready to play.
So someone might be able to say that after the 321st prime, you are
actually more likely to get 1 than 2, considering p_321 mod 3.
In the limit, the primes show no bias towards 1 or 2, but that is insufficient
to show that the sequence is random.
PREDICTION is key here.
Yes, it is. I've got my quarters all ready.
If p mod 3 is NOT random, then knowing rules governing the behavior
could allow you to predict, that say, 2 is more likely to be next.
We just did that. Weren't you paying attention?
So remember, and this is crucial in this debate, that if I am wrong,
then there are some rules that could help you figure out which
number--1 or 2--would come next in that sequence.
Yes. There is. I showed you how.
There can be no other way, logically.
My fear, which is why I'm making yet another post, is that too many of
you have been beaten down by the easy tactic that mathematicians and
math people have of simply being abstruse.
There is nothing abstruse about this argument. It's simply a matter of
running an experiment.
If things get complicated, people zone out or fear that they're just
too stupid to get it.
I fear someone is too stupid to get it, but it isn't me I'm worried about.
Well let me be the one who looks stupid here. I'll ask the questions
no matter how stupid I look, and notice that people from math circles
have no problems with the put-downs.
If you don't fear looking stupid, you shouldn't be surprised when people
call you stupid.
And neither do I.
The politics here are HUGE. If I can convince some of you that math
people routinely lie in this area it could be an immense swing.
I'm not sure how demonstrating your ignorance and stupidity will achieve
that end, but by all means, continue.
If I am wrong, then, oh well, I'll learn something myself about primes.
Now you are straining credulity.
My credibility is not an issue.
It is also not in dispute.
Math people have already marked me as a crackpot across the web.
With some just cause, it must be ntoed.
So now you know the stakes. I say math people are vicious and
ruthless, quite capable of lying on a huge scale, and of course, they
would be mad as hornets and ornery as cornered rattlesnakes for that to
come out, so I expect it to get very brutal.
Even if this were true, the sequence you describe isn't random.
But make no mistake, if p mod 3 is NOT random, then damn it, somebody
better start talking about how you predict the next number in the
sequence beyond saying there is a 50% chance it is 1, or 2.
Look at the last number generated in the sequence.
If it is a 1, guess 2.
If it is a 2, guess 1.
Concrete tests are CRUCIAL when dealing with intelligent people who
have a lot to lose, and a history of successfully lying to a lot of
people all over the world.
I can't make it more concrete than that.
Mark
James Harris
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| User: "Tim Peters" |
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| Title: Re: JSH: Randomness debate, some ground rules |
31 Aug 2006 07:54:33 AM |
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[added "JSH:" to subject]
[jstevh@msn.com]
So now I'm in a debate about whether or not p mod 3, with p an odd
prime greater than 3, is random or not, when I say the result of
marching up the primes is random, like coin flips.
For your sake, I hope you stop saying that soon :-)
That is crucial for my more political claim that mathematicians wrongly
ignore randomness that can be found with primes.
But you have no real idea what mathematicians have /done/ in this area, even
100 years back. Why would anyone imagine you know what they do now, let
alone whether they're right or wrong?
But hey, maybe I'm wrong, but what does that mean exactly?
Umm ... it would mean you were wrong?
Well, I say primes show no preference for a particular residue modulo a
lesser prime,
In a certain precise sense, that's true.
so the behavior is random because no rules are around to make it not
be random.
QED -- LOL. Son, that's not even an inbred distant cousin of a coherent
argument's shadow.
If I am wrong, then that statement is what is wrong, and primes DO show
some preference modulo a lesser prime.
It's actually the case that:
the behavior is random because no rules are around to make it not
be random
reaches "not even wrong" status.
To understand what that means consider yet again what I showed with the
first 23 primes after 3:
Na, but thanks for offering. Repetition doesn't make it "more true". Never
does. The problem here isn't that people didn't understand your claim the
first time around.
...
That is crucial, if I am wrong, then there are rules that slant whether
you get 1 or 2, one way or the other and rules mean PREDICTION is
possible.
How many days have people been telling you that's exactly the case here?
And showing you why? Forget my chi square stuff: while that's a rigorous
way to show non-randomness here, all you really have to do is /look/ at the
counts of adjacent pairs multiple people have posted. It's dead obvious.
So someone might be able to say that after the 321st prime, you are
actually more likely to get 1 than 2, considering p_321 mod 3.
PREDICTION is key here.
If p mod 3 is NOT random, then knowing rules governing the behavior
could allow you to predict, that say, 2 is more likely to be next.
So remember, and this is crucial in this debate, that if I am wrong,
then there are some rules that could help you figure out which
number--1 or 2--would come next in that sequence.
There can be no other way, logically.
All of this was addressed in my first reply to you, you know.
My fear, which is why I'm making yet another post, is that too many of
you have been beaten down by the easy tactic that mathematicians and
math people have of simply being abstruse.
Like this?
Look at adjacent-prime pairs through 10000019 and it gets worse
again:
number with residues (1, 1): 71286
number with residues (1, 2): 94939
number with residues (2, 1): 94683
number with residues (2, 2): 71381
If you didn't understand the
chi = 6635.6 with 3 degrees of freedom
following, so what, just /look at the counts/. Is it "abstruse" to expect
you to notice that 94939 and 94683 are more than an ignorably little bit
larger than 71286 and 71381? 3-i follows i way more often than i follows i
to hold a rational belief that the sequence is random. The chi-square test
is a standard way to /quantify/ just how "way more often" that is, but it's
freakin' obvious by mere eyeball inspection.
If things get complicated, people zone out or fear that they're just
too stupid to get it.
Is anyone here really "too stupid" to look at the counts above and make the
obvious inference? Hmm. I see there is one person here who hasn't figured
that out yet, and who has started a seemingly endless number of new and
irresponsibly cross-posted threads advertising insistently that he just
can't see it.
Well let me be the one who looks stupid here. I'll ask the questions
no matter how stupid I look, and notice that people from math circles
have no problems with the put-downs.
But you don't ask any questions, James, apart from long strings of
rhetorical counterfactual hypotheticals ("but just consider, if I'm right,
which mathematicians should be executed first?"). Asking an /honest/
question is tantamount to admitting you don't know the answer -- which has
always been my guess for why you so rarely ask questions about what's
already known (or, indeed, so rarely read a relevant technical paper or
book).
And neither do I.
The politics here are HUGE. If I can convince some of you that math
people routinely lie in this area it could be an immense swing.
Then you'd be better off picking an area where you're not so obviously
wrong, eh?
If I am wrong, then, oh well, I'll learn something myself about primes.
My credibility is not an issue.
:-)
Math people have already marked me as a crackpot across the web.
Harangue people on sci.physics instead about, oh, how Stephen Hawking is
engaged in a conspiracy to suppress your breakthrough theory that gravity is
/really/ just a temperature inversion, and maybe you'll get a break from
math people calling you names ;-)
So now you know the stakes.
Curious how you always set conflicts up so that /you/ have nothing at stake.
If you're right, your loyal opposition are lying scum who deserve to be
killed by a righteously outraged public, but if you're wrong ... oh well,
you'll learn something about primes. And people who resented your prolific
production of new threads screaming that you were right? Oh well, screw
them, that's their problem. After all, you learned something about primes
and didn't even need to open a book to do it!
I say math people are vicious and ruthless, quite capable of lying on
a huge scale,
You say a lot of goofy things.
and of course, they would be mad as hornets and ornery as cornered
rattlesnakes
LOL -- I do like the imagery.
for that to come out, so I expect it to get very brutal.
Na, this one has already gotten as bad as it's going to get. Your claim
this time is so obviously false that even I'm getting tired of repeating why
:-)
But make no mistake, if p mod 3 is NOT random, then damn it, somebody
better start talking about how you predict the next number in the
sequence beyond saying there is a 50% chance it is 1, or 2.
See below.
Concrete tests are CRUCIAL when dealing with intelligent people who
have a lot to lose,
Heh -- from a fellow legitimately notorious for making false claims easily
refuted by /obvious/ (even to him) simple tests he refuses to do, that's
rich.
and a history of successfully lying to a lot of people all over the world.
Nobody like that here I know of. This was an oblique reference to your
oft-repeated belief that "math people" lie about your research (or even more
deviously pretend to ignore it despite that they know it's correct)? Some
day this will sink in: when you say things like that, people question your
sanity. The belief is really that far removed from reality.
Oh well, just so long as you learn something about primes ;-)
Just because it was fun, here's a program that tries to predict the next
value of the i'th prime mod p, given p as input, and basing its prediction
on the residue that was most often seen previously following the most recent
n residues in the sequence. I'm not going to take time to explain details,
except to note that the "prime(i)" function is one I'm not going to
reproduce here -- it returns the i'th prime, so that prime(1) = 2, prime(2)
= 3, and so on:
def simple_learning_guesser(limit, p, n=1):
from primes import prime
number_tried = number_correct = 0
def dump():
print "%d correct in %d tries ~= %6.3f%% at prime %d" % (
number_correct, number_tried,
1e2 * number_correct / number_tried,
prime(i))
# Map n-tuple of previous n values to list of p-1 counts:
# [# times followed by 1, # times followed by 2, ...]
counts = {}
# Skip over the primes <= p.
i = 1
while prime(i) <= p:
i += 1
# The last n residue classes seen. Consume n primes to get this
# started.
previous = tuple(prime(i+j) % p for j in range(n))
i += n
while i <= limit:
number_tried += 1
# Predict: look at the counts for the n previous primes, and
# predict the residue class with maximal count (ties favor the
# smallest residue, for lack of an obvious better idea).
cs = counts.get(previous)
if cs:
predicted = cs.index(max(cs)) + 1
else:
# Haven't seen this sequence of n previous residues before
# -- make something up.
counts[previous] = cs = [0] * (p-1)
predicted = 1 # arbitrary
# Get next prime and see whether we're right.
pmod = prime(i) % p
if predicted == pmod:
number_correct += 1
if number_tried % 100000 == 0:
dump()
# Learn: bump the count on the actual pmod seen.
cs[pmod - 1] += 1
previous = previous[1:] + (pmod,)
i += 1
dump()
That's a Python program. It's not particularly efficient, and I don't care.
Note that the /only/ thing it remembers about the primes is their residues
mod p:
pmod = prime(i) % p
doesn't even store prime(i) anywhere. So it's emulating what would happen
if someone just fed in the sequence of residues, giving no knowledge of the
primes themselves.
I ran this with p=101 because that was most interesting to me at the time.
If you want to see it with other p's, fine, run it yourself (and write a
prime(i) function for it to use). The non-randomness of the residue
sequence becomes "more obvious more quickly" the larger p; OTOH, it takes
O((p-1)^(n+1)) memory to remember the (p-1)^n possible n-tuples of residue
classes and the p-1 counts associated with each, and that can be fatal for
large p or large n.
Note that at p=101, there are p-1 = 100 possible residue classes, so if the
sequence is truly random this stupid little program /shouldn't/ be able to
guess right significantly more (or less!) than 1% of the time. Here's a
sample run:
simple_learning_guesser(10**6, p=101, n=1)
16670 correct in 100000 tries ~= 16.670% at prime 1300097
32144 correct in 200000 tries ~= 16.072% at prime 2750491
47265 correct in 300000 tries ~= 15.755% at prime 4256617
62075 correct in 400000 tries ~= 15.519% at prime 5800591
76792 correct in 500000 tries ~= 15.358% at prime 7369133
91105 correct in 600000 tries ~= 15.184% at prime 8960867
105206 correct in 700000 tries ~= 15.029% at prime 10571269
119267 correct in 800000 tries ~= 14.908% at prime 12195629
133309 correct in 900000 tries ~= 14.812% at prime 13834573
147219 correct in 999973 tries ~= 14.722% at prime 15485867
So remembering /just/ (n=1) which j followed which i in the past most often
allowed it to predict about 15% of the next values correctly across this
range, instead of the 1% we'd expect if the sequence were random. That's
pretty spectacular. As a sanity check, add
from random import randrange
near the top and replace
pmod = prime(i) % p
with
pmod = randrange(1, p)
Then the pmods chosen /are/ (pseudo-)random, and the output in one run
changed to:
1006 correct in 100000 tries ~= 1.006% at prime 1300097
1973 correct in 200000 tries ~= 0.987% at prime 2750491
2973 correct in 300000 tries ~= 0.991% at prime 4256617
3957 correct in 400000 tries ~= 0.989% at prime 5800591
4950 correct in 500000 tries ~= 0.990% at prime 7369133
5869 correct in 600000 tries ~= 0.978% at prime 8960867
6870 correct in 700000 tries ~= 0.981% at prime 10571269
7894 correct in 800000 tries ~= 0.987% at prime 12195629
8919 correct in 900000 tries ~= 0.991% at prime 13834573
9872 correct in 999973 tries ~= 0.987% at prime 15485867
So, as expected, it didn't get any benefit from remembering "j followed i
this many times" info when the sequence was really close to being random.
Back to the above, let's try remembering how often each k followed each (i,
j) pair (n=2):
simple_learning_guesser(10**6, p=101, n=2)
15299 correct in 100000 tries ~= 15.299% at prime 1300111
30563 correct in 200000 tries ~= 15.281% at prime 2750509
45856 correct in 300000 tries ~= 15.285% at prime 4256639
60983 correct in 400000 tries ~= 15.246% at prime 5800603
75996 correct in 500000 tries ~= 15.199% at prime 7369151
90799 correct in 600000 tries ~= 15.133% at prime 8960893
105673 correct in 700000 tries ~= 15.096% at prime 10571273
120549 correct in 800000 tries ~= 15.069% at prime 12195637
135190 correct in 900000 tries ~= 15.021% at prime 13834619
149630 correct in 999972 tries ~= 14.963% at prime 15485867
This was a mixed bag: it started worse but ended better. A real drawback
is that there are now 100^2 = 10000 possible pairs to remember, and the
first time a pair is encountered the code stupidly guesses "1" no matter
what:
predicted = 1 # arbitrary
That's going to be right only about 1% of the time it's tried, and that
drags the average down at first. Picking n=3 makes this effect much
worse -- then there are 100^3 = a million possible residue triples to
remember:
simple_learning_guesser(10**6, p=101, n=3)
11820 correct in 100000 tries ~= 11.820% at prime 1300127
25540 correct in 200000 tries ~= 12.770% at prime 2750513
39623 correct in 300000 tries ~= 13.208% at prime 4256653
53685 correct in 400000 tries ~= 13.421% at prime 5800637
67951 correct in 500000 tries ~= 13.590% at prime 7369171
82432 correct in 600000 tries ~= 13.739% at prime 8960909
96996 correct in 700000 tries ~= 13.857% at prime 10571299
111591 correct in 800000 tries ~= 13.949% at prime 12195641
126410 correct in 900000 tries ~= 14.046% at prime 13834621
141004 correct in 999971 tries ~= 14.101% at prime 15485867
So that starts much worse, but keeps getting better, in part likely because
the total number of tries was less than the total number of possible residue
triples(!). While I won't show it here, that slowly continues improving its
winning percentage through about the first 7.2 million primes.
Cool.
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