Counting primes, and why I'm pissed
mathedman
Who cares?? Name ONE!
>I'm an admitted amateur when it comes to the math thing. I started
>posting as I was looking for a simple approach to FLT, realizing that
>I'd run into flack, and yes, I got tons of flack. I'm not here to
>talk about that yet again, as along the way I figured out how to count
>primes, and mathematicians cheated.
>
>Supposedly they just care about math, even "pure math", but when I
>talked about my methods for counting primes, I kept hearing it was
>irrelevant, and they kept bugging me about whether or not it was
>faster than the current methods for counting primes.
>
>That's an especially egregious kind of cheat to me, because they're
>putting my neat amateur discovery against the best that mathematicians
>have managed to do in over a hundred years of effort, and besides,
>people who know anything about algorithms versus math formulas, know
>that coming up with an efficient algorithm can be quite different from
>having a math formula that provides the reason.
>
>For instance, consider Fast Fourier Transforms (FFT) versus Fourier's
>own work.
>
>So now I'm going to show you a formula which is algorithmic in that it
>counts primes from N equal 50 to 120 (actually it's a little wider
>than that but I'm keeping things simple), but you have to use even N.
>
>With even N, 49 < N <121,
>
>pi(N) =
>
> floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
>
> floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
>
>and if you don't believe that mathematicians are cheats, ask *any*
>mathematician in the world to produce a shorter function for that
>region.
>
>Better yet, ask any mathematician in the ENTIRE WORLD to reproduce
>that expression without using my work.
>
>That's it. And if you think you know math, why don't you test your
>own knowledge or go pick up a math text.
>
>That's right, no matter who you are, what you think you know, or where
>you are in the world, I sit here defining knowledge outside of the
>limits of what you could have learned without looking at my research.
>
>You are limited.
>
>
>James Harris
Christian Bau
jst...@msn.com (James Harris) wrote:
> With even N, 49 < N <121,
>
> pi(N) =
>
> floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
>
> floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
>
> and if you don't believe that mathematicians are cheats, ask *any*
> mathematician in the world to produce a shorter function for that
> region.
pi(N) =
N/2 - floor((N-4)/6) - floor((N-16)/10) +
floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
It is shorter. You are so obsessed with yourself, you post this formula
and claim that no mathematician could improve on it, and even an average
ten year old could do it.
Yifei Chen
"mathedman" <math...@hotmail.CUT.com> wrote in message
news:3e4abde...@netnews.worldnet.att.net...
Zachary Turner
"James Harris" <jst...@msn.com> wrote in message
news:3c65f87.03021...@posting.google.com...
> I'm an admitted amateur when it comes to the math thing. I started
> posting as I was looking for a simple approach to FLT, realizing that
> I'd run into flack, and yes, I got tons of flack. I'm not here to
> talk about that yet again, as along the way I figured out how to count
> primes, and mathematicians cheated.
>
> Supposedly they just care about math, even "pure math", but when I
> talked about my methods for counting primes, I kept hearing it was
> irrelevant, and they kept bugging me about whether or not it was
> faster than the current methods for counting primes.
>
> That's an especially egregious kind of cheat to me, because they're
> putting my neat amateur discovery against the best that mathematicians
> have managed to do in over a hundred years of effort, and besides,
> people who know anything about algorithms versus math formulas, know
> that coming up with an efficient algorithm can be quite different from
> having a math formula that provides the reason.
>
> For instance, consider Fast Fourier Transforms (FFT) versus Fourier's
> own work.
>
> So now I'm going to show you a formula which is algorithmic in that it
> counts primes from N equal 50 to 120 (actually it's a little wider
> than that but I'm keeping things simple), but you have to use even N.
>
> With even N, 49 < N <121,
>
> pi(N) =
>
> floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
>
> floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
So, this function only works for N an even integer between 50 and 120
inclusive. bahahaha. You've produced a function which returns pi(N) for
exactly 41 values of N. I could one up you effortlessly by simply
constructing a 42nd degree polynomial that returns these exact same values.
It works for the first 500 values? Great, I could one up it again by
producing a 501st degree polynomial that evaluates to pi(N). Ooooh, and
mine would work for _any_ N in that range, not just even ones. How is a
function that successfully calculates 41 values of out _infinity_ values
even remotely interesting?
Dirk Van de moortel
"Christian Bau" <christ...@cbau.freeserve.co.uk> wrote in message
news:christian.bau-5DA...@slb-newsm1.svr.pol.co.uk...
Very nice :-)
http://users.pandora.be/vdmoortel/dirk/Physics/ImmortalGems.html#AnyMath
Dirk Vdm
dull...@sprynet.com
>I'm an admitted amateur when it comes to the math thing. I started
>posting as I was looking for a simple approach to FLT, realizing that
>I'd run into flack, and yes, I got tons of flack.
You didn't get any flak for looking for a simple approach to FLT.
You got flak for insisting that you'd proved FLT, over and over,
after errors in various versions of the proofs had been clearly
explained.
>I'm not here to
>talk about that yet again,
Then why are you talking about it?
>as along the way I figured out how to count
>primes, and mathematicians cheated.
>
>Supposedly they just care about math, even "pure math", but when I
>talked about my methods for counting primes, I kept hearing it was
>irrelevant, and they kept bugging me about whether or not it was
>faster than the current methods for counting primes.
>
>That's an especially egregious kind of cheat to me, because they're
>putting my neat amateur discovery against the best that mathematicians
>have managed to do in over a hundred years of effort,
_You're_ cheating by putting it this way. Because it was _you_ who
insisted on comparing your work to the best that has been done in
hundreds of years, and claimed it was a huge improvement. Just
a few days (weeks?) ago in alt.writing(!) you were explaining how
one amazing thing about your formula was that it was the first
ever to involve a function of two variables, ignoring the fact
that exactly the same sort of function of two variables comes up
in many of the standard algorithms, and explaining that another
amazing thing about it is that it led to that "PDE", although you've
_never_ given any reason to think that the solution to the PDE
has anything to do with pi(n), and claiming that it seemed likely
that your work would show that the Riemann Hypothesis was
false, without giving _any_ hint why this should be.
I like that, "cheating":
JSH: Look. I'm one of the greatest number theorists on the
planet - my stuff is better than anything that's ever been done
before.
Everyone else: Uh, no, actually it's a lot worse than a lot
of things that have been done before.
JSH: That's cheating, comparing my work to the best that
mathematicians have been able to do in hundreds of years.
>and besides,
>people who know anything about algorithms versus math formulas, know
>that coming up with an efficient algorithm can be quite different from
>having a math formula that provides the reason.
>
>For instance, consider Fast Fourier Transforms (FFT) versus Fourier's
>own work.
>
>So now I'm going to show you a formula which is algorithmic in that it
>counts primes from N equal 50 to 120 (actually it's a little wider
>than that but I'm keeping things simple), but you have to use even N.
>
>With even N, 49 < N <121,
>
>pi(N) =
>
> floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
>
> floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
>
>and if you don't believe that mathematicians are cheats, ask *any*
>mathematician in the world to produce a shorter function for that
>region.
>
>Better yet, ask any mathematician in the ENTIRE WORLD to reproduce
>that expression without using my work.
>
>That's it. And if you think you know math, why don't you test your
>own knowledge or go pick up a math text.
>
>That's right, no matter who you are, what you think you know, or where
>you are in the world, I sit here defining knowledge outside of the
>limits of what you could have learned without looking at my research.
See, if you want people to say your neat amateur discoveries are neat,
what you should do is say hey, look at that formula - isn't it neat?
People might agree that it was neat, who knows. When you instead
make this sort of absurd claim about how you can do things that
nobody else in the ENTIRE WORLD can do you get flak. That's the
way it works.
>You are limited.
>
>
>James Harris
David C. Ullrich
Michael van Opstall
[non-mathematical content snipped]
>
> So now I'm going to show you a formula which is algorithmic in that it
> counts primes from N equal 50 to 120 (actually it's a little wider
> than that but I'm keeping things simple), but you have to use even N.
>
> With even N, 49 < N <121,
>
> pi(N) =
>
> floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
>
> floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
>
> and if you don't believe that mathematicians are cheats, ask *any*
> mathematician in the world to produce a shorter function for that
> region.
>
> Better yet, ask any mathematician in the ENTIRE WORLD to reproduce
> that expression without using my work.
>
[non-math snipped]
This really caught my attention. I've always felt a little weak in the
area of prime numbers, so I thought I'd work it out. So the formula breaks
because it "counts" 121 as a prime since it doesn't account for factors of
11, since the denominators are 2, 2x3, 2x5, 2x3x5, 2x7, 2x3x7. OK.
Empirical testing shows that the formula is actually good down to about
23... Also, I think it works without the evenness assumption.
So why does it work? The first term:
floor(N/2)
is the number of numbers up to N not divisible by 2, so they've got
a chance to be prime. Next
floor((N-4)/6)
throws out the remaining numbers divisible by 3, but not 3 itself, because
of the N-4...
floor((N-16)/30)
throws out the remaining numbers divisible by 5, except 5 and 15, 5
because it is prime, and 15 because it was thrown out in the previous
step.
floor((N-16)/30)
needs to be there to put back the numbers divisible by 15 but greater than
15 (hence the -16) which were discarded twice...
floor((N-36)/14)
throws out the numbers divisible by 7 which are greater than 35, since 7
ir prime, 21 is already thrown out, and 35 is already thrown out.
and floor((N-22)/42) puts back those divisible by 7 but greater than 21
which were thrown out twice. Interesting.
Example: for 70: (floors omitted, thinking like C)
70/2=35 : #{2}union{odds<70} included
66/6=11 : 9,15,21,27,33,39,45,51,57,63,69 excluded
54/10=5 : 25,35,45,55,65 excluded
54/30=1 : 45 adjust, was excluded twice
44/14=3 : 35,49,63 excluded
48/22=2 : 35,63 adjust, were excluded twice.
> James Harris
>
======================================================================
Michael A. Van Opstall
Padelford C-113
ops...@math.washington.edu
http://www.math.washington.edu/~opstall/
The Last Danish Pastry
news:%Zz2a.6546$in1.3...@twister.austin.rr.com...
41? Do you mean 36?
--
Clive Tooth
http://www.clivetooth.dk
Jim Ferry
> So now I'm going to show you a formula which is algorithmic in that it
> counts primes from N equal 50 to 120 (actually it's a little wider
> than that but I'm keeping things simple), but you have to use even N.
>
> With even N, 49 < N <121,
>
> pi(N) =
>
> floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
>
> floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
>
> and if you don't believe that mathematicians are cheats, ask *any*
> mathematician in the world to produce a shorter function for that
> region.
>
> Better yet, ask any mathematician in the ENTIRE WORLD to reproduce
> that expression without using my work.
Your expression does what you claim. Good work.
However, your implication that it's significant is misguided. We
can achieve something like it by using Legendre's formula (which
itself is a trivial result, being a simple application of the
inclusion-exclusion formula).
Legendre's formula gives
pi(N) = 3 + [N] - [N/2] - [N/3] - [N/5] - [N/7] + [N/6] + [N/10] +
[N/14] + [N/15] + [N/21] + [N/35] - [N/30] - [N/42] - [N/70] - [N/105].
for 6 < N < 121. Here [x] denotes the floor of x.
This can be simplified by noting that [x] = [x/2] + [(x+1)/2], and
by bringing the 3 inside one of the resulting terms:
pi(N) = [(N+7)/2] - [(N+3)/6] - [(N+5)/10] - [(N+7)/14] + [(N+15)/30] +
[(N+21)/42] + [(N+35)/70] - [N/105],
for 6 < N < 121.
This expression has the denominators you use, plus 70 and 105.
We can eliminate these last two terms by noting that
[(N+35)/70] - [N/105] = 1 for 34 < N < 175.
This yields
pi(N) = [(N+9)/2] - [(N+3)/6] - [(N+5)/10] - [(N+7)/14] +
[(N+15)/30] + [(N+21)/42],
for 34 < N < 121.
We now have an expression as simple as yours, but which extends
over a broader range and works for odd N as well. It has been
achieved via trivial manipulations. This makes your grandiose
claims look rather foolish.
> That's right, no matter who you are, what you think you know, or where
> you are in the world, I sit here defining knowledge outside of the
> limits of what you could have learned without looking at my research.
>
> You are limited.
Actually, you struggle to produce low-quality results, which anyone
with the least glimmer of mathematical talent can surpass easily.
You think your results must be profound because they are so
difficult for you to produce, but, in reality, they are crappy --
because you are so bad at math.
Did you watch the first round of "American Idol"? I'm referring
specifically to the scenes of atrocious singers who think they're
going to be the next big pop star. Apparently television audiences
are drawn to these embarrassing spectacles of performances so bad
that you cringe at the thought of singing like that in front of
another person let alone millions. The mindblowing part is that
the "singers" actually believe they're good. I can't deny that
I'm drawn to these travesties. I'll wager that all of your regular
readers would be too.
| Jim Ferry | Center for Simulation |
+------------------------------------+ of Advanced Rockets |
| http://www.uiuc.edu/ph/www/jferry/ +------------------------+
| jferry@[delete_this]uiuc.edu | University of Illinois |
Virgil
> That's an especially egregious kind of cheat to me, because they're
> putting my neat amateur discovery against the best that mathematicians
> have managed to do in over a hundred years of effort, and besides,
> people who know anything about algorithms versus math formulas, know
> that coming up with an efficient algorithm can be quite different from
> having a math formula that provides the reason.
If your computer methods are no faster than the paper and
pencil methods of 100 years ago, your methods deserve being
put down.
And if you get so pissed so easily, maybe you ought to swear
off the sauce for a bit.
Zachary Turner
"The Last Danish Pastry" <TheLastDa...@yahoo.com> wrote in message
news:b2ek73$1c3h98$1...@ID-11651.news.dfncis.de...
Oops, I counted the even numbers from 120-130 as well
Dik T. Winter
Let's analyse:
> pi(N) =
> N/2 - floor((N-4)/6) - floor((N-16)/10) +
> floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
1: N/2: remove multiples of 2.
2: floor((N-4)/6): remove multiples of 3 larger than 3.
3: floor((N-16)/10): remove multiples of 5 larger than 15.
4: floor((N-16)/30): add in multiples removed both in steps 2 and 3.
5: floor((N-36)/14): remove multiples of 7 larger than 35.
6: floor((N-22)/42): add in multiples removed both in steps 2 and 5.
This is again getting pretty close to Legendre's algorithm.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Dr. Michael Ulm
--snip--
>
> So now I'm going to show you a formula which is algorithmic in that it
> counts primes from N equal 50 to 120 (actually it's a little wider
> than that but I'm keeping things simple), but you have to use even N.
>
> With even N, 49 < N <121,
>
> pi(N) =
>
> floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
>
> mathematician in the world to produce a shorter function for that
> region.
>
I realize that this has nothing to do with anything really,
but I find the following result nice:
For even N and 43 < N < 121,
pi(N) = floor(91/19 + 5/23 N
+ 3/13 (cos(Pi (N-2)/3) - cos(Pi (N-2)/12) + 2 sin(Pi (N+4)/18)))
which under a suitable metric can be considered shorter than
the other function. My formula has still a bit of "wiggle space",
so with some additional effort I believe it can be shortened further.
Michael.
--
&&&&&&&&&&&&&&&&#@#&&&&&&&&&&&&&&&&
Dr. Michael Ulm
FB Mathematik, Universitaet Rostock
micha...@mathematik.uni-rostock.de
Jim Ferry
> pi(N) = [(N+9)/2] - [(N+3)/6] - [(N+5)/10] - [(N+7)/14] +
> [(N+15)/30] + [(N+21)/42],
>
> for 34 < N < 121.
I should have written this in the following simpler form. Let
f(x) = floor((x+1)/2). Then
pi(N) = 4 + f(N) - f(N/3) - f(N/5) - f(N/7) + f(N/15) + f(N/21),
for 34 < N < 121.
I've already noted that this formula extends over a larger range
than James's, works for odd as well as even numbers, and can be
derived in a few simple steps. Now it's simpler as well.
David Ames
> I'm an admitted amateur when it comes to the math thing. I started
> posting as I was looking for a simple approach to FLT, realizing that
> I'd run into flack, and yes, I got tons of flack. I'm not here to
> talk about that yet again, as along the way I figured out how to count
> primes, and mathematicians cheated.
>
> Supposedly they just care about math, even "pure math", but when I
> talked about my methods for counting primes, I kept hearing it was
> irrelevant, and they kept bugging me about whether or not it was
> faster than the current methods for counting primes.
>
> That's an especially egregious kind of cheat to me, because they're
> putting my neat amateur discovery against the best that mathematicians
> have managed to do in over a hundred years of effort, and besides,
> people who know anything about algorithms versus math formulas, know
> that coming up with an efficient algorithm can be quite different from
> having a math formula that provides the reason.
>
> For instance, consider Fast Fourier Transforms (FFT) versus Fourier's
> own work.
>
> counts primes from N equal 50 to 120 (actually it's a little wider
> than that but I'm keeping things simple), but you have to use even N.
>
> With even N, 49 < N <121,
>
> pi(N) =
>
> floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
>
> floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
>
> and if you don't believe that mathematicians are cheats, ask *any*
> mathematician in the world to produce a shorter function for that
> region.
>
> that expression without using my work.
>
> own knowledge or go pick up a math text.
>
> you are in the world, I sit here defining knowledge outside of the
> limits of what you could have learned without looking at my research.
>
> You are limited.
>
>
This is perhaps not on point, but when I took Theory of Numbers back
in the '60's, the professor remarked that there was indeed a formula
for the n-th prime, BUT knowledge of (n + k) primes was needed in
order to obtain the desired result.
My point, however, is similar to that of many others who have posted
to this thread. My point, by analogy(?)/analogic(?)/whatever(?)
regards Mr. Harris' claim with an emphatic "So what? His claim is not
useful."
David Ames
James Harris
> On 12 Feb 2003 12:56:33 -0800, James Harris <jst...@msn.com> wrote:
> --snip--
> >
> > So now I'm going to show you a formula which is algorithmic in that it
> > counts primes from N equal 50 to 120 (actually it's a little wider
> > than that but I'm keeping things simple), but you have to use even N.
> >
> > With even N, 49 < N <121,
> >
> > pi(N) =
> >
> > floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
> >
> > floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
> >
> > and if you don't believe that mathematicians are cheats, ask *any*
> > mathematician in the world to produce a shorter function for that
> > region.
> >
>
> I realize that this has nothing to do with anything really,
> but I find the following result nice:
>
> For even N and 43 < N < 121,
>
> pi(N) = floor(91/19 + 5/23 N
> + 3/13 (cos(Pi (N-2)/3) - cos(Pi (N-2)/12) + 2 sin(Pi (N+4)/18)))
>
> which under a suitable metric can be considered shorter than
> the other function. My formula has still a bit of "wiggle space",
> so with some additional effort I believe it can be shortened further.
>
> Michael.
Nope. Your problem has to do with the length of the cosine and sine
functions based on number of operations.
Here shortness has to do with number of operations necessary to get an
answer.
Unfortunately for mathematicians, what I gave is the shortest
possible, and there's not a mathematician in the world that can even
reproduce it, without using my research.
And unfortunately for the world, mathematicians dance away from the
truth, even with such a clear demonstration of their shortcomings.
They have learned to cheat. And like the taste of it.
Remember, there's nothing for mathematicians to lose from telling the
truth, in comparison to what they lose by cheating--thinking that my
gain is their loss.
They *think* they're cheating me, but they're showing what rules they
*really* play by versus what they say, when they ask you to trust
them.
James Harris
C. Bond
> <snip>
> They depend on a glorious history of past *great* mathematicians to
> justify their current food--mostly your money, your tax money--and now
> happily try to cheat me out of the fruits of my research.
Perhaps you would do yourself a service by expiring. Then you could become a 'past *great* mathematician'
and the idol of future generations.
> But to do that they must cheat the world out of knowledge by ignoring
> its value.
If they're that pompous, arrogant and deceitful, would you *really* want their praise? You seem very
anxious to be lauded by those who you regard as scoundrels. Wouldn't it be better to seek the company of
your peers? (We all wish you would, by the way.)
--
There are two things you must never attempt to prove: the unprovable -- and the obvious.
http://www.crbond.com
Christian Bau
> > > So now I'm going to show you a formula which is algorithmic in that it
> > > counts primes from N equal 50 to 120 (actually it's a little wider
> > > than that but I'm keeping things simple), but you have to use even N.
> > >
> > > With even N, 49 < N <121,
> > >
> > > pi(N) =
> > >
> > > floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
> > >
> > > floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
> > >
> > > and if you don't believe that mathematicians are cheats, ask *any*
> > > mathematician in the world to produce a shorter function for that
> > > region.
I posted a shorter one earlier. Wasn't difficult because you are just
sooooo stupid. But here is a C function that returns pi (n) for even n,
50 <= n <= 120:
int pi (int n) {
return "323222322222322222212110101111212110" [n/2-25]-'0'-n/4;
}
Beat that. Two divisions instead of your six. Three subtractions instead
of your eight subtractions and two additions. And my formula can be
trivially extended to even values 0 <= n <= 120:
int pi (int n) {
return "0112222323333323332323333"
"323222322222322222212110101111212110" [n/2]-'0'-n/4;
}
Now you find a shorter formula that is valid for all even n, 0 <= n <=
120.
Here another version. Looks slightly more complicated, but the compiler
that I use produces a total amount of SIX assembler instructions plus
one return instruction for this function:
unsigned int pi (unsigned int n) {
return ((unsigned char *) "x112222323333323332323333"
"323222322222322222212110101111212110") [n/2]-'0'-n/4;
}
Christian Bau
> > > >With even N, 49 < N <121,
> > > >
> > > >pi(N) =
> > > >
> > > > floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
> > > >
> > > > floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
> > > >
> > > >and if you don't believe that mathematicians are cheats, ask *any*
> > > >mathematician in the world to produce a shorter function for that
> > > >region.
>
> It's actually rather beautiful, yes it works,
> yes it's shorter than anything any other mathematician can produce,
This is a blatant lie, and misleading. First, I posted a shorter
formula. So your formula is _not_ shorter than anything any
mathematician can produce; you are lying. When you first posted this,
you were just wrong. But since you were proven to be wrong, and you
still insist on it, that makes you a liar.
Second, you use the words "any other mathematician". That would imply
that you are a mathematican, which is wrong. You are a poor misguided
child suffering from narcissistic personality disorder who mostly posts
when he is pissed, not a mathematician.
> no they can't even
> reproduce, and no they won't give me the proper credit for my
> research.
> Modern "mathematicians" are cheats.
You are a cheat. I proved it.
> They depend on a glorious history of past *great* mathematicians to
> justify their current food--mostly your money, your tax money--and now
> happily try to cheat me out of the fruits of my research.
Are you sincerely suggesting that you would deserve _money_ for what you
call "research"?
Steve Leibel
> In my case, I have the histories of John Nash and Evariste Galois
> among others to tell me how cheaters seek to destroy a discoverer.
>
That earns a whole heap o' crank points right there.
fwi...@sbcglobal.net
"Christian Bau"
> Are you sincerely suggesting that you would deserve _money_ for what you
> call "research"?
Indeed.
'A mathematician,' Erdös was fond of saying,
'is a machine for turning coffee into theorems.'
Note conspicuous absence of mention of money.
If Erdös wasn't smothered in gold, who was so
talented, cooperative and prolific of theorems, how
could one expect much reward today for mere
untrained labor, even for years of it?
Umm, anyone know if he ever expressed an opinion
about how hard the Goldbach Conjecture would be
to prove? I read one web site that said Gauss
probably just didn't think it worth proving....
Terrence Stanley
>
> With even N, 49 < N <121,
>
> pi(N) =
>
> floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
>
> floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
>
>
> James Harris
Here's another function; but if you or anybody else criticizes this or
points out errors I'm going to be hopping mad. The only reply I will
be satisified with is "By jove I think he's got something here - stop
the presses!"
for natural numbers n such that 42 < n < 127
pi(N) =
floor [C * (10 ^ {2(n-41)}) ] - floor [C * (10 ^ {2(n-42)}) ] * 100
where the constant C is
00141414141515151515151616161616161717181818181818191919192020212121212121
22222222232323232323242424242424242425252525262627272727
2828292929293030303030303030303030303030 / [ 10^ (2(127-42)) ]
Brian Quincy Hutchings
that's the integer part of a number, but
it looks as if I'll have to, in order\
to grok biquadratic residues (and nonresidues).
Jim Ferry <jferry@[delete_this]uiuc.edu> wrote in message news:<eXP2a.22646$Vf3.2...@vixen.cso.uiuc.edu>...
> I should have written this in the following simpler form. Let
> f(x) = floor((x+1)/2). Then
>
> pi(N) = 4 + f(N) - f(N/3) - f(N/5) - f(N/7) + f(N/15) + f(N/21),
>
> for 34 < N < 121.
>
> I've already noted that this formula extends over a larger range
> than James's, works for odd as well as even numbers, and can be
> derived in a few simple steps. Now it's simpler as well.
--les duc d'Enron!
http://www.tarpley.net
James Waldby
[with 49<n<121, n even]
> int pi (int n) {
> return "323222322222322222212110101111212110" [n/2-25]-'0'-n/4;
> }
[with 0 <=n<121, n even]
> int pi (int n) {
> return "0112222323333323332323333"
> "323222322222322222212110101111212110" [n/2]-'0'-n/4;
> }
>
> Now you find a shorter formula that is valid for all even n, 0 <= n <=
> 120.
Ok, see below. BTW, you have a typo in all the versions you
posted. -n/4 gives negative results and should be +n/4.
// A shorter program for 0 <= n < 129, n even:
int pin(int n) {
long long i=2,p=0,b=0x816d129a64b4cb6f;
for (;i<=n;i+=2,b>>=1) p+=b&1; return p;
}
IIRC, the C standard doesn't define whether >> is an
arithmetic or logical shift so the result pin(130) is
implementation dependent and may be 31 or 32. My
athlon gives pin(130)=32, pin(132)=33, etc.
-jiw
Dr. Michael Ulm
> "Dr. Michael Ulm" <ta...@news.uni-rostock.de> wrote in message news:<slrnb4nbg...@hades.math.uni-rostock.de>...
>> On 12 Feb 2003 12:56:33 -0800, James Harris <jst...@msn.com> wrote:
>> --snip--
>> >
>> > So now I'm going to show you a formula which is algorithmic in that it
>> > counts primes from N equal 50 to 120 (actually it's a little wider
>> > than that but I'm keeping things simple), but you have to use even N.
>> >
>> > With even N, 49 < N <121,
>> >
>> > pi(N) =
>> >
>> > floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
>> >
>> > floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
>> >
>> > and if you don't believe that mathematicians are cheats, ask *any*
>> > mathematician in the world to produce a shorter function for that
>> > region.
>> >
>>
>> I realize that this has nothing to do with anything really,
>> but I find the following result nice:
>>
>> For even N and 43 < N < 121,
>>
>> pi(N) = floor(91/19 + 5/23 N
>> + 3/13 (cos(Pi (N-2)/3) - cos(Pi (N-2)/12) + 2 sin(Pi (N+4)/18)))
>>
>> which under a suitable metric can be considered shorter than
>> the other function. My formula has still a bit of "wiggle space",
>> so with some additional effort I believe it can be shortened further.
>>
>
> functions based on number of operations.
>
Please notice that I wrote "under a suitable metric" (e.g. the length
it takes to write down the formula in ascii). One point here is of
course that shortness is not a well defined term. One other point
is that for any reasonable definition of shortness there will almost
certainly be a formula that will be shorter than yours (Christian
Bau gave you one for your definition of shortness above, so I do
not have to).
It is also only a small curiosity to find a short formula for a fixed
intervall. Of course your method can give "short" formulas for
intervalls of arbitrary length (your next claim for grandiosity
should be formulated in that way), but still there are other methods
that would give shorter methods still (my secret method if one
wants something easy to write down as a mathematical formula, or
Christians method if one wants something easy to compute).
HTH,
Dik T. Winter
...
> // A shorter program for 0 <= n < 129, n even:
> int pin(int n) {
> long long i=2,p=0,b=0x816d129a64b4cb6f;
> for (;i<=n;i+=2,b>>=1) p+=b&1; return p;
> }
>
> IIRC, the C standard doesn't define whether >> is an
> arithmetic or logical shift so the result pin(130) is
> implementation dependent and may be 31 or 32.
You should make b 'unsigned long long', in that case >> is
properly defined.
Dr. Michael Ulm
> --snip--
>
> For even N and 43 < N < 121,
>
> pi(N) = floor(91/19 + 5/23 N
> + 3/13 (cos(Pi (N-2)/3) - cos(Pi (N-2)/12) + 2 sin(Pi (N+4)/18)))
>
> which under a suitable metric can be considered shorter than
> the other function. My formula has still a bit of "wiggle space",
> so with some additional effort I believe it can be shortened further.
>
Just for the record, I just came up with the slightly shorter
pi(N) = floor( 44/9 + 3/14 N + sin[pi (N+2)/18]/2
- pin[pi N/12]/4 + Sin[pi N/3]/4 )
for even N and 47 < N < 121. I particularily like the additional
referenze to pi (i.e. 3/14) in this formula :-)
James Waldby
>
> In article <3E4C8325...@pat7.com> j-wa...@pat7.com writes:
> ...
> > // A shorter program for 0 <= n < 129, n even:
> > int pin(int n) {
> > long long i=2,p=0,b=0x816d129a64b4cb6f;
> > for (;i<=n;i+=2,b>>=1) p+=b&1; return p;
> > }
> >
> > IIRC, the C standard doesn't define whether >> is an
> > arithmetic or logical shift so the result pin(130) is
> > implementation dependent and may be 31 or 32.
>
> You should make b 'unsigned long long', in that case >> is
> properly defined.
Quite so, but those nine characters "unsigned " seriously
chew up the margin of 14 characters shorter than Christian's
program. :)
-jiw
Phil Carmody
92 characters?
Your pin(2)=pin(3) worries me.
A function which does the same job (thus equally wrong, or with an equally
shrunk validity range) is
int pin(int n){int p=0;n/=2;while(n-->0)p+=0x816d129a64b4cb6f>>n&1;return p;}
77 characters. Yes, I know that K&R would shrink it, but I don't like
dropping ints.
Phil
($6 pledge, as rules is rules.)
C. Bond
> I'm an admitted amateur when it comes to the math thing. I started
> posting as I was looking for a simple approach to FLT, realizing that
> I'd run into flack, and yes, I got tons of flack. I'm not here to
> talk about that yet again, as along the way I figured out how to count
> primes, and mathematicians cheated.
>
> Supposedly they just care about math, even "pure math", but when I
> talked about my methods for counting primes, I kept hearing it was
> irrelevant, and they kept bugging me about whether or not it was
> faster than the current methods for counting primes.
>
> That's an especially egregious kind of cheat to me, because they're
> putting my neat amateur discovery against the best that mathematicians
> have managed to do in over a hundred years of effort, and besides,
> people who know anything about algorithms versus math formulas, know
> that coming up with an efficient algorithm can be quite different from
> having a math formula that provides the reason.
>
> For instance, consider Fast Fourier Transforms (FFT) versus Fourier's
> own work.
>
> So now I'm going to show you a formula which is algorithmic in that it
> counts primes from N equal 50 to 120 (actually it's a little wider
> than that but I'm keeping things simple), but you have to use even N.
>
> With even N, 49 < N <121,
>
> pi(N) =
>
> floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
>
> floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
>
> and if you don't believe that mathematicians are cheats, ask *any*
> mathematician in the world to produce a shorter function for that
> region.
>
> Better yet, ask any mathematician in the ENTIRE WORLD to reproduce
> that expression without using my work.
>
You think you've got problems! Consider this: I've discovered the shortest
possible expression which shows that 5 can be written as the sum of two
integers. Namely,
4 + 1 = 5
I defy any mathematician in the ENTIRE WORLD, nay the ENTIRE GALAXY, nay
the ENTIRE UNIVERSE to find a shorter solution! One may argue that here
are other expressions using other integers, of course, but this particular
one has immediate value for anyone trying to get change for a 5 dollar
bill and is willing to accept a 1 dollar bill and a 4 dollar bill.
Terrence Stanley
>
> With even N, 49 < N <121,
>
> pi(N) =
>
> floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
>
> floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
>
>
One might note:
Since a sloppy upper bound of pi(x) is 2x/ln x;
and 2x/ln x < 2^x;
the number of base 10 digits of pi(n) < 2^n,
and so;
There is a C such that 1/1000 < C < 2/1000 such that for
all integers n > 0,
pi(n) = floor [C * (10^(2^n-1)) ] - floor [C * (10 ^ (2^(n-1) - 1) ]
* (10^(2^n-1))
C = .0010002000000000000000000000003.........(a bunch of other
digits).
and of couse C is exactly =
pi(1)/(10^(2^1-1)) + pi(2)/(10^(2^2-1)) + pi(3)/(10^(2^3-1)) + ...
Matt Gutting
"C. Bond" <cb...@ix.netcom.com> wrote in message
news:3E4E78E5...@ix.netcom.com...
How about 4+1=5 ? -- No spaces :-)
C. Bond
> > You think you've got problems! Consider this: I've discovered the shortest
> > possible expression which shows that 5 can be written as the sum of two
> > integers. Namely,
> >
> > 4 + 1 = 5
> >
> > I defy any mathematician in the ENTIRE WORLD, nay the ENTIRE GALAXY, nay
> > the ENTIRE UNIVERSE to find a shorter solution!
>
> How about 4+1=5 ? -- No spaces :-)
Unlike the OP, I am eager to concede defeat when confronted with superior
intelligence. You are now the smartest mathematician in the universe! Perhaps
you could use your proven capabilites to knock some sense into JHS.
James Harris
> > pi(N) = [(N+9)/2] - [(N+3)/6] - [(N+5)/10] - [(N+7)/14] +
> > [(N+15)/30] + [(N+21)/42],
> >
> > for 34 < N < 121.
>
> I should have written this in the following simpler form. Let
> f(x) = floor((x+1)/2). Then
>
> pi(N) = 4 + f(N) - f(N/3) - f(N/5) - f(N/7) + f(N/15) + f(N/21),
>
> for 34 < N < 121.
>
> I've already noted that this formula extends over a larger range
> than James's, works for odd as well as even numbers, and can be
> derived in a few simple steps. Now it's simpler as well.
You're deluded. My formula has six terms. Yours has seven. Six is
less than seven.
>
> | Jim Ferry | Center for Simulation |
> +------------------------------------+ of Advanced Rockets |
> | http://www.uiuc.edu/ph/www/jferry/ +------------------------+
> | jferry@[delete_this]uiuc.edu | University of Illinois |
You're an angry man Ferry. But at least you're predictable, and
sometimes entertaining.
James Harris
James Harris
> In article <christian.bau-5DA...@slb-newsm1.svr.pol.co.uk> Christian Bau <christ...@cbau.freeserve.co.uk> writes:
>
> Let's analyse:
>
> > pi(N) =
> > N/2 - floor((N-4)/6) - floor((N-16)/10) +
> > floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
>
> 1: N/2: remove multiples of 2.
> 2: floor((N-4)/6): remove multiples of 3 larger than 3.
> 3: floor((N-16)/10): remove multiples of 5 larger than 15.
> 4: floor((N-16)/30): add in multiples removed both in steps 2 and 3.
> 5: floor((N-36)/14): remove multiples of 7 larger than 35.
> 6: floor((N-22)/42): add in multiples removed both in steps 2 and 5.
>
> This is again getting pretty close to Legendre's algorithm.
But Legendre's is inferior as it gives over twice as many terms.
Hey, I'm just saying that Legendre's went halfway, and I finished the
job.
Mathematicians should welcome my research and add it to references,
but instead I get to argue with bottom feeders on the outskirts of the
Information Highway.
James Harris
James Harris
Really? Why?
James Harris
> In article <3c65f87.03021...@posting.google.com>,
> jst...@msn.com (James Harris) wrote:
>
> > > > >With even N, 49 < N <121,
> > > > >
> > > > >pi(N) =
> > > > >
> > > > > floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
> > > > >
> > > > > floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
> > > > >
> > > > >and if you don't believe that mathematicians are cheats, ask *any*
> > > > >mathematician in the world to produce a shorter function for that
> > > > >region.
> >
> > It's actually rather beautiful, yes it works,
> > yes it's shorter than anything any other mathematician can produce,
>
> This is a blatant lie, and misleading. First, I posted a shorter
> formula. So your formula is _not_ shorter than anything any
> mathematician can produce; you are lying. When you first posted this,
> you were just wrong. But since you were proven to be wrong, and you
> still insist on it, that makes you a liar.
<deleted>
I don't necessarily read your posts Christian Bau. But I did just
read the post you mention, where you changed floor(N/2) to N/2, as N
being even makes it unnecessary to use the floor function.
So, yes, you did in some sense manage to shorten the expression.
And I'm not surprised at your posting as if that's a big deal.
James Harris
James Harris
> "The Last Danish Pastry" <TheLastDa...@yahoo.com> wrote in message
> news:b2ek73$1c3h98$1...@ID-11651.news.dfncis.de...
> > "Zachary Turner" <_NOzturner...@hotmail.com> wrote in message
> > news:%Zz2a.6546$in1.3...@twister.austin.rr.com...
> >
> > > "James Harris" <jst...@msn.com> wrote in message
<deleted>
> > > >
> > > > With even N, 49 < N <121,
> > > >
> > > > pi(N) =
> > > >
> > > > floor(N/2) - floor((N-4)/6) - floor((N-16)/10) +
> > > >
> > > > floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
> > >
> > > So, this function only works for N an even integer between 50 and 120
> > > inclusive. bahahaha. You've produced a function which returns pi(N)
> for
> > > exactly 41 values of N.
> >
> > 41? Do you mean 36?
>
> Oops, I counted the even numbers from 120-130 as well
Interesting.
Christian Bau
"In some sense"? Good example of intellectual dishonesty, which anyone
expected from you. Fact is that you claimed doing this was beyond what
any mathematician could achieve. I just showed that you are a complete
idiot.
> And I'm not surprised at your posting as if that's a big deal.
I explicitely posted that it was _not_ a big deal, that any ten year old
should be capable of doing it. So you are lying again. And to be honest,
showing that you are a moron is no big deal. You just make it too easy.
Jim Ferry
> "Dik T. Winter" <Dik.W...@cwi.nl> wrote in message news:<HA89y...@cwi.nl>...
>
>>In article <christian.bau-5DA...@slb-newsm1.svr.pol.co.uk> Christian Bau <christ...@cbau.freeserve.co.uk> writes:
>>
>>Let's analyse:
>>
>> > pi(N) =
>> > N/2 - floor((N-4)/6) - floor((N-16)/10) +
>> > floor((N-16)/30) - floor((N-36)/14) + floor((N-22)/42)
>>
>>1: N/2: remove multiples of 2.
>>2: floor((N-4)/6): remove multiples of 3 larger than 3.
>>3: floor((N-16)/10): remove multiples of 5 larger than 15.
>>4: floor((N-16)/30): add in multiples removed both in steps 2 and 3.
>>5: floor((N-36)/14): remove multiples of 7 larger than 35.
>>6: floor((N-22)/42): add in multiples removed both in steps 2 and 5.
>>
>>This is again getting pretty close to Legendre's algorithm.
>
>
> But Legendre's is inferior as it gives over twice as many terms.
One gets rid of those terms by using floor(x) = floor(x/2) + floor((x+1)/2).
> Hey, I'm just saying that Legendre's went halfway, and I finished the
> job.
Legendre's formula is pretty trivial. So is simplifying it by applying
the aforementioned identity for floor. But you didn't even do this:
instead you can up with a botched version. Even if you'd done it right
it would be only a little less trivial than Christian Bau's object lesson
of simplifying floor(N/2) to N/2 for even N.
> Mathematicians should welcome my research and add it to references,
> but instead I get to argue with bottom feeders on the outskirts of the
> Information Highway.
It amuses me to respond to you even though all evidence suggests I'm in
your kill file. The thing is, it doesn't matter whether you read this
post or not. It's not like anything ever sinks in.
David C. Ullrich
<jferry@[delete_this]uiuc.edu> wrote:
>James Harris wrote:
[...]
>
>> Mathematicians should welcome my research and add it to references,
>> but instead I get to argue with bottom feeders on the outskirts of the
>> Information Highway.
>
>It amuses me to respond to you even though all evidence suggests I'm in
>your kill file.
No need to worry. It's become clear through the years that he doesn't
have a kill file. He reads everything, just pretends not to notice the
things he pretends to have killfiled.
> The thing is, it doesn't matter whether you read this
>post or not. It's not like anything ever sinks in.
******************
David C. Ullrich
Willow Schlanger
pi, limits, and prime #'s", posted recently to alt.math. Skip the
"aside" sections in it. You will probably be very interested in it.
It's impressive what you have done, and it shows an interest in math,
which is very nice. But the 49 < N <121 foruma was constructed after
already knowing what the answer was, so it's not really related to prime
numbers, but instead compressing a sequence of pseudo-prandom numbers
down to ordinary mathematical symbols. In any case, here's an exerpt
from the former post:
[[begin excerpt]]
Now here's some observations and thoughts on prime numbers.
f(x) = g(g(g(...g(x)...)))
g(x) = x + sin(x)
y(x, n) = sin(pi * x / n)
Observe that y(x, n) = 0 if x is evenly divisible by n, and nonzero
otherwise.
h(x, n) = 1 - f(y(x, n)) ^ 2 / pi
("^ 2" means "to the power of two").
Observe that h(x, n) = 0 if x is not evenly divisible by n, and = 1 if x
is evenly divisible by n
z(x) = (SUMMATION from n = 2 to n = N) h(x, n)
N is any large number that x will not exceed. It may be infinity, and
then z(x) is defined for all integers > 1, or it may be a large integer,
in which case a restriction is imposed on the defined values of x
(namely x <= N).
Observe that z(x) = 1 if x is prime, and z(x) > 1 if x is nonprime.
In particular, z(x) tells the number of integers > 1 (and <= N) that x
is evenly divisible by. Clearly this is 1 for all prime numbers, and for
prime numbers only.
[[end exerpt]]
Also take a look at the pattern in the factors of numbers (see that
post), and see if you can compress that sequence of numbers (0, 1, 0, 2,
0, 1, 0, 3, ...) etc. to some function f(x, n) that works for some n.
If you could do that, it would be interesting and fun.
Yours turly,
Willow Schlanger
willow.schlanger.name
Fishfry
wrote:
> James, see my post with the subject "Some Observations and thoughts on
> pi, limits, and prime #'s", posted recently to alt.math. Skip the
> "aside" sections in it. You will probably be very interested in it.
>
> It's impressive what you have done, and it shows an interest in math,
> which is very nice.
JSH shows an interest in math in the same way that Osama bin Laden shows
an interest in airplanes. Or for that matter, in the same way that John
Ashcroft shows an interest in the Constitution.