Mersenne Digest Wednesday, May 3 2000 Volume 01 : Number 727
----------------------------------------------------------------------
Date: Mon, 1 May 2000 11:46:08 -0700
From: "Scott Kurowski" <[EMAIL PROTECTED]>
Subject: Mersenne: Error 404s fixed
Hi all,
We moved PrimeNet, Mersenne.org and the Entropia.com sites from Silicon
Valley to our new San Diego/La Jolla office.
All of the web content should be appearing properly now - seems we
overlooked forwarding subfolders of the root www service. We put in a
forwarding server until the DNS servers throughout the Internet catch up
with the change of IP addresses. Mostly we focused upon ensuring the client
transactions reached PrimeNet, which worked out great.
You should have seen us on the plane early Saturday AM with our servers
entombed in bubblewrap and strapped into the adjacent seats! (Of course, we
did take several other precautions. :-)
regards,
scott
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------------------------------
Date: Mon, 1 May 2000 19:25:29 EDT
From: [EMAIL PROTECTED]
Subject: Subj: Re: Mersenne: A conjecture on Perfect numbers
In a message dated 4/29/00 6:31:10 AM Pacific Daylight Time,
[EMAIL PROTECTED] writes:
<< Good. Now that we are on the topic of perfect numbers, I have proven that
any odd perfect number must have at least 9 factors. Another boy at my
school is trying to tell me that all perfect numbers are in the from
(a^(p-1))(a^p-1), thus showing that there are no odd perfect numbers. Can
someone please explain this to me? I am quite sure that he is wrong.
>>
The kid at school probably was talking about even perfect numbers. (a = 2)
Anyway, the proof I had goes like this, in case you are wondering, because,
like I said, I was pretty busy.
Anyway: Here are a couple of things you need to know.
(2(n) + 1) is the formula for odd numbers. (2^(n) - 1) is always odd, (2^n -
2) is an odd times 2. Another is that even + even = even, and even + odd =
odd. There is also this Pascal's triangle.
64
32 32
16 32 16
8 24 24 8
4 16 24 16 4
2 10 20 20 10 2
1 6 15 20 15 6 1
You'll notice that this is the same as the regular triangle, except you
multiply powers of two. Now for the proof:
_________________________________________________
By definition of an odd, perfect number, with only distinct factors, the
following must be true, if the odd, perfect number has only two factors:
(2)(2a + 1)(2b + 1) = (2a + 1)(2b + 1) + (2a + 1) + (2b + 1) + 1
(2a + 1)(2b + 1) = 2(a + b) + 2 + 1
4ab + 2(a + b) + 1 = 2(a + b) + 2 + 1
4ab = 2
2ab = 1
This is the same as saying (even = odd), which is an impossibility.
If you go to three factors, four factors, or even five factors, you'll notice
a pattern, having to do with the triangle above. You keep on dividing the
starting number in half for that triangle. Watch with 6 factors how it
relates to the triangle above.
X1 = sum of combinations formed from 6 (abcdef)
X2 = sum of combinations formed from 5 (abcde+abcdf+abcef+etc)
X3 = sum of combinations formed from 4 (abcd+abce+etc)
...
X6 = sum of combinations formed from 1 (a+b+c+d+e+f)
64 * X1 + 32 * X2 + 16 * X3 + 8 * X4 + 4 * X5 + 2 * X6 + 1 =
(32(X2) + 32(X3) + 24(X4) + 16(X5) + 10(X6) + 6) + (16(X3) + 24(X4) +
24(X5) + 20(X6) + 15) + (8(X4) + 16(X5) + 20(X6) + 20)
+ (4(X5) + 10(X6) + 15) +(2(X6) + 6) + 1.
Notice the similarities between this and the Pascal's triangle? (write the
triangle if necessary) Now, every row of that triangle equals 64, because of
the reason in the e-mail sent by the other guy. Anyway, after subtracting,
from the above equation gives the middle of the triangle:
32
24 24
16 24 16
10 20 20 10
6 15 20 15 6
And the rows add up to: (64 - 64), (64 - 32) , (64 - 16), (64 - 8), (64 - 4)
, and (64 - 2), and this was proven by the last guy's email, since the first
triangle is just the regular triangle times descending powers of 2:
1 * 64 = 64
1 1 * 32 = 32 32
1 2 1 * 16 = 16 32 16
1 3 3 1 * 8 = 8 24 24 8
etc..
So, this goes to:
64(X1) = 32(2-1)(X3) + 16(4-1)(X4) + (8)(8-1)(X5) + 4(16-1)(X6) + 2(32-1) * 1
This can be divided by two, giving:
32(X1) = 16(2-1)(X3) + 8(4-1)(X4) + 4(8-1)(X5) + 2(16-1)(X6) + (32-1)(1)
which is saying:
even = even + even + even + even + odd
even = even + odd
even = odd
Which is a good enough contradiction.
I explained a whole lot rather than just doing algebra, because we can take
the same route for the general equation rather than multiplying out, however,
I think it may have several problems, because it might not apply to the
general equation. Any way, I hope this helps. Maybe someone can do the
rest, and so the whole world can confidently say that if an odd, perfect
number exists, then it must be 0 mod ((2n+1)^2). Then we can go on to
squares times odd numbers, cubes time odd numbers, etc. I just hope the
whole thing is provable, after this.
Hoping and praying that 2^(9025267) - 1 is prime!!!!!!!
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------------------------------
Date: Tue May 02 00:25:18 2000
From: [EMAIL PROTECTED]
Subject: Mersenne: Re: look before you leap
Steinar Gunderson <[EMAIL PROTECTED]> wrote:
> OK, to complete the mess (I saw your message saying `ignore this',
> but I want to throw in my own errors as well ;-) ):
>
> If year % 4 = 0 then leap year
> If year % 100 = 0 then not leap year
> If year % 400 = 0 then leap year
> If year % 10000 = 0 then not leap year
I'm not sure about your 10000-year correction, for reasons I'll
give below (see the section about the anomalistic year below).
A favorite way of torturing beginning programmers is to ask
them to effect the above in a fashion that minimizes redundancy,
i.e. via a nested conditional structure. At the risk of grabbing
the leap year programming bug baton myself, I believe that the
resulting pseudocode segment should read as follows (I'm going
to only explicitly include corrections up to the 400-year one):
if(year % 4 = 0) then
if(year % 100 = 0) then
if(year % 400 = 0) then
leap year = true
{or add higher-order corrections here}
else
leap year = false
endif
else
leap year = true
endif
else
leap year = false
endif
This kind of thing probably causes thousands of first-time
programming students to curse Pope Gregory's name each year,
although in all fairness he is only responsible for the mod
100 and 400 part of it; the mod 4 part (the so-called Julian
calendar) goes back to the reign of Julius Caesar in 45 BC.
Of course all of these are simply corrections to account for
the fact that the 'true' time for the Earth to complete one
orbit relative to the 'fixed' stars, the so-called sidereal
year, is slightly longer than 365 days, and is not (at least
in general) a rational number.
Whoops - what I really wanted was not the sidereal year (which
astronomers prefer), but the tropical year, the length of time
between successive vernal equinoxes (more interesting back
during those predominantly agrarian economic times). Silly me!
My Norton's Star Atlas (vintage 1950) lists the length of the
tropical year as 365.24219... years (ellipses mine), so one
can argue that Gregory could heve done much better by chucking
out the old one-in-four rule and finding a parsimonious rational
approximation to 0.24219..., rather than giving us the start
of the recursive nightmare listed above, but from a practical
point of view, it is desirable to be able to add corrections
to the pre-existing ones, i.e. one wants each higher modulus
to be divisible by all smaller ones. From a human-nature point
of view, one wants things to be in terms of numbers people (or
at least 16th-century popes :) are comfortable with, hence the
bias toward base-10. So you calendrically oriented recreational
math enthusiasts can work out the next few corrections, starting
with:
- - the civil year, of necessity a whole number of days, 365;
- - adding one leap day every 4th year gives year = 365.25 days;
- - subtracting one leap day every 100 years gives year = 365.24 days.
- - adding one leap day every 400 years gives year = 365.2425 days.
Now, if we subtract one leap day every 10000 years we get
year = 365.2424 days, but if we instead subtract three leap
days every 10000 years, we get year = 365.2422 days, which is
much closer to 365.24219 days. So either my Norton's has it wrong
or we need to ... yes, you guessed it ... modify the calendar yet
again!
What does any of this have to do with Mersenne primes? Let's
see if I can whip up a quick ex post facto connection:
- - Pope Gregory was a Catholic; so was (arguments about which
sect and such aside); so was friar Mersenne;
- - Gregory devised his modifications to the Julian calendar
in 1582; Mersenne was born in 1588, which is just six (the
smallest perfect number!) years (civil years, that is) later.
Gosh, what an amazing set of coincidences.
Gosh, look how much time I just wasted composing this.
- -Ernst
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------------------------------
Date: Tue May 02 00:25:51 2000
From: [EMAIL PROTECTED]
Subject: Mersenne: what's up with the server?
Stefan Struiker wrote:
> A garland of 404s beyond the / of /ips. Yet results are still received,
> and the Entropia Home page displays. Never knew I'd miss the weekend
> stats so much.
>
> Sigh,
> Stefanovic
Martijn Kruithof wrote:
> prepend www to entropia.com and it works
Yeah, I noticed this yesterday, too, while attempting to
access the manual testing forms.
Is it the nameserver used by Entropia's ISP (or perhaps by
Entropia itself) that's responsible for whether a www prefix
is needed?
Whatever caused the change, if a www is now needed, I suggest
Entropia's webmaster prepend a www to all of the links between
the PrimeNet (and other) webpages on their site: right now some
links have a www, but others don't, and it becomes tedious to
click on a frequently-used link, see the 404 ERROR message,
then have to add the www manually to get to the desired page.
Thanks,
Ernst
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------------------------------
Date: Tue, 02 May 2000 15:54:33 +0200
From: Harald Tveit Alvestrand <[EMAIL PROTECTED]>
Subject: Re: Mersenne: Re: Richard's Y2.1K error !!
At 17:29 29.04.2000 -0400, [EMAIL PROTECTED] wrote:
>In a message dated 4/29/00 2:14:03 PM Pacific Daylight Time,
>[EMAIL PROTECTED] writes:
>
><< I don't think there are any more conditions than this, but who knows? :-)
> Of course, we're forgetting about the leap SECONDS now...
> >>
>Like anyone's going to care about a second or so. Just write it using these
>rules. You won't have to worry for another couple of century's. So, If
>another rule is found, their is ample time to fix it. So I think it's all
>right : - )
one of the amazing things I discovered some years ago when browsing the
website of the Institute for Earth Rotation (!) is that leap seconds can't
be predicted more than approximately 5 years in advance.
it's a strange world.
- --
Harald Tveit Alvestrand, EDB Maxware, Norway
[EMAIL PROTECTED]
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------------------------------
Date: Tue, 2 May 2000 16:12:57 +0100
From: "David Hoyle" <[EMAIL PROTECTED]>
Subject: Mersenne: Re: Problems with Iteration Time Prime 95
I am running v19.2.1 on a P3/500 and can achieve 0.34 second iteration time
with Prime running alone. However, because of our dodgy electricity supply
out here in the country, I have to run the m/c with an APC UPS which has
Powerchute 5.0.1 installed to achieve graceful shutdown if there is a
protracted power failure. Even despite setting Prime95 to priority 9, a
program in the UPS suite - ICONCLNT.EXE - takes 78% to Prime's 17% of the
cycles according to Wintop98. Before getting in touch with the manufacturer
of the UPS system, who is not going to see his program as malfunctioning, I
thought I would check if anyone else in the GIMPS programme has had a
similar problem and found a way round it. The only things taking any
signficant resources for the vast proportion of the time are Prime 95 and
the UPS to ensure the power supply integrity is guaranteed. I hope this is
not a Catch 22 situation.
David Hoyle
[EMAIL PROTECTED]
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------------------------------
Date: Tue, 2 May 2000 16:31:08 -0700 (PDT)
From: John R Pierce <[EMAIL PROTECTED]>
Subject: Mersenne: sumouts...
I have a machine that generated a bunch of illegal sumouts before.
I quit primenet because of this, later identified the cause, repaired
it and restarted primenet. I've had zero errors since then. I'm using
the same machine ID's, will this somehow 'discount' the quality of
any results I generate? If this could be manually overridden in the database,
user 'jp', machines 'dell4400' and 'dell4400b' are now 100% AOK and no
erroneous or suspect results contaminate any calculations.
- -jrp
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------------------------------
Date: Tue, 02 May 2000 17:00:47 -0700
From: Eric Hahn <[EMAIL PROTECTED]>
Subject: Mersenne: Re: Problems with Iteration Time Prime 95
David Hoyle wrote:
>I am running v19.2.1 on a P3/500 and can achieve 0.34 second
>iteration time with Prime running alone. However, because
>of our dodgy electricity supply out here in the country, I
>have to run the m/c with an APC UPS which has Powerchute
>5.0.1 installed to achieve graceful shutdown if there is a
>protracted power failure. Even despite setting Prime95 to
>priority 9, a program in the UPS suite - ICONCLNT.EXE - takes
>78% to Prime's 17% of the cycles according to Wintop98. Before
>getting in touch with the manufacturer of the UPS system, who
>is not going to see his program as malfunctioning, I thought
>I would check if anyone else in the GIMPS programme has had a
>similar problem and found a way round it. The only things
>taking any signficant resources for the vast proportion of
>the time are Prime 95 and the UPS to ensure the power supply
>integrity is guaranteed. I hope this is not a Catch 22
>situation.
David,
Good news!! I ran across the same problem when I first
installed my UPS several years ago. I *did* call APC and
was told that ICONCLNT.EXE only manages the icon in the
system tray (that's the *ONLY* thing it does!!). It you
can live without it (ie: not having the power status of the
UPS always displaying in the system tray), you can get rid
of it. It will *not* affect the reliability of the UPS in
any way. The system tray icon is a visual indicator for the
user. The UPS makes noise when the status changes, so there
isn't any real need for it.
In Windows 98, go into System Information tool (located
under Programs | Accessories | System Tools. Select the
Tools menu and then System Configuration Utility. Click on
the Startup tab. Find ICONCLNT.EXE in the list, and remove
the check in the box. Click OK, exit the program, and
restart Windows. BAM!! No more system tray icon, Prime95
can get the CPU's time, and the UPS is running fine!!
Eric Hahn
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------------------------------
Date: Tue, 02 May 2000 20:19:56 EDT
From: "Nathan Russell" <[EMAIL PROTECTED]>
Subject: Mersenne: d.net IRC channel
I have been talking with the distributed.net IRC channel, #distributed on
EFnet. I am known there as Trekkie. I've been telling them about the
greater chance of a successful result. The channel operators are okay with
this, and it's provoking a lively discussion.
If anyone wants to join the channel and is checking their mail now, this'd
be a golden oppertunity...
Nathan
P.S. I am being polite, and several people seem sincerely interested.
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------------------------------
Date: Tue, 02 May 2000 17:48:00 -0700
From: Spike Jones <[EMAIL PROTECTED]>
Subject: Re: Mersenne: Re: Richard's Y2.1K error !!
Harald Tveit Alvestrand wrote:
> one of the amazing things I discovered some years ago when browsing the
> website of the Institute for Earth Rotation (!) is that leap seconds can't
> be predicted more than approximately 5 years in advance.
>
> it's a strange world.
Not so strange. The rule that always holds is the conservation of
angular momentum. The uncertainty is in the moment of inertia of
the earth. Movements in the tectonic plates cause slight variations,
enough to throw off predictions of leap seconds. spike
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------------------------------
Date: Wed, 3 May 2000 18:27:38 -0700
From: Todd Sauke <[EMAIL PROTECTED]>
Subject: Re: Mersenne: Re: look before you leap
Warning! Continuation of off-topic leap year posts!
Do not read if you are uninterested in the leap-year discussion!
1) The once per 10000 year leap-year exception referred to by Steinar IS
NOT part of the Gregorian calendrical system. As Ernst mentioned, it would
not even correctly fix the next order of error. Ernst suggested that 3
exceptions per 10000 years would come closer to "fixing" the next order of
error. But. . .
2) The main problem with trying to determine a permanent system of leap
year "fixes" is that the average number of days per tropical year IS
CHANGING with time. 3 leap-year exceptions per 10000 years would work out
to an average of 365.2422 days per year which is pretty close to the value
Ernst noted of 365.24219, but by the time 10000 years have elapsed this
value will be far from the actual case! I'm not sure what the current best
estimate of the rate of change is or even if it is a constant rate, but one
old reference puts it at minus 6.14 times 10^-8 days per year. This value
would add up to a cumulative need for 3 MORE leap-year exceptions over the
same 10000 year period. We could "fix" this problem by making ~6 exceptions
over 10000 years, or say one exception every 1600 years. But the
accumulated drift error would be greater than 1/2 a day at about the year
7000. To make a long-lasting rule that keeps the cumulative error below
1/2 a day for as long as possible, we should make an exception only every,
say 2000 years, rather than 1600 years. This rule would work until about
the year 10500 with less than 1/2 a day error over the whole period
(ASSUMING a constant value of rate of change of -6.14 x 10^-8 days per
year). BUT this works out on average to what we would get if the 400 year
exception rule were made to be a 500 year rule instead! IF ONLY THE
GREGORIAN CALENDAR would have made the 400 year rule to be a 500 year rule
instead it would have been SIMPLER AND MORE ACCURATE for a LONGER PERIOD of
time, until the year 10500!!
3) Global warming, which would expand the earth's atmosphere and change
earth's moment of inertia, and thus the length of a day, may make all of
the above irrelevant before the year 10500 anyway. . .
Cheers,
Todd Sauke
Ernst wrote:
>Steinar Gunderson <[EMAIL PROTECTED]> wrote:
>
>> OK, to complete the mess (I saw your message saying `ignore this',
>> but I want to throw in my own errors as well ;-) ):
>>
>> If year % 4 = 0 then leap year
>> If year % 100 = 0 then not leap year
>> If year % 400 = 0 then leap year
>> If year % 10000 = 0 then not leap year
>
>I'm not sure about your 10000-year correction, for reasons I'll
>give below (see the section about the anomalistic year below).
>
snip
>
>Of course all of these are simply corrections to account for
>the fact that the 'true' time for the Earth to complete one
>orbit relative to the 'fixed' stars, the so-called sidereal
>year, is slightly longer than 365 days, and is not (at least
>in general) a rational number.
>
>Whoops - what I really wanted was not the sidereal year (which
>astronomers prefer), but the tropical year, the length of time
>between successive vernal equinoxes (more interesting back
>during those predominantly agrarian economic times). Silly me!
>
>My Norton's Star Atlas (vintage 1950) lists the length of the
>tropical year as 365.24219... years (ellipses mine), so one
>can argue that Gregory could heve done much better by chucking
>out the old one-in-four rule and finding a parsimonious rational
>approximation to 0.24219..., rather than giving us the start
>of the recursive nightmare listed above, but from a practical
>point of view, it is desirable to be able to add corrections
>to the pre-existing ones, i.e. one wants each higher modulus
>to be divisible by all smaller ones. From a human-nature point
>of view, one wants things to be in terms of numbers people (or
>at least 16th-century popes :) are comfortable with, hence the
>bias toward base-10. So you calendrically oriented recreational
>math enthusiasts can work out the next few corrections, starting
>with:
>
>- the civil year, of necessity a whole number of days, 365;
>- adding one leap day every 4th year gives year = 365.25 days;
>- subtracting one leap day every 100 years gives year = 365.24 days.
>- adding one leap day every 400 years gives year = 365.2425 days.
>
>Now, if we subtract one leap day every 10000 years we get
>year = 365.2424 days, but if we instead subtract three leap
>days every 10000 years, we get year = 365.2422 days, which is
>much closer to 365.24219 days. So either my Norton's has it wrong
>or we need to ... yes, you guessed it ... modify the calendar yet
>again!
>
>What does any of this have to do with Mersenne primes? Let's
>see if I can whip up a quick ex post facto connection:
>
>- Pope Gregory was a Catholic; so was (arguments about which
>sect and such aside); so was friar Mersenne;
>
>- Gregory devised his modifications to the Julian calendar
>in 1582; Mersenne was born in 1588, which is just six (the
>smallest perfect number!) years (civil years, that is) later.
>
>Gosh, what an amazing set of coincidences.
>
>Gosh, look how much time I just wasted composing this.
>
>-Ernst
>
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------------------------------
End of Mersenne Digest V1 #727
******************************
