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$100,000 and 2^p -1
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Patrick Hamlyn
science forum beginner


Joined: 03 May 2005
Posts: 45

PostPosted: Tue May 16, 2006 12:48 am    Post subject: Re: $100,000 and 2^p -1 Reply with quote

"Dan in NY" <Someone@NY.net> wrote:

Quote:
Posting
on Monday, May 15, 2006
Greetings,

I have a recreational, on-the-side puzzle suggested to me by a $100,000
prize for computing a prime number. I am curious, I propose this puzzle:
"What is the smallest prime, p, such that 2^p -1 has ten million decimal
digits?" Please note that I am asking p to be prime but not 2^p -1. This
background information indicates my interest in the answer:-

The web site http://www.eff.org/awards/coop.php tells of prize money in
computing. Here are selected quotes from the site:- "50,000 to the first
individual or group who discovers a prime number with at least 1,000,000
decimal digits (awarded Apr. 6, 2000). $100,000 to the first individual or
group who discovers a prime number with at least 10,000,000 decimal digits."
See the site, http://www.mersenne.org/prime.htm for information about GIMPS,
the Great Internet Mercenne Prime Search. If you use GIMPS to find a
winning prime, you will share the prize.

As far as I know, the largest prime now published is 2^30,402,457 -1. It is
a Mersenne prime having 9,152,052 decimal digits; in binary, all its
30,402,457 bits are ones. The number 30,402,457 is itself a prime number.
It is possible -- I suppose it is very likely -- that the winning number for
this $100,000 prize will be a Mercenne prime. A Mercenne number is a
number, 2^p -1, where p is prime. If 2^p -1 is prime the number is a
Mercenne prime.

I would like to know the value of p for the "smallest" Mercenne number of at
least a million decimal digits, written as 2^p -1. I want to know a the
prime number, p, not any of the ten million digits of the Mersenne number.

AFAIK, only 43 Mersenne numbers are known to be prime. Please note that I
am asking a much simpler question than that of the prize; I am asking for a
Mercenne number, not a Mersenne prime.

I call this a puzzle because I have calculated an approximate answer to this
but I don't want to reveal my answer yet, rather I want to give this hint:
my number, p, is between 33,000,000 and 34,000,000. Also I want to learn of
the number and/or the method you use to find the answer to this puzzle.

What's wrong with just taking (log 2) of 10^10 000 000+1?

You might need an extended precision program is all.

Eg in Mathematica you type Log[2,10^10000000+1] and get
33219280.94887362347870319429489390175864831393024580 plus another 10 million or
so digits of precision if you want.
--
Patrick Hamlyn posting from Perth, Western Australia
Windsurfing capital of the Southern Hemisphere
Moderator: polyforms group (polyforms-subscribe@egroups.com)
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bert
science forum addict


Joined: 04 Jan 2006
Posts: 54

PostPosted: Tue May 16, 2006 9:28 am    Post subject: Re: $100,000 and 2^p -1 Reply with quote

Patrick Hamlyn wrote:
Quote:
"Dan in NY" <Someone@NY.net> wrote:

Posting
on Monday, May 15, 2006
Greetings,

I have a recreational, on-the-side puzzle suggested to me by a $100,000
prize for computing a prime number. I am curious, I propose this puzzle:
"What is the smallest prime, p, such that 2^p -1 has ten million decimal
digits?" Please note that I am asking p to be prime but not 2^p -1. This
background information indicates my interest in the answer:-

The web site http://www.eff.org/awards/coop.php tells of prize money in
computing. Here are selected quotes from the site:- "50,000 to the first
individual or group who discovers a prime number with at least 1,000,000
decimal digits (awarded Apr. 6, 2000). $100,000 to the first individual or
group who discovers a prime number with at least 10,000,000 decimal digits."
See the site, http://www.mersenne.org/prime.htm for information about GIMPS,
the Great Internet Mercenne Prime Search. If you use GIMPS to find a
winning prime, you will share the prize.

As far as I know, the largest prime now published is 2^30,402,457 -1. It is
a Mersenne prime having 9,152,052 decimal digits; in binary, all its
30,402,457 bits are ones. The number 30,402,457 is itself a prime number.
It is possible -- I suppose it is very likely -- that the winning number for
this $100,000 prize will be a Mercenne prime. A Mercenne number is a
number, 2^p -1, where p is prime. If 2^p -1 is prime the number is a
Mercenne prime.

I would like to know the value of p for the "smallest" Mercenne number of at
least a million decimal digits, written as 2^p -1. I want to know a the
prime number, p, not any of the ten million digits of the Mersenne number.

AFAIK, only 43 Mersenne numbers are known to be prime. Please note that I
am asking a much simpler question than that of the prize; I am asking for a
Mercenne number, not a Mersenne prime.

I call this a puzzle because I have calculated an approximate answer to this
but I don't want to reveal my answer yet, rather I want to give this hint:
my number, p, is between 33,000,000 and 34,000,000. Also I want to learn of
the number and/or the method you use to find the answer to this puzzle.

What's wrong with just taking (log 2) of 10^10 000 000+1?

You might need an extended precision program is all.

Eg in Mathematica you type Log[2,10^10000000+1] and get
33219280.94887 . . .
--
Patrick Hamlyn posting from Perth, Western Australia

I think the OP wants to know the first prime which is greater
than 33,219,280, not just the first integer greater than it.
But if he can't even do that for himself, it seems unlikely that
he could have anything of much import to reveal afterwards.
--
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Patrick Hamlyn
science forum beginner


Joined: 03 May 2005
Posts: 45

PostPosted: Tue May 16, 2006 10:09 am    Post subject: Re: $100,000 and 2^p -1 Reply with quote

"bert" <bert.hutchings@btinternet.com> wrote:

Quote:
I think the OP wants to know the first prime which is greater
than 33,219,280, not just the first integer greater than it.
But if he can't even do that for himself, it seems unlikely that
he could have anything of much import to reveal afterwards.

Yeah it's not exactly a long way to search:
33 219 281 is prime.
--
Patrick Hamlyn posting from Perth, Western Australia
Windsurfing capital of the Southern Hemisphere
Moderator: polyforms group (polyforms-subscribe@egroups.com)
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Jens Kruse Andersen
science forum beginner


Joined: 23 Jul 2005
Posts: 40

PostPosted: Tue May 16, 2006 8:53 pm    Post subject: Re: $100,000 and 2^p -1 Reply with quote

Patrick Hamlyn wrote:
Quote:
"Dan in NY" <Someone@NY.net> wrote:
"What is the smallest prime, p, such that 2^p -1 has ten million decimal
digits?"

I would like to know the value of p for the "smallest" Mercenne number of
at least a million decimal digits, written as 2^p -1. I want to know a
the prime number, p, not any of the ten million digits of the Mersenne
number.

What's wrong with just taking (log 2) of 10^10 000 000+1?

It leads to a number with 10 000 001 digits.
(log 2) of 10^9 999 999 = 33219277.6269
The next prime is 33219281.
But 2^33219281 also has 10 000 001 digits.
So there is no answer to the first question which didn't say "at least".

The second question asks for at least a million digits.
(log 2) of 10^999 999 = 3321924.7730.
The next prime is 3321937.
So the answer is 2^3321937-1 which has 1 000 003 digits.
Good thing he didn't want "the ten million digits".
I would have been 8 999 997 short.

--
Jens Kruse Andersen
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Dan in NY
science forum beginner


Joined: 26 Mar 2005
Posts: 20

PostPosted: Tue May 16, 2006 9:02 pm    Post subject: Re: $100,000 and 2^p -1 Reply with quote

"Patrick Hamlyn" <path@multipro.N_OcomSP_AM.au> wrote in message
news:949j6296tjm62gibjksd77gjhnr1h5p9f8@4ax.com...
Quote:
"bert" <bert.hutchings@btinternet.com> wrote:

I think the OP wants to know the first prime which is greater
than 33,219,280, not just the first integer greater than it.
But if he can't even do that for himself, it seems unlikely that
he could have anything of much import to reveal afterwards.

Yeah it's not exactly a long way to search:
33 219 281 is prime.
--
Patrick Hamlyn posting from Perth, Western Australia
Windsurfing capital of the Southern Hemisphere
Moderator: polyforms group (polyforms-subscribe@egroups.com)

&&&
Thanks for your replies, Patrick. You verified that my answer of 33,219,281
(as I write it in NY) is the answer to the puzzle. It is almost out of my
reach to determine whether numbers that large are prime.

I know that many things posted on Math usenet sites are requests to help
with tests and/or homework but I thought such protests from me would be
regarded with skepticism. In this case, I tried to turn a personal
curiosity of mine into a puzzle.

My desire to find this number is to be able to compare it to the next larger
prime announced and be able to tell whether it is large enough to have at
least 10,000,000 digits.

I have been following the new large primes being found using GIMPS related
to the $100,000 prize and part of my interest was to kindle interest in
anyone who reads the post and may be so inclined.

Before I posted the puzzle, I used a web site to find a list of primes
showing these four numbers as consecutive primes:-
33,219,253
33,219,281
33,219,283
33,219,313

I only found an approximate method for determining the number of digits for
2^p-1 using these primes. If Log[10,2^p] is rounded down to the nearest
integer does it give an exact answer?

You suggested that Log[2,10^10000000+1] be evaluated then gave its value
using Mathematica to evaluate it. Usually I use a spreadsheet, Quatro Pro,
to assist me but I have access to a computer that has Mathematica and (after
problems with a balky firewall) I was able to verify your answer to a few
digits beyond the decimal point. It seems as well to use Log[2,10^10000000]
(without +1) because it would take lots of precision to tell the difference.

Of course, as pointed out in the reply by bert, my puzzle asked for a prime
so it was sufficient to only include enough digits to be sure it wasn't
rounded up to reach 10,000,000 digits. Even though the puzzle asked for a
prime, an integer suffices to satisfy my curiosity. Rather by chance, the
smallest integer is prime, so if I had asked for the smallest such integer
exceeding a power of 2, the answer would be the same.
--
Dan in NY
(for email change t with g in
dKlinkenbert at hvc dot rr dot com)
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Dan in NY
science forum beginner


Joined: 26 Mar 2005
Posts: 20

PostPosted: Tue May 16, 2006 9:36 pm    Post subject: Re: $100,000 and 2^p -1 Reply with quote

"bert" <bert.hutchings@btinternet.com> wrote in message
news:1147771720.773656.119590@g10g2000cwb.googlegroups.com...
[omited]
Quote:
I call this a puzzle because I have calculated an approximate answer to
this
but I don't want to reveal my answer yet, rather I want to give this
hint:
my number, p, is between 33,000,000 and 34,000,000. Also I want to
learn of
the number and/or the method you use to find the answer to this puzzle.

What's wrong with just taking (log 2) of 10^10 000 000+1?

You might need an extended precision program is all.

Eg in Mathematica you type Log[2,10^10000000+1] and get
33219280.94887 . . .
--
Patrick Hamlyn posting from Perth, Western Australia

I think the OP wants to know the first prime which is greater
than 33,219,280, not just the first integer greater than it.
But if he can't even do that for himself, it seems unlikely that
he could have anything of much import to reveal afterwards.

&&&

Greetings bert,

Thanks for your reply regarding my puzzle. You are correct that I asked for
a prime, not a number with a great deal of precision. I could have asked
for the exponent of the smallest integral power of 2 with 10,000,000 decimal
digits. It seems that the answer is the same as for my puzzle. I think it
would have been better for me to give an example using, say, 10 digits, then
asked for one using a million times as many digits..

As for doing it for myself, I wanted to pose a recreational puzzle, so you
are correct that I don't have anything of much import to be revealed now.
However, I am curious, how would you go about finding the smallest prime
larger than 33,219,280.94887?

About the only thing I can say now is that if a new Mersenne prime is
announced and the exponent of 2^p -1 is given, I can compare it to
33,219,281 to know whether it is large enough to have at least 10,000,000
digits.
--
Dan in NY
(for email change t with g in
dKlinkenbert at hvc dot rr dot com)
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bert
science forum addict


Joined: 04 Jan 2006
Posts: 54

PostPosted: Wed May 17, 2006 12:30 pm    Post subject: Re: $100,000 and 2^p -1 Reply with quote

Dan in NY wrote:
Quote:
"bert" <bert.hutchings@btinternet.com> wrote in message
news:1147771720.773656.119590@g10g2000cwb.googlegroups.com...

I think the OP wants to know the first prime which is greater
than 33,219,280, not just the first integer greater than it.
But if he can't even do that for himself, it seems unlikely that
he could have anything of much import to reveal afterwards.

Greetings bert,

Thanks for your reply regarding my puzzle. You are correct that I asked for
a prime, not a number with a great deal of precision. I could have asked
for the exponent of the smallest integral power of 2 with 10,000,000 decimal
digits. It seems that the answer is the same as for my puzzle. I think it
would have been better for me to give an example using, say, 10 digits, then
asked for one using a million times as many digits..

As for doing it for myself, I wanted to pose a recreational puzzle, so you
are correct that I don't have anything of much import to be revealed now.
However, I am curious, how would you go about finding the smallest prime
larger than 33,219,280.94887?

About the only thing I can say now is that if a new Mersenne prime is
announced and the exponent of 2^p -1 is given, I can compare it to
33,219,281 to know whether it is large enough to have at least 10,000,000
digits.
--
Dan in NY

The web must offer dozens of good interactive calculator programs
for multiple-precision integers, so it would just be a case of
choosing one and using it. It would have several number-theoretic
functions more complicated than "next prime", do most things with
integers up to 50 digits faster than you could blink, and more slowly
up to at least 300 digits. Writing one for yourself, say in Visual
Basic, is good fun.
--
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Dan in NY
science forum beginner


Joined: 26 Mar 2005
Posts: 20

PostPosted: Thu May 18, 2006 12:11 am    Post subject: Re: $100,000 and 2^p -1 Reply with quote

"Jens Kruse Andersen" <jens.k.a@NOSPAMget2net.dk> wrote in message
news:sYqag.116$c47.85@news.get2net.dk...
Quote:
Patrick Hamlyn wrote:
"Dan in NY" <Someone@NY.net> wrote:
"What is the smallest prime, p, such that 2^p -1 has ten million decimal
digits?"
I would like to know the value of p for the "smallest" Mercenne number
of at least a million decimal digits, written as 2^p -1. I want to
know a the prime number, p, not any of the ten million digits of the
Mersenne number.
What's wrong with just taking (log 2) of 10^10 000 000+1?

It leads to a number with 10 000 001 digits.
(log 2) of 10^9 999 999 = 33219277.6269
The next prime is 33219281.
But 2^33219281 also has 10 000 001 digits.
So there is no answer to the first question which didn't say "at least".

The second question asks for at least a million digits.
(log 2) of 10^999 999 = 3321924.7730.
The next prime is 3321937.
So the answer is 2^3321937-1 which has 1 000 003 digits.
Good thing he didn't want "the ten million digits".
I would have been 8 999 997 short.
--
Jens Kruse Andersen
&&&


Greetings Jens,

Thank you for your reply. Because of it, I see an error in the my first
post of this thread, that I missed until now. I can't be sure, but you
probably figured that out. In the quote of my post above (from the
background information for the puzzle) where I said, "at least a million
decimal digits", I meant to write, "at least ten million decimal digits."

Also, you are correct that in my puzzle, I missed writing, "at least". I
didn't think it was an error because if a number has more than ten million
digits, it has ten million, but I should have been more clear. [That is, if
I have 15 eggs and I am asked whether I have a dozen eggs, I would likely
say, "Yes," even if the questioner didn't say, "at least."]

In your reply to Patrick's comment, "What's wrong with just taking (log 2)
of 10^10 000 000+1?", I was surprised at your reply, "It leads to a number
with 10 000 001 digits." It took me a few moments to realize what you
meant. "10^10 000 000+1" is a bit ambiguous but that hadn't occurred to me
when I read what Patrick wrote. I figured that the one really shouldn't be
added at all.

To clear up the ambiguity I am referring to, add parentheses after the "^".
But there are two ways to do this. Either "10^(10 000 000)+1 or "10^(10 000
000+1). If the "+1" is included inside as part of the exponent, I thought
it should have been added to make it "10^10 000 001" so I didn't consider
that idea. However, doing it that way does do what you said, it, "It leads
to a number with 10 000 001 digits."

I assumed I should put the "+1" after both parentheses. However, in doing
that, the "+1" becomes rather redundant because after taking Log 2, seeing
the difference between adding 1 or not adding it would require a lot more
precision than a few digits after the decimal point.

When you say, "But 2^33219281 also has 10 000 001 digits," I disagree with
you but I really do want to know whether you are correct. I think it has
exactly ten million digits and I could be wrong. In fact I posted my puzzle
partly in an attempt to verify that result of mine. Do you know of a
smaller integer (or prime) than 2^33219281 that has at least ten million
digits? I think the integer, 2^33219280, has 9 999 999 digits.

It is different to write this using a keyboard rather than a pencil and
paper; may lead to ambiguities. (With more complicated expressions that is
even more likely.) I should have stopped without keying in so many words
because the longer this is, the more likely I have made another mistake. If
anyone spots an error I would like to know but if we keep this going, I must
be careful because I might be beating it to death.
--
Dan in NY
(for email change t with g in
dKlinkenbert at hvc dot rr dot com)
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Dan in NY
science forum beginner


Joined: 26 Mar 2005
Posts: 20

PostPosted: Thu May 18, 2006 1:36 am    Post subject: Re: $100,000 and 2^p -1 Reply with quote

"bert" <bert.hutchings@btinternet.com> wrote in message
news:1147869023.102499.325610@i39g2000cwa.googlegroups.com...
Quote:
Dan in NY wrote:
"bert" <bert.hutchings@btinternet.com> wrote in message
news:1147771720.773656.119590@g10g2000cwb.googlegroups.com...

The web must offer dozens of good interactive calculator programs
for multiple-precision integers, so it would just be a case of
choosing one and using it. It would have several number-theoretic
functions more complicated than "next prime", do most things with
integers up to 50 digits faster than you could blink, and more slowly
up to at least 300 digits. Writing one for yourself, say in Visual
Basic, is good fun.

Greetings bert,

Because of this suggestion, I have used Google to attempt to find an
interactive web page that would do any sort of multiple-precision
operations. I found some programs to buy and lots of sites that mentioned
packages but no web page that does it. After I read this note, I thought it
would be easy to find. However, I don't have enough use for one to buy the
program. I have access to a computer having Mathematica so I might try
using that. It might well have a next prime function that counts the digits
of a number without showing those digits.

Your mention of using Visual Basic reminds me of years ago when I had an
Apple + computer. It booted up in a basic program called Applesoft written
by a (then) little known company now called Microsoft. (I don't even know
whether Bill Gates was part of the company at that time.) But I digress: I
did write an Applesoft basic program called Bignum with arbitrary precision
integers (up to a program set maximum number of digits). It was based on a
program I found but I added other functions. One was to calculate the power
of a number; another could perform a Mod function on the power of a number.

I don't remember what all it could do but I don't have any programming
language on this computer other than Quatro Pro with a precision of about 15
digits (including preliminary results). My basic program did things related
to RSA messaging but it didn't do much else regarding prime numbers. I
think I had to limit it to 999 digits, much smaller than ten million!

Thanks for the memories. (I would be interested to have a URL for an
interactive web site that can do multiple-precision operations. If I find
one, I might even buy a low cost program that has functions I am interested
to have.
--
Dan in NY
(for email change t with g in
dKlinkenbert at hvc dot rr dot com)
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Jens Kruse Andersen
science forum beginner


Joined: 23 Jul 2005
Posts: 40

PostPosted: Thu May 18, 2006 11:28 am    Post subject: Re: $100,000 and 2^p -1 Reply with quote

Dan in NY wrote:
Quote:
"Jens Kruse Andersen" <jens.k.a@NOSPAMget2net.dk> wrote in message
news:sYqag.116$c47.85@news.get2net.dk...
Patrick Hamlyn wrote:
"Dan in NY" <Someone@NY.net> wrote:
"What is the smallest prime, p, such that 2^p -1 has ten million decimal
digits?"
I would like to know the value of p for the "smallest" Mercenne number
of at least a million decimal digits, written as 2^p -1. I want to
know a the prime number, p, not any of the ten million digits of the
Mersenne number.
What's wrong with just taking (log 2) of 10^10 000 000+1?

It leads to a number with 10 000 001 digits.
(log 2) of 10^9 999 999 = 33219277.6269
The next prime is 33219281.
But 2^33219281 also has 10 000 001 digits.
So there is no answer to the first question which didn't say "at least".

The second question asks for at least a million digits.
(log 2) of 10^999 999 = 3321924.7730.
The next prime is 3321937.
So the answer is 2^3321937-1 which has 1 000 003 digits.
Good thing he didn't want "the ten million digits".
I would have been 8 999 997 short.

Thank you for your reply. Because of it, I see an error in the my first
post of this thread, that I missed until now. I can't be sure, but you
probably figured that out. In the quote of my post above (from the
background information for the puzzle) where I said, "at least a million
decimal digits", I meant to write, "at least ten million decimal digits."

Yes, I figured out what you meant. On Usenet, people may tease you by taking
badly formulated questions literally.

Quote:
Also, you are correct that in my puzzle, I missed writing, "at least". I
didn't think it was an error because if a number has more than ten million
digits, it has ten million, but I should have been more clear. [That is, if
I have 15 eggs and I am asked whether I have a dozen eggs, I would likely
say, "Yes," even if the questioner didn't say, "at least."]

That's a non-mathematical context where the questioner probably just
wants to know whether you have enough eggs for something.
If you were asked "Do you have 9 fingers?", I'm guessing you would say
"No, 10" (assuming you actually do!).
Mathematics usually require precision to avoid such speculation.
Your question was: "What is the smallest prime, p, such that 2^p -1 has
ten million decimal digits?"
I think most mathematicians would interpret this as exactly ten million.

Quote:
In your reply to Patrick's comment, "What's wrong with just taking (log 2)
of 10^10 000 000+1?", I was surprised at your reply, "It leads to a number
with 10 000 001 digits." It took me a few moments to realize what you
meant. "10^10 000 000+1" is a bit ambiguous but that hadn't occurred to me
when I read what Patrick wrote. I figured that the one really shouldn't be
added at all.

To clear up the ambiguity I am referring to, add parentheses after the "^".
But there are two ways to do this. Either "10^(10 000 000)+1 or "10^(10 000
000+1). If the "+1" is included inside as part of the exponent, I thought
it should have been added to make it "10^10 000 001" so I didn't consider
that idea. However, doing it that way does do what you said, it, "It leads
to a number with 10 000 001 digits."

No, it would give 10 000 002 digits.
How many digits in 10^2? In 10^3? See a pattern?

Quote:
When you say, "But 2^33219281 also has 10 000 001 digits," I disagree with
you but I really do want to know whether you are correct. I think it has
exactly ten million digits and I could be wrong. In fact I posted my puzzle
partly in an attempt to verify that result of mine. Do you know of a
smaller integer (or prime) than 2^33219281 that has at least ten million
digits? I think the integer, 2^33219280, has 9 999 999 digits.

No, it has 10 000 000.
(log 10) (2^33219280) = 33219280*(log 10) 2 ~= 9 999 999.714
So 2^33219280 ~= 10^9 999 999.714
Remember the pattern?

--
Jens Kruse Andersen
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Andreas Leitgeb
science forum beginner


Joined: 09 Jan 2006
Posts: 10

PostPosted: Thu May 18, 2006 2:37 pm    Post subject: Re: $100,000 and 2^p -1 Reply with quote

Dan in NY <Someone@NY.net> wrote:
Quote:
You verified that my answer of 33219281 [...] is the answer
to the puzzle.
It is almost out of my reach to determine whether numbers
that large are prime.

Oh gosh, do you do it with paper&pencil ?
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Dan in NY
science forum beginner


Joined: 26 Mar 2005
Posts: 20

PostPosted: Fri May 19, 2006 6:32 am    Post subject: Re: $100,000 and 2^p -1 Reply with quote

"Jens Kruse Andersen" <jens.k.a@NOSPAMget2net.dk> wrote in message
news:jTYag.69$Eo2.27@news.get2net.dk...
Quote:
Dan in NY wrote:
"Jens Kruse Andersen" <jens.k.a@NOSPAMget2net.dk> wrote in message
news:sYqag.116$c47.85@news.get2net.dk...
Patrick Hamlyn wrote:
"Dan in NY" <Someone@NY.net> wrote:
"What is the smallest prime, p, such that 2^p -1 has ten million
decimal
digits?"
-------------------------------------------------------------------------------

[I should have written the above question with "at least": "What is the
smallest
prime, p, such that 2^p -1 has at least ten million decimal digits?"]
-------------------------------------------------------------------------------
[text omited]
Quote:
When you say, "But 2^33219281 also has 10 000 001 digits," I disagree
with
you but I really do want to know whether you are correct. I think it has
exactly ten million digits and I could be wrong. In fact I posted my
puzzle
partly in an attempt to verify that result of mine. Do you know of a
smaller integer (or prime) than 2^33219281 that has at least ten million
digits? I think the integer, 2^33219280, has 9 999 999 digits.

No, it has 10 000 000.
(log 10) (2^33219280) = 33219280*(log 10) 2 ~= 9 999 999.714
So 2^33219280 ~= 10^9 999 999.714
Remember the pattern?
--
Jens Kruse Andersen

&&&

Greetings Jen (and all who are still reading this thread),

Thank you again. Now we have a disagreement that I would like to pursue. I
might be about to learn that my pre-determined answer to this puzzle was
incorrect. Did I incorrectly pronounce someone the winner? Oh, well, I
said I posted it partly to verify my answer, so that is OK with me. However
it is after 2 AM. here and I realize I am not functioning well enough. I
need to think about this more after some sleep. If you are correct about
2^33219280 having ten million digits then I haven't yet seen the answer to
my puzzle.

If I don't see a smaller answer here, I may post this question in a new
thread.
What smallest possible value of an integer, n, so that 2^n has at least ten
million decimal digits?
--
Dan in NY
(for email change to with go in
dKlinkenbert at hvc dot rr dot com)
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Mike Amling
science forum Guru


Joined: 05 May 2005
Posts: 525

PostPosted: Mon Jun 12, 2006 5:24 pm    Post subject: Re: $100,000 and 2^p -1 Reply with quote

In article <1147869023.102499.325610@i39g2000cwa.googlegroups.com>,
bert <bert.hutchings@btinternet.com> wrote:
Quote:

The web must offer dozens of good interactive calculator programs
for multiple-precision integers,

Could you recommend one? For that matter, I can think of some
fun things to do with a very extended precision floating point
calculator as well.


--
Please reply to: | "Any sufficiently advanced incompetence is
pciszek at panix dot com | indistinguishable from malice."
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moriman
science forum beginner


Joined: 06 Apr 2006
Posts: 26

PostPosted: Mon Jun 12, 2006 6:01 pm    Post subject: Re: $100,000 and 2^p -1 Reply with quote

"Paul Ciszek" <nospam@nospam.com> wrote in message
news:e6k80r$pm2$1@reader2.panix.com...

Quote:
Could you recommend one? For that matter, I can think of some
fun things to do with a very extended precision floating point
calculator as well.


Ya naughty little thing :-D


Quote:

--
Please reply to: | "Any sufficiently advanced incompetence is
pciszek at panix dot com | indistinguishable from malice."
Autoreply is disabled |
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Bob Delaney
science forum beginner


Joined: 05 Apr 2005
Posts: 16

PostPosted: Tue Jun 13, 2006 12:51 am    Post subject: Re: $100,000 and 2^p -1 Reply with quote

In article <e6k80r$pm2$1@reader2.panix.com>, Paul Ciszek
<nospam@nospam.com> wrote:

Quote:
In article <1147869023.102499.325610@i39g2000cwa.googlegroups.com>,
bert <bert.hutchings@btinternet.com> wrote:

The web must offer dozens of good interactive calculator programs
for multiple-precision integers,

Could you recommend one? For that matter, I can think of some
fun things to do with a very extended precision floating point
calculator as well.

Try my freeware RPN calculator:

http://homepage.mac.com/delaneyrm/MPCalcRB.html

Bob
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Google

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