Conference rusure::math

rticle 1465 of sci.math
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From: [email protected] (Bernie Cosell)
Newsgroups: sci.math,sci.crypt
Subject: Re: New Largest Known Prime
Keywords: prime,record
Message-ID: <[email protected]>
Date: 22 Aug 89 12:12:31 GMT
References: <[email protected]>
Sender: [email protected]
Reply-To: [email protected] (Bernie Cosell)
Organization: Bolt Beranek and Newman Inc., Cambridge MA
Lines: 25
Xref: mountn.dec.com sci.math:1465 sci.crypt:418

In article <[email protected]> [email protected] (Landon C. Noll) writes:
}
}
}                                         216193
}                               391581 * 2      -1
}
}                                    is prime
}
}
 [mucho numbers]

}This number, at the time of this posting is the largest known prime.  It is
}65087 digits long.  It was discovered on 6 Augist 1989 at 00:53 PDT by a 
 ^^^^^
 !!!!!
}team consisting of Joel Smith, John Brown, Landon Curt Noll, Bodo Parady, 
}Gene Smith and Sergio Zarantonello.

Is it just me: is there *any* utility whatsoever in actually sending the
digits of the number about.  The results are interesting, to be sure, but
to my eyes sending 20 screensful of mostly-random digits is just a complete
waste.  Is there some purpose served by having the actual digits that I'm
missing?

  /Bernie\


Article 1469 of sci.math
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From: [email protected] (Gene W. Smith)
Newsgroups: sci.math,sci.crypt
Subject: Re: New Largest Known Prime
Keywords: prime,record
Message-ID: <[email protected]>
Date: 23 Aug 89 03:47:35 GMT
References: <[email protected]> <[email protected]> <[email protected]>
Sender: [email protected] (USENET Administrator;;;;ZU44)
Reply-To: [email protected] (Gene W. Smith)
Followup-To: sci.math
Organization: Garnet Gang Gems of Wisdom, Inc.
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Xref: mountn.dec.com sci.math:1469 sci.crypt:422

In article <[email protected]>, sean@ms (Sean Casey) writes:

>What kind of primality test does one use for a 67000 digit number?

  That depends on the number. A generic 67000 digit number N
would be very difficult to prove prime unless one had some
special knowledge, e.g. a partial factorization of N-1 or N+1.
This particular prime is such that factoring N+1 is easy:

  N = 391581 * 2^216193 - 1, so
  N+1 = 2^216193 * 3^3 * 14503.

  Not only is N+1 highly composite, by far the largest
prime-power factor is a power of 2. The conditions are right for
a generalized Lucas-Lehmer test. For more info, read:

  H. Riesel, "Lucasian Criteria for the Primality of N = h*2^n-1",
Math. Comp., v. 23, 1969, pp. 869-875.

  Don't try this on your Mac.
--
ucbvax!garnet!gsmith     Gene Ward Smith/Brahms Gang/Berkeley CA 94720
"To name the unnamable, to point at frauds, to take sides, start arguments,
shape the world and stop it from going asleep". -- 'The Satanic Verses'


Article 1475 of sci.math
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From: [email protected] (Doug Davis)
Newsgroups: sci.math,sci.crypt
Subject: Re: New Largest Known Prime
Summary: question
Keywords: prime,record
Message-ID: <[email protected]>
Date: 23 Aug 89 21:28:45 GMT
References: <[email protected]>
Reply-To: [email protected] (Doug Davis)
Followup-To: sci.math
Organization: Logic Process  Dallas, Texas.
Lines: 13
Xref: mountn.dec.com sci.math:1475 sci.crypt:426

In article <[email protected]> [email protected] (Landon C. Noll) writes:
[ ... ] 
> 587848116810659383244926076364867553666818690988820055185630787017896301365577
              ^ Waita minute!?!? Isn't this supposed to be a 7??? ;-)


Seriously, though I would be interested seeing the program used to
test this monster, any thoughts on posting it?

doug
--
Doug Davis/1030 Pleasant Valley Lane/Arlington/Texas/76015/817-467-3740
{sys1.tandy.com, motown!sys1, uiucuxc!sys1 lawnet, attctc, texbell} letni!doug


Article 1476 of sci.math
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From: [email protected] (George Reimer)
Newsgroups: sci.math,sci.crypt
Subject: Re: New Largest Known Prime
Keywords: prime,record
Message-ID: <[email protected]>
Date: 23 Aug 89 16:57:39 GMT
References: <[email protected]> <[email protected]>
Reply-To: [email protected] (George Reimer)
Organization: Electrohome Ltd., Kitchener, ON
Lines: 37
Xref: mountn.dec.com sci.math:1476 sci.crypt:427

In article <[email protected]> [email protected] (Bernie Cosell) writes:
>
>}This number, at the time of this posting is the largest known prime.  It is
>}65087 digits long.  It was discovered on 6 Augist 1989 at 00:53 PDT by a 
> ^^^^^
 [ stuff delted ]

>Is it just me: is there *any* utility whatsoever in actually sending the
>digits of the number about.  The results are interesting, to be sure, but
>to my eyes sending 20 screensful of mostly-random digits is just a complete
>waste.  Is there some purpose served by having the actual digits that I'm
>missing?
>
>  /Bernie\


Well I for one, haven't seen a 65087 digits long number for quite a while. 
It was refreshing to be reminded of how big such a number really is!
I mean, after a stint of working with mega, micro, nano, and giga numbers,
seeing the voyager billions of miles out traveling at thousands of miles/hr 
etc, etc, etc.. . then seeing this printout, I can really appreciate the simple things
in life, like 7, 3, 12, koolaid, mild cheese and inverse matrix multiplication.
 
			  216193
		8^) *       -1 



8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^) 
8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^) 

[ stuff deleted  ;^)    ]

8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^)8^) 

George
Reimer

	Yes, there's an important reason to publish the number itself,
	all 800 lines of digits thereof.

	The important reason is to encourage at LEAST one person to
	verify the result by some independent method.  Personally,
	I doubt the number really isn't prime until an independent
	test of some sort is done.

/Eric

    re -1
    
    Why are all the digits needed for the varification?
    
    Eugene


	2

(Anyone care to verify?)

Andrew

	from .0

>> Primality was demonstrated by a program implementing the Lacasian h=3A test.
>> An Amdahl 1200 takes 1987 seconds to test primality.  We wish to thank
>> Jeff Young for confirming our result.

	Of course they had confirmation from at least one
	outside source before posting their result.  Do
	enough of you want the digits posted here or should
	I just mail them to Eric?

	Dan

re: .4

If it's prime, it's only divisible by itself and 1.  So wouldn't 1 be the
smallest prime?
					>>>==>PStJTT

    Re: -.1
    
    I think, by convention, 1 is ruled out as a prime. It may also be
    ruled out so that theorems work out nicely. For example, unique prime
    factorization of an integer would not be true any more if one includes
    1 as a prime.

    Re: .6, .7
    
    1 is not prime.  1 is a unit.  What you learned in high school was wrong.
    
    Integers come in 3 categories:  prime, composite, unit.
    
    (Other examples of units are -1, �i in the complex numbers, etc.)

Re: .8

Ah.  Well if it was the only thing I ever learned that was wrong, I suppose
I'd be more upset.  Is the distinction of "unit" valuable for some reason,
or is it, as .7 suggested, just becuase the value of 1 made things uncomfor-
table?
					>>>==>PStJTT

    Re: .8
    
    I think a unit is an element that has a multiplicative inverse. 
    Is that correct? If so, the concept of unit depends on the ring you
    are talking about. In your example, I take it that you meant -1 is
    a unit in Z and +/- i are units in C.
    
    Since the complex numbers constitute a field C, any complex number is
    a unit in C. Therefore, your example that +/- i is a unit is not too
    meaningful. Unless you mean +/- i in the ring of gaussian integers.
    (or is that a field?)

>    Since the complex numbers constitute a field C, any complex number is
>    a unit in C. Therefore, your example that +/- i is a unit is not too
>    meaningful. 
    
    	Excluding zero, that is correct.  So let's consider:
    
>    Unless you mean +/- i in the ring of gaussian integers.
    
    	... which is not a field because any inverse of, say, 4 would have
    to be isomorphic to 1/4 which is not a member of the Gaussian integers.
    
    As to the more general question of "Well, is this notion of a `unit'
    something real or is it merely a convenience so's we can have unique
    factorization domains?", I plead ignorance.  The same question can be
    applied to the definition of a field, in which every element except the
    additive identity has a multiplicative inverse.  Boy if that doesn't
    look like a "setup" to avoid division by zero, I don't know what does.
    I complained when my teacher (not high school :-) wrote down that
    definition and was told "But that's what you want to model the Reals".
    Somehow answers like that aren't very satisfying; you would like to
    think there's some deeper truth involved.  Would the Foundations experts
    care to comment?
    
      John

        You give names to the useful concepts.  If you are lucky,
        the more important concepts (used more often) have the
        shorter names. :-)  When studying rings, if there is a
        multiplicative identity then units are those ring
        elements that divide it.  Primes are defined in a way
        that excludes units and the additive identity (zero).
        That way theorems are worded "For every prime p ..."
        instead of "For every prime p > 1, ...".  You also will
        see a lot of "For every odd prime p ..." but overall it
        is more useful to keep 2 a prime and also have "odd primes"
        than it is to not have 2 be a prime and have "prime or 2".
        
        Fields are what they are, and are a useful concept.  If
        you like division by zero, make a new definition and go
        off and study that new creature.  You will probably find
        fewer "interesting" examples of them, but who knows? :-)
        
        Dan

	Thank you.

	Dan

    I thought I heard of a recent discovery of a new "largest prime
    number". I didn't catch the size, other than it took 32 pages to print
    it! Does anyone have a pointer to the number, and some of the
    background as to how it was found, and any other interesting
    properties? I think I heard that it was a Mersenne prime number, but
    I'm not sure.

    See topic 2.
    
    
    				-- edp

    This looks like as good a place as any to deposit this.
    
    Could somebody with the capability do the "finger" command below and
    post the response here?
    
      John
    -----------------------------------------------------------------------
    

                             THE LARGEST KNOWN PRIMES
                       (All primes with 1000 or more digits)
        Originally Compiled by Samuel Yates -- Continued by Chris Caldwell
                                (19 September 1994)
    
    So that I can maintain this list, please send any new gigantic or titanic
    primes, comments and/or corrections, to Chris K. Caldwell at
    
      Mathematics/Computer Science         or better, e-mail:
      University of Tennessee at Martin          [email protected]
      Martin, TN 38238, USA                      [email protected]
      (901) 587-7360                             [email protected]
    
    Finger [email protected] for ways to receive this list.  The letters
    after the rank indicate the most recently added primes.  See the last
    pages for information about these and other notations.
    
    rank  prime                        digits who year
             *** Gigantic Primes (those with over 10,000 digits) ***
        1  2^859433-1                    258716 SG 94 Mersenne ?
        2  2^756839-1                    227832 SG 92 Mersenne ??
        3  391581*2^216193-1              65087 Z  89
        4  2^216091-1                     65050 S  85 Mersenne 31
        5  2^132049-1                     39751 S  83 Mersenne 30
    
    ... and so on

    
>    Could somebody with the capability do the "finger" command below and
>    post the response here?

Finger is passed through the ftp gateways so anyone can do it.  This
particular one requires the command:

	finger [email protected]@ftp-gw.pa.dec.com

Which gives:
    
[gatekeeper.dec.com]
[unix1.utm.edu]
Login name: primes                      In real life: Chris Caldwell
Site Info:  Math and CS
Directory: /u/staff/primes              Shell: /bin/ksh
Plan:

Some Record Primes
----------------------    ------ --- ---- -------------------------
prime                     digits who when what
----------------------    ------ --- ---- -------------------------
2^859433-1                258716  SG  94  Largest Prime (a Mersenne)
2^756839-1                227832  SG  92  Next Largest (also Mersenne)
391581*2^216193-1          65087  Z   89  Largest Non-Mersenne
10^11810+1465641*10^5902+1 11811  D   94  Largest Palindrome
3610!-1                    11277  C   93  Largest Factorial minus one
10^11010+3242423*10^5502+1 11011  D   94  Next Palindrome
24029#+1                   10387  C   93  Largest Primorial Plus One
5415312903*10^4526-1        4536  D   94  Largest Sophie Germain
1692923232*10^4020+/-1      4030  D   93  Largest Known Twin Primes
4655478828*10^3429+/-1      3439  D   93  Next Twin Primes
----------------------    ------ --- ---- -------------------------

Ways to get partial or complete lists of "The Largest Known Primes":

by gopher to
   unix1.utm.edu  (directory 1/user/Public_FTP/pub/math/primes or
     choose Departments; Mathematics; then Largest Known Primes) 

by anonymous ftp to   
   math.utm.edu  (directory /pub/math/primes)

by e-mail (least preferable method!)
   [email protected]
   [email protected] 
   [email protected]

Please help me maintain this list by sending any corrections, new
titanic primes (greater that 1000 digits), or new gigantic primes 
(greater than 10,000 digits) to any of the above e-mail address.

    Thanks, Monty. I run VMS and don't have a finger command, though surely
    there are utilities and suchlike if you've got a TCP address. I don't.
    
    The entire file is 1089 blocks, 500+ kbytes so is a bit too large to
    post here. Plus it changes monthly.
    
    FYI, the smallest prime listed was proven prime in 1990:
    
    #9753  	(212+10^500)*10^499+1
    
      John

Hey, did any of you notice from the report:

prime                     digits who when what

10^11810+1465641*10^5902+1 11811  D   94  Largest Palindrome
10^11010+3242423*10^5502+1 11011  D   94  Next Palindrome


The number of *digits* in the palindromic primes are *also* palindromes.
Is there a theorem lurking here or is this just amazing kowinky dinky
(conincidence) ?

/Eric

    A quick way to see if there's a theorem lurking here is to express the
    numbers in a different base and see if the number of digits is also a
    palindrome in the new base.
    
      John

If the palindromic prime converted into hex is no longer palindromic, then
we have no information about whether a theorem is lurking.

> If the palindromic prime converted into hex is no longer palindromic, then
> we have no information about whether a theorem is lurking.
    
    Not true.
    
    Nearly every great theorem in arithmetic is independent of the base 
    of the number system. Unique factorization is true regardless of your
    choice of bases: decimal, hex, binary, whatever.
    
    So when you observe something which is true only in decimal, it's not
    terribly interesting because it only applies to a notation-specific
    representation of numbers. There is no isomorphism to the integers.
    
    Now if you think you have something independent of representation, then
    you've got a theorem lurking. If not, no theorem.
    
      John

    Re .23:
    
    I think the theorem to be considered is:
    
    	Are the base B representations of the numbers of digits in
    	the base B representations of the numbers in the set S
    	palindromic, where S is the set of primes that are palindromic
    	in base B or some "interesting" subset thereof?
    
    As Eric points out, no contradiction to this theorem is evidenced by a
    decimal-palindromic prime that is not hexadecimal-palindromic.
    
    
    				-- edp
    
    
Public key fingerprint:  8e ad 63 61 ba 0c 26 86  32 0a 7d 28 db e7 6f 75
To find PGP, read note 2688.4 in Humane::IBMPC_Shareware.
    

    re .23
    
    Or to look at it slightly differently...
    
    While a theorem is inherently more interesting if it is independent of
    base, since the entire concept of a palindrome relies on some assumed
    base (a number that is a palindrome in one base is not generally a
    palindrome in other bases), it is reasonable to allow the possibility
    that some theorem about palindromes applies only in one or more
    specific bases.
    
    As for the original assertion that using hexadecimal disproves the
    existence of a theorem, that depends entirely on what Eric's implied
    theorem was. If the theorem was intended to apply only in base 10 or is
    generalised as stated in .24 then converting the listed primes to
    hexadecimal and finding them not even to be palindromic does not disprove
    the theorem.
    
    One thing to bear in mind when looking at the list of large palindromic
    primes is that they may not be representative i.e. may reflect
    currently available techniques for proving a number prime. Certain
    "forms" are more amenable to testing.
    

Some Record Primes
----------------------    ------ --- ---- -------------------------
prime                     digits who when what
----------------------    ------ --- ---- -------------------------
2^859433-1                258716  SG  94  Largest Prime (a Mersenne)
2^756839-1                227832  SG  92  Next Largest (also Mersenne)
391581*2^216193-1          65087  Z   89  Largest Non-Mersenne
10^11810+1465641*10^5902+1 11811  D   94  Largest Palindrome
3610!-1                    11277  C   93  Largest Factorial minus one
10^11010+3242423*10^5502+1 11011  D   94  Next Palindrome
24029#+1                   10387  C   93  Largest Primorial Plus One
5415312903*10^4526-1        4536  D   94  Largest Sophie Germain
1692923232*10^4020+/-1      4030  D   93  Largest Known Twin Primes
4655478828*10^3429+/-1      3439  D   93  Next Twin Primes
----------------------    ------ --- ---- -------------------------

I was noticing how all of these famous prime numbers are representable in
full accurace in 26 columns !

How "fortunate" we are that the largest prime, for example, isn't

	nnnn...nnnn * mmmmmm...mmmmm + 1

in other words the product of 2 115000-digit primes plus 1.

It's almost as though the algorithm for searching for these primes was chosen
so as to find primes whose representation fits the requirements of press
releases !

/Eric

   Special tricks are always used for primes of the forms shown
   that don't apply for ones that are near to the product of two
   "random" huge prime factors.

   - Jim