Discussion:
Is 2^(2^13466918-4)*(2^13466917-1) a big number?
mountain man
2007-09-18 10:25:02 UTC
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It has been conjectured (April 2004) that the largest known balanced*
number is the number: 2^(2^13466918-4)*(2^13466917-1).

This is a number with about 10^4053946 digits,
(not 4053946 digits but 10^4053946 digits!!!),
but perfectly balanced, having precisely
2^(2^13466918-4)*(2^13466917-1) ordered factorizations.

* BALANCED Number?

The ancient notion of aliquot type historically (Euclid) defined by
summation of the component aliquots, and notes the three separate threads of
number so formalised. Abundant numbers have the sum of their divisors
greater than themselves, deficient numbers have the sum of their divisors
less than themselves. In the middle, with the sum of their divisors exactly
equal to the number are the "Perfect Numbers".

A redefinition in terms of the number of ordered factorisations (divisors)
for any given number (rather than the summation of the divisors) is
investigated.
That is, we define the following categories in regard to the number of
ordered factorizations.
Let the number of ordered factors of n be H(n), then:

a.. Deficient Ordered Factorizations ... H(n) < n
Note that this set includes the primes for which by definition H(n) = 1.

b.. Balanced Ordered Factorizations ... H(n) = n
The term balanced was used in preference to "perfect".
The first six balanced numbers are: 1, 48, 1280, 2496, 28672, 29808.
These numbers are scant and reclusive like their cousins the Perfect
Numbers. In the following analyses these are classed as abundant.

c.. Abundant Ordered Factorizations ... H(n) > n
The abundant numbers are those numbers for which
the number of ordered factors is greater than the number itself.

http://www.mountainman.com.au/harmonics_01.htm

Best wishes,

P.R.F. Brown
m***@hotmail.com.CUT
2007-09-18 20:53:12 UTC
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Define what you mean by "big". That's not a standard math term!!
mountain man
2007-09-19 00:58:51 UTC
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Post by m***@hotmail.com.CUT
Define what you mean by "big". That's not a standard math term!!
Good question!
But that is exactly the purpose of the post.

I am wondering what types of BIG NUMBERS
are being examined in the fields of mathematics
and how this one fits in the line-up.

Can anyone assist?
Ta.

Pete
Dan in NY
2007-09-19 12:48:46 UTC
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Post by mountain man
Post by m***@hotmail.com.CUT
Define what you mean by "big". That's not a standard math term!!
Good question!
But that is exactly the purpose of the post.
I am wondering what types of BIG NUMBERS
are being examined in the fields of mathematics
and how this one fits in the line-up.
Can anyone assist?
Ta. Pete

I have seen this question in several newsgroups so I have cross-posted this
reply to three others. In the group of web sites called the "Caldwell Prime
Pages" I found a Glossary with some terms used to categorize prime numbers
as big. I suppose you could use the same terms to categorize all numbers as
has been done for primes.

The web site, http://primes.utm.edu/glossary/page.php?sort=Megaprime, has
these meanings listed (I have paraphrased):

a {titanic prime} has 1000 or more digits
a {gigantic prime} has 10,000 or more digits
a {megaprime} has 1,000,000 or more digits

I suggest that what is meant by a "big" number depends on your context. For
any particular purpose, to define a "big" number, first decide what is the
{smallest "big" number}. A "big" number is then defined to be any number
that big or larger. As suggested by the web site, most numbers are big.
Note that most numbers are bigger than any number that is named.

The symbols 2^(2^13466918 -4) * (2^13466917 -1) can be said to name a
number. Most numbers are bigger than this number. To decide whether
2^(2^13466918 -4) * (2^13466917 -1) is a {big number}, any number you choose
could be defined as the smallest {big number}.
--
Dan in NY
(for email, exchange y with g in
dKlinkenbery at hvc dot rr dot com)
Jens Kruse Andersen
2007-09-21 15:14:15 UTC
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Post by mountain man
It has been conjectured (April 2004) that the largest known balanced*
number is the number: 2^(2^13466918-4)*(2^13466917-1).
perfectly balanced, having precisely 2^(2^13466918-4)*(2^13466917-1)
ordered factorizations.
If p is an odd prime then 2^(2p-2)*p is balanced.
The record history for the largest known prime is at
http://primes.utm.edu/notes/by_year.html

The 6 latest records and still largest known:
Prime Digits Month Year
2^13466917-1 4053946 November 2001
2^20996011-1 6320430 November 2003
2^24036583-1 7235733 May 2004
2^25964951-1 7816230 February 2005
2^30402457-1 9152052 December 2005
2^32582657-1 9808358 September 2006
All are Mersenne primes found by GIMPS.

The "conjectured (April 2004)" balanced record was beaten for the same
form in November 2003, and 4 more times since then. I don't know
whether larger balanced numbers are known for other forms of numbers.
Post by mountain man
I am wondering what types of BIG NUMBERS are being examined in the fields
of mathematics and how this one fits in the line-up.
"Big number" is not mathematically defined but without special context I
would call 2^(2^13466918-4)*(2^13466917-1) big.
http://en.wikipedia.org/wiki/Large_numbers mentions some numbers most
people would probably call big.
Post by mountain man
The web site, http://primes.utm.edu/glossary/page.php?sort=Megaprime, has
a {titanic prime} has 1000 or more digits
a {gigantic prime} has 10,000 or more digits
a {megaprime} has 1,000,000 or more digits
I suggest that what is meant by a "big" number depends on your context.
The terms titanic prime and gigantic prime were coined at a time where
such primes were hard to find and few were known:
http://primes.utm.edu/glossary/page.php?sort=TitanicPrime
http://primes.utm.edu/glossary/page.php?sort=GiganticPrime

Today a titanic prime can be found in seconds on a normal PC and a
gigantic prime in less than an hour. The names for those sizes would
probably
not have been chosen today - at least not by the person who made them.

"Megaprime" for 1,000,000 digits is of course because mega means
1,000,000 and not just because "mega" is sometimes used about big things.
--
Jens Kruse Andersen
m***@aol.com
2007-09-21 17:11:57 UTC
Raw Message
Post by Jens Kruse Andersen
Post by mountain man
It has been conjectured (April 2004) that the largest known balanced*
number is the number: 2^(2^13466918-4)*(2^13466917-1).
perfectly balanced, having precisely 2^(2^13466918-4)*(2^13466917-1)
ordered factorizations.
If p is an odd prime then 2^(2p-2)*p is balanced.
The record history for the largest known prime is athttp://primes.utm.edu/notes/by_year.html
Prime Digits Month Year
2^13466917-1 4053946 November 2001
2^20996011-1 6320430 November 2003
2^24036583-1 7235733 May 2004
2^25964951-1 7816230 February 2005
2^30402457-1 9152052 December 2005
2^32582657-1 9808358 September 2006
All are Mersenne primes found by GIMPS.
The "conjectured (April 2004)" balanced record was beaten for the same
form in November 2003, and 4 more times since then. I don't know
whether larger balanced numbers are known for other forms of numbers.
Post by mountain man
I am wondering what types of BIG NUMBERS are being examined in the fields
of mathematics and how this one fits in the line-up.
"Big number" is not mathematically defined but without special context I
would call 2^(2^13466918-4)*(2^13466917-1) big.http://en.wikipedia.org/wiki/Large_numbersmentions some numbers most
people would probably call big.
Post by mountain man
The web site,http://primes.utm.edu/glossary/page.php?sort=Megaprime, has
a {titanic prime} has 1000 or more digits
a {gigantic prime} has 10,000 or more digits
a {megaprime} has 1,000,000 or more digits
I suggest that what is meant by a "big" number depends on your context.
The terms titanic prime and gigantic prime were coined at a time where
such primes were hard to find and few were known:http://primes.utm.edu/glossary/page.php?sort=TitanicPrimehttp://primes.utm.edu/glossary/page.php?sort=GiganticPrime
Today a titanic prime can be found in seconds on a normal PC and a
gigantic prime in less than an hour. The names for those sizes would
probably
not have been chosen today - at least not by the person who made them.
"Megaprime" for 1,000,000 digits is of course because mega means
1,000,000 and not just because "mega" is sometimes used about big things.
So, if named today, titanic prime would be kiloprime and
gigantic prime would be...decakiloprime?
Post by Jens Kruse Andersen
--
Jens Kruse Andersen
mountain man
2007-09-29 21:17:21 UTC
Raw Message
Thanks Dan et al,
Post by Dan in NY
Post by mountain man
Post by m***@hotmail.com.CUT
Define what you mean by "big". That's not a standard math term!!
Good question!
But that is exactly the purpose of the post.
I am wondering what types of BIG NUMBERS
are being examined in the fields of mathematics
and how this one fits in the line-up.
Can anyone assist?
Ta. Pete
I have seen this question in several newsgroups so I have cross-posted
this reply to three others. In the group of web sites called the
"Caldwell Prime Pages" I found a Glossary with some terms used to
categorize prime numbers as big. I suppose you could use the same terms
to categorize all numbers as has been done for primes.
The web site, http://primes.utm.edu/glossary/page.php?sort=Megaprime, has
a {titanic prime} has 1000 or more digits
a {gigantic prime} has 10,000 or more digits
a {megaprime} has 1,000,000 or more digits
I suggest that what is meant by a "big" number depends on your context.
For any particular purpose, to define a "big" number, first decide what is
the {smallest "big" number}. A "big" number is then defined to be any
number that big or larger. As suggested by the web site, most numbers are
big. Note that most numbers are bigger than any number that is named.
The symbols 2^(2^13466918 -4) * (2^13466917 -1) can be said to name a
number. Most numbers are bigger than this number. To decide whether
2^(2^13466918 -4) * (2^13466917 -1) is a {big number}, any number you
choose could be defined as the smallest {big number}.
The number: 2^(2^13466918-4)*(2^13466917-1)
has about 10^4053946 digits. That's not four million
digits (as per the megaprime example above) but
10^4053946 digits!!!). That's huge.

And that's why I was wondering how this type of
number stacks up to the "world's largest numbers"
however they might like to be defined.

Thanks and best wishes,

Pete Brown

It is perfectly balanced,
having precisely 2^(2^13466918-4)*(2^13466917-1)
ordered factorizations.