Catalan_s Constant to 1_500_000 Places 321 downloads.pdf

The Project Gutenberg eBook of Catalan's Constant to
1,500,000 Places

This eBook is for the use of anyone anywhere in the United States and most
other parts of the world at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms of
the Project Gutenberg License included with this eBook or online at
www.gutenberg.org. If you are not located in the United States, you will
have to check the laws of the country where you are located before using
this eBook.

Title: Catalan's Constant to 1,500,000 Places

Release date: March 1, 1997 [eBook #812]
Most recently updated: April 14, 2013

Language: English

Other information and formats: www.gutenberg.org/ebooks/812

Credits: Produced by Thomas Papanikolaou

*** START OF THE PROJECT GUTENBERG EBOOK CATALAN'S
CONSTANT TO 1,500,000 PLACES ***

Produced by Thomas Papanikolaou

See also: Catalan's Constant [Ramanujan's] PG#682

The algorithm used is the one presented
by Greg Fee on ISSAC '90 (using only integer arithmetic in the main loop).
Again, I have verified the previous 1000100-digit value using
LiDIA and text comparison.

Best
Thomas Papanikolaou

Catalan constant to 1500000 digits computed on October 26, 1996 by
using a Sun Sparc20 in 7 day 7 hour 14 min 52 sec 71 hsec The algorithm
used is the standard series for Catalan, accelerated by an Euler transform as
shown by

Greg Fee, ACM 1990, Proceedings of the ISAAC conference, 1990, p.
157

The algorithm was implemented using the LiDIA library for
computational number theory and it will be part of the multiprecision
floating-point arithmetic of the package in release 1.4. LiDIA is available
from

ftp://crypt1.cs.uni-sb.de/pub/systems/LiDIA/LiDIA-1.2.1.tgz
http://www-jb.cs.uni-sb.de/LiDIA/linkhtml/lidia/lidia.html

Here is the output of the program:

Calculating Catalan's constant to 1500000 decimals

Time required: 7 day 7 hour 14 min 52 sec 71 hsec

*** END OF THE PROJECT GUTENBERG EBOOK CATALAN'S
CONSTANT TO 1,500,000 PLACES ***

Updated editions will replace the previous one—the old editions will be
renamed.

Creating the works from print editions not protected by U.S. copyright law
means that no one owns a United States copyright in these works, so the
Foundation (and you!) can copy and distribute it in the United States
without permission and without paying copyright royalties. Special rules,
set forth in the General Terms of Use part of this license, apply to copying
and distributing Project Gutenberg™ electronic works to protect the
PROJECT GUTENBERG™ concept and trademark. Project Gutenberg is a
registered trademark, and may not be used if you charge for an eBook,
except by following the terms of the trademark license, including paying
royalties for use of the Project Gutenberg trademark. If you do not charge
anything for copies of this eBook, complying with the trademark license is
very easy. You may use this eBook for nearly any purpose such as creation
of derivative works, reports, performances and research. Project Gutenberg
eBooks may be modified and printed and given away—you may do
practically ANYTHING in the United States with eBooks not protected by
U.S. copyright law. Redistribution is subject to the trademark license,
especially commercial redistribution.

START: FULL LICENSE

THE FULL PROJECT GUTENBERG™ LICENSE
PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK

To protect the Project Gutenberg™ mission of promoting the free
distribution of electronic works, by using or distributing this work (or any
other work associated in any way with the phrase “Project Gutenberg”), you
agree to comply with all the terms of the Full Project Gutenberg License
available with this file or online at www.gutenberg.org/license.

Section 1. General Terms of Use and Redistributing
Project Gutenberg electronic works

1.A. By reading or using any part of this Project Gutenberg electronic work,
you indicate that you have read, understand, agree to and accept all the
terms of this license and intellectual property (trademark/copyright)
agreement. If you do not agree to abide by all the terms of this agreement,
you must cease using and return or destroy all copies of Project Gutenberg
electronic works in your possession. If you paid a fee for obtaining a copy
of or access to a Project Gutenberg electronic work and you do not agree to
be bound by the terms of this agreement, you may obtain a refund from the
person or entity to whom you paid the fee as set forth in paragraph 1.E.8.

1.B. “Project Gutenberg” is a registered trademark. It may only be used on
or associated in any way with an electronic work by people who agree to be
bound by the terms of this agreement. There are a few things that you can
do with most Project Gutenberg electronic works even without complying
with the full terms of this agreement. See paragraph 1.C below. There are a
lot of things you can do with Project Gutenberg electronic works if you
follow the terms of this agreement and help preserve free future access to
Project Gutenberg electronic works. See paragraph 1.E below.

1.C. The Project Gutenberg Literary Archive Foundation (“the Foundation”
or PGLAF), owns a compilation copyright in the collection of Project
Gutenberg electronic works. Nearly all the individual works in the
collection are in the public domain in the United States. If an individual
work is unprotected by copyright law in the United States and you are

located in the United States, we do not claim a right to prevent you from
copying, distributing, performing, displaying or creating derivative works
based on the work as long as all references to Project Gutenberg are
removed. Of course, we hope that you will support the Project Gutenberg
mission of promoting free access to electronic works by freely sharing
Project Gutenberg works in compliance with the terms of this agreement for
keeping the Project Gutenberg name associated with the work. You can
easily comply with the terms of this agreement by keeping this work in the
same format with its attached full Project Gutenberg License when you
share it without charge with others.

1.D. The copyright laws of the place where you are located also govern
what you can do with this work. Copyright laws in most countries are in a
constant state of change. If you are outside the United States, check the
laws of your country in addition to the terms of this agreement before
downloading, copying, displaying, performing, distributing or creating
derivative works based on this work or any other Project Gutenberg work.
The Foundation makes no representations concerning the copyright status
of any work in any country other than the United States.

1.E. Unless you have removed all references to Project Gutenberg:

1.E.1. The following sentence, with active links to, or other immediate
access to, the full Project Gutenberg License must appear prominently
whenever any copy of a Project Gutenberg work (any work on which the
phrase “Project Gutenberg” appears, or with which the phrase “Project
Gutenberg” is associated) is accessed, displayed, performed, viewed, copied
or distributed:

This eBook is for the use of anyone anywhere in the United States
and most other parts of the world at no cost and with almost no
restrictions whatsoever. You may copy it, give it away or re-use it
under the terms of the Project Gutenberg™ License included with
this eBook or online at www.gutenberg.org. If you are not located in
the United States, you will have to check the laws of the country
where you are located before using this eBook.

1.E.2. If an individual Project Gutenberg electronic work is derived from
texts not protected by U.S. copyright law (does not contain a notice
indicating that it is posted with permission of the copyright holder), the
work can be copied and distributed to anyone in the United States without
paying any fees or charges. If you are redistributing or providing access to a
work with the phrase “Project Gutenberg” associated with or appearing on
the work, you must comply either with the requirements of paragraphs
1.E.1 through 1.E.7 or obtain permission for the use of the work and the
Project Gutenberg trademark as set forth in paragraphs 1.E.8 or 1.E.9.

1.E.3. If an individual Project Gutenberg electronic work is posted with the
permission of the copyright holder, your use and distribution must comply
with both paragraphs 1.E.1 through 1.E.7 and any additional terms imposed
by the copyright holder. Additional terms will be linked to the Project
Gutenberg License for all works posted with the permission of the
copyright holder found at the beginning of this work.

1.E.4. Do not unlink or detach or remove the full Project Gutenberg License
terms from this work, or any files containing a part of this work or any
other work associated with Project Gutenberg.

1.E.5. Do not copy, display, perform, distribute or redistribute this
electronic work, or any part of this electronic work, without prominently
displaying the sentence set forth in paragraph 1.E.1 with active links or
immediate access to the full terms of the Project Gutenberg License.

1.E.6. You may convert to and distribute this work in any binary,
compressed, marked up, nonproprietary or proprietary form, including any
word processing or hypertext form. However, if you provide access to or
distribute copies of a Project Gutenberg work in a format other than “Plain
Vanilla ASCII” or other format used in the official version posted on the
official Project Gutenberg website (www.gutenberg.org), you must, at no
additional cost, fee or expense to the user, provide a copy, a means of
exporting a copy, or a means of obtaining a copy upon request, of the work
in its original “Plain Vanilla ASCII” or other form. Any alternate format
must include the full Project Gutenberg License as specified in paragraph
1.E.1.

1.E.7. Do not charge a fee for access to, viewing, displaying, performing,
copying or distributing any Project Gutenberg works unless you comply
with paragraph 1.E.8 or 1.E.9.

1.E.8. You may charge a reasonable fee for copies of or providing access to
or distributing Project Gutenberg electronic works provided that:

• You pay a royalty fee of 20% of the gross profits you derive from the
use of Project Gutenberg works calculated using the method you
already use to calculate your applicable taxes. The fee is owed to the
owner of the Project Gutenberg trademark, but he has agreed to donate
royalties under this paragraph to the Project Gutenberg Literary
Archive Foundation. Royalty payments must be paid within 60 days
following each date on which you prepare (or are legally required to
prepare) your periodic tax returns. Royalty payments should be clearly
marked as such and sent to the Project Gutenberg Literary Archive
Foundation at the address specified in Section 4, “Information about
donations to the Project Gutenberg Literary Archive Foundation.”

• You provide a full refund of any money paid by a user who notifies
you in writing (or by e-mail) within 30 days of receipt that s/he does
not agree to the terms of the full Project Gutenberg™ License. You
must require such a user to return or destroy all copies of the works
possessed in a physical medium and discontinue all use of and all
access to other copies of Project Gutenberg™ works.

• You provide, in accordance with paragraph 1.F.3, a full refund of any
money paid for a work or a replacement copy, if a defect in the
electronic work is discovered and reported to you within 90 days of
receipt of the work.

• You comply with all other terms of this agreement for free distribution
of Project Gutenberg™ works.

1.E.9. If you wish to charge a fee or distribute a Project Gutenberg™
electronic work or group of works on different terms than are set forth in
this agreement, you must obtain permission in writing from the Project

Gutenberg Literary Archive Foundation, the manager of the Project
Gutenberg™ trademark. Contact the Foundation as set forth in Section 3
below.

1.F.

1.F.1. Project Gutenberg volunteers and employees expend considerable
effort to identify, do copyright research on, transcribe and proofread works
not protected by U.S. copyright law in creating the Project Gutenberg™
collection. Despite these efforts, Project Gutenberg™ electronic works, and
the medium on which they may be stored, may contain “Defects,” such as,
but not limited to, incomplete, inaccurate or corrupt data, transcription
errors, a copyright or other intellectual property infringement, a defective or
damaged disk or other medium, a computer virus, or computer codes that
damage or cannot be read by your equipment.

1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except
for the “Right of Replacement or Refund” described in paragraph 1.F.3, the
Project Gutenberg Literary Archive Foundation, the owner of the Project
Gutenberg™ trademark, and any other party distributing a Project
Gutenberg™ electronic work under this agreement, disclaim all liability to
you for damages, costs and expenses, including legal fees. YOU AGREE
THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT
LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT
EXCEPT THOSE PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE
THAT THE FOUNDATION, THE TRADEMARK OWNER, AND ANY
DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE LIABLE
TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL,
PUNITIVE OR INCIDENTAL DAMAGES EVEN IF YOU GIVE
NOTICE OF THE POSSIBILITY OF SUCH DAMAGE.

1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you
discover a defect in this electronic work within 90 days of receiving it, you
can receive a refund of the money (if any) you paid for it by sending a
written explanation to the person you received the work from. If you
received the work on a physical medium, you must return the medium with
your written explanation. The person or entity that provided you with the
defective work may elect to provide a replacement copy in lieu of a refund.

If you received the work electronically, the person or entity providing it to
you may choose to give you a second opportunity to receive the work
electronically in lieu of a refund. If the second copy is also defective, you
may demand a refund in writing without further opportunities to fix the
problem.

1.F.4. Except for the limited right of replacement or refund set forth in
paragraph 1.F.3, this work is provided to you ‘AS-IS’, WITH NO OTHER
WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING
BUT NOT LIMITED TO WARRANTIES OF MERCHANTABILITY OR
FITNESS FOR ANY PURPOSE.

1.F.5. Some states do not allow disclaimers of certain implied warranties or
the exclusion or limitation of certain types of damages. If any disclaimer or
limitation set forth in this agreement violates the law of the state applicable
to this agreement, the agreement shall be interpreted to make the maximum
disclaimer or limitation permitted by the applicable state law. The invalidity
or unenforceability of any provision of this agreement shall not void the
remaining provisions.

1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the
trademark owner, any agent or employee of the Foundation, anyone
providing copies of Project Gutenberg™ electronic works in accordance
with this agreement, and any volunteers associated with the production,
promotion and distribution of Project Gutenberg™ electronic works,
harmless from all liability, costs and expenses, including legal fees, that
arise directly or indirectly from any of the following which you do or cause
to occur: (a) distribution of this or any Project Gutenberg work, (b)
alteration, modification, or additions or deletions to any Project Gutenberg
work, and (c) any Defect you cause.

Section 2. Information about the Mission of Project
Gutenberg

Project Gutenberg is synonymous with the free distribution of electronic
works in formats readable by the widest variety of computers including
obsolete, old, middle-aged and new computers. It exists because of the

efforts of hundreds of volunteers and donations from people in all walks of
life.

Volunteers and financial support to provide volunteers with the assistance
they need are critical to reaching Project Gutenberg’s goals and ensuring
that the Project Gutenberg collection will remain freely available for
generations to come. In 2001, the Project Gutenberg Literary Archive
Foundation was created to provide a secure and permanent future for
Project Gutenberg and future generations. To learn more about the Project
Gutenberg Literary Archive Foundation and how your efforts and donations
can help, see Sections 3 and 4 and the Foundation information page at
www.gutenberg.org.

Section 3. Information about the Project Gutenberg
Literary Archive Foundation

The Project Gutenberg Literary Archive Foundation is a non-profit 501(c)
(3) educational corporation organized under the laws of the state of
Mississippi and granted tax exempt status by the Internal Revenue Service.
The Foundation’s EIN or federal tax identification number is 64-6221541.
Contributions to the Project Gutenberg Literary Archive Foundation are tax
deductible to the full extent permitted by U.S. federal laws and your state’s
laws.

The Foundation’s business office is located at 41 Watchung Plaza #516,
Montclair NJ 07042, USA, +1 (862) 621-9288. Email contact links and up
to date contact information can be found at the Foundation’s website and
official page at www.gutenberg.org/contact

Section 4. Information about Donations to the Project
Gutenberg Literary Archive Foundation

Project Gutenberg™ depends upon and cannot survive without widespread
public support and donations to carry out its mission of increasing the
number of public domain and licensed works that can be freely distributed
in machine-readable form accessible by the widest array of equipment

including outdated equipment. Many small donations ($1 to $5,000) are
particularly important to maintaining tax exempt status with the IRS.

The Foundation is committed to complying with the laws regulating
charities and charitable donations in all 50 states of the United States.
Compliance requirements are not uniform and it takes a considerable effort,
much paperwork and many fees to meet and keep up with these
requirements. We do not solicit donations in locations where we have not
received written confirmation of compliance. To SEND DONATIONS or
determine the status of compliance for any particular state visit
www.gutenberg.org/donate.

While we cannot and do not solicit contributions from states where we have
not met the solicitation requirements, we know of no prohibition against
accepting unsolicited donations from donors in such states who approach us
with offers to donate.

International donations are gratefully accepted, but we cannot make any
statements concerning tax treatment of donations received from outside the
United States. U.S. laws alone swamp our small staff.

Please check the Project Gutenberg web pages for current donation methods
and addresses. Donations are accepted in a number of other ways including
checks, online payments and credit card donations. To donate, please visit:
www.gutenberg.org/donate.

Section 5. General Information About Project Gutenberg
electronic works

Professor Michael S. Hart was the originator of the Project Gutenberg
concept of a library of electronic works that could be freely shared with
anyone. For forty years, he produced and distributed Project Gutenberg
eBooks with only a loose network of volunteer support.

Project Gutenberg eBooks are often created from several printed editions,
all of which are confirmed as not protected by copyright in the U.S. unless a

copyright notice is included. Thus, we do not necessarily keep eBooks in
compliance with any particular paper edition.

Most people start at our website which has the main PG search facility:
www.gutenberg.org.

This website includes information about Project Gutenberg, including how
to make donations to the Project Gutenberg Literary Archive Foundation,
how to help produce our new eBooks, and how to subscribe to our email
newsletter to hear about new eBooks.