Fermat is a computer algebra system for Macintosh, Windows, Linux, and Unix by me, Robert H. Lewis of Fordham University, that does arithmetic of arbitrarily long integers and fractions, symbolic calculations, matrices over polynomial rings, graphics, and other numerical calculations. It is extremely fast and extremely economical of space. The main version that I care most about is oriented toward polynomial and matrix algebra over the rationals Q and finite fields. On the Mac side, there is a version for OS X and versions for OS 9. There is also a "float" version for graphics. All versions are available here.


Fermat is a state-of-the-art research tool for real problems.

See for example:

Compare Your Computer Algebra System. Take the Fermat Tests!

There are now four tests. One test involves evaluation of rational functions, the second involves Smith Normal Form, the third resultants, the fourth rational function arithmetic.

Fermat is shareware, $US 60. The only exception is mathematicians earning less than $US 45000 per year. Institutions must pay $US 60 per machine.


Fermat is Overall Best in the World at Polynomial GCD

According to two independent researchers.

Mac OS X Intel Version of Fermat Now Available. 3.9.7, revised May 6, 2008

Mac OS X G3-G5 Version of Fermat Now Available. 3.9.7, revised May 6, 2008

Linux Version of Fermat Now Available. 3.9.7, revised May 6, 2008

   Unix Version of Fermat Now Available. 3.6.9, revised December 15, 2005

   Windows Version of Fermat Now Available. 3.6.9, revised October 29, 2007

   Mac OS 9 Version of Fermat Now Available. 3.7.0, revised October 29, 2007

   Fermat Very Impressive in Comparison of Computer Algebra Systems

Look at this preprint of a comparison of small polynomial-oriented computer algebra systems. This paper appeared in the SIGSAM Bulletin in January 2000.

   HTML and PDF Versions of the Manuals

There are revised (June 10, 2005) HTML versions of the Fermat manual, for Windows, U/Linux, and Mac, and of the FFermat manual (FFermat, the float version, is Mac only). In addition, there are revised ps and pdf versions of the manuals. See below.

Version 3.9.7 is here, May 6, 2008

3.9.7 Fixed two old bugs and refined the multivariate polynomial gcd algorithm.

3.9.2 New transpose function and new option in determinant command.

3.8.8 improved the arithmetic of quolynomials (rational functions).

3.8.7 for Mac OS X and Linux: (1) several speed improvements in multivariate polynomial gcd. (2) Cosmetic improvement of not listing hundreds of identical names on the terminal after an &v command.


3.8.4 for Linux and OSX has some user convenience improvements and a bug fix. The bug did not occur in the Windows version.

3.8.1 (March 2007) has a few small new features an a bug fix (OS X and Linux). The bug fix was also done in the Windows version.

3.7.6 - 3.7.8 (OSX and Linux) have various speedups and bug fixes. The Windows version has also been updated.

3.7.4 (OSX and Linux) has further refinements to polynomial gcd for very sparse polys. 3.7.3, here very briefly in early March, was a half way step. 3.7.0 - 3.7.2 have various bug fixes for polynomial g.c.d., and new features, as did 3.6.9. 3.6.0 contains more speed improvements. 3.6.1 has a bux fix. 3.6.3 is available for OSX and Linux (Intel); it has speed improvements and bug fixes. 3.6.4, in all versions but Unix, fixes a small memory leak.

3.5.5 and 3.5.8 had small feature improvements, but this release was mainly a speed improvement in polynomial gcd. See the Fermat Tests page.

The major improvement in 3.3 was a significant gain in the speed of one variable polynomial arithmetic and GCD. Some computations increase in speed 60%. 3.4.7 implements ground ring Z/p for primes p up to 2^31 - 1 (in Windows the upper limit is a bit less). 3.4.7 implements efficiently builtin finite fields GF(2^8) and GF(2^16), and special feature LaGrange interpolation for determinant. 3.4.8 is a minor improvement, but extends all of 3.4.7 to Windows, Linux, Mac OS X and OS 9. More details are here.


Fermat FAQ frequently asked questions.




Why should I use this system?

Because it works. Fermat has proven to be extremely good at what it does. If you tackle real problems with computer algebra you have probably found that some well known systems are too slow, use too much space, crash too often, or have weird limitations built into them. If all you do is make up examples for your undergraduate students, you probably don't need Fermat. If you try to compute the characteristic polynomial of a 400 X 400 matrix over Q, you need Fermat or something like it. Time and again, Fermat has bested the well known expensive systems in both time and space, often by enormous, almost unbelievable, ratios. Look at this paper that later appeared in the SIGSAM Bulletin. Look at what these independent researchers have to say.

Fermat is especially good at polynomial and rational function arithmetic; Smith normal form; determinant, normal forms, and inverse of matrices with multivariate polynomial entries over Z, Q, Zp, finite fields, or more complex fields; sparse matrices; characteristic polynomials; and gcd of multivariate polynomials over Z, Zp, or finite fields. In addition, I have striven to make it easy to use. For example, in the Mac version it is very easy to edit the output of Fermat (on the screen) and make it the input. This is a great boon in experiments with matrices. Extensive facilities exist for saving data to files and reading such data. Fermat has the ability to be interrupted and then later return to the computation, picking up where it left off.

To retrieve the two ReadMe files click here and here.

The ReadMes contain the latest update information, especially the second one. The first one has some history. The second is about the Mac, Windows, Linux, and Unix versions.


Fermat Features Summary

To get a quick idea of the polynomial and matrix features of Fermat (the rational version), look at this html version of Appendix 4 of the Windows manual. If you viewing on a Mac, you may need to go to the "View" menu of your browser, then under "Character Set" set it to "Western(Mac Roman)."



Fermat in Actual Use

Here is a documented collection of functions that implement the Dixon-EDF method for solving systems of polynomial equations in Fermat.


Some research projects that used Fermat -- and needed it:

Bela Palancz, Robert H. Lewis, Piroska Zaletnyik, and Joseph Awange, "Computational Study of the 3D Affine Transformation Part I. 3-point Problem."
First version online at the Mathematica website, revised version available here. March 2008.


S. Bekavac, D. Seidel, "On-shell renormalisation constants including two different nonzero masses." October 2007. http://aps.arxiv.org/list/hep-ph/0710


Nazar Khan, "Silhouette-Based 2D-3D Pose Estimation Using Implicit Algebraic Surfaces", Master Thesis, Saarland University, Germany, 2007. advisor: Bodo Rosenhahn. download pdf version.


Lewis, Robert H. and E. A. Coutsias, "Algorithmic Search for Flexibility using Resultants of Polynomial Systems." to appear, Proceedings of the ADG 2006 Conference (Automatic Deduction in Geometry), Pontevedra Spain, August 31 - September 2, 2006. download pdf version.


Bozoki, Sandor and Robert H. Lewis, "Solving the Least Squares Method Problem in the AHP for 3 x 3 and 4 x 4 Matrices," Central European Journal for Operations Research, September 2005. download pdf version.


K.G. Chetyrkin, M. Faisst, C. Sturm, and M. Tentyukov. e-Finite Basis of Master Integrals for the Integration-By-Parts Method. January 2006. 28pp. http://arxiv.org/pdf/hep-ph/0601165


M. Czakon. "The Four-loop QCD Beta-Function and Anomalous Dimensions." DESY-04-223, SFB-CPP-04-62, Nov 2004. 14pp. Published in Nucl.Phys.B710:485-498, 2005. http://arxiv.org/pdf/hep-ph/0411261


M. Czakon, J. Gluza, T. Riemann. "Master Integrals for Massive Two-Loop BHABHA Scattering in QED." DESY-04-222, SFB-CPP-04-61, Dec 2004. 21pp. Published in Phys.Rev.D71:073009, 2005. http://arxiv.org/pdf/hep-ph/0412164


Little, John. "Solving the Selesnick-Burrus Filter Design Equations Using Computational Algebra and Algebraic Geometry", Advances in Applied Mathematics, 31 (2003), p. 463-500.


Brumer, Armand. "The Rank of Jo(N)," Asterisque 228 (1995) p. 41-68.


Yuen, David S. Siegel modular cusp forms (2004). personal communication.


Lewis, Robert H. and Stephen Bridgett, "Conic Tangency Equations and Apollonius Problems in Biochemistry and Pharmacology," Mathematics and Computers in Simulation 61(2) (2003) p. 101-114. download pdf version.


Lewis, Robert H. and Peter F. Stiller, "Solving the recognition problem for six lines using the Dixon resultant," Mathematics and Computers in Simulation 49 (1999) p. 205-219. download pdf version.


Lewis, Robert H. and George Nakos, "Solving the Six Line Problem with Resultants," presented to the "Grand Challenges" session at IMACS, Prague, August 1998.


Lewis, Robert H. "The Six Line Problem and Resultants," presented to the "Grand Challenges" session at IMACS, Hawaii, July 1997.


Lewis, Robert H. and Guy D. Moore. "Computer Search for Nilpotent Complexes," Experimental Mathematics 6:3 (1997) p. 239-246.


Lewis, Robert H. and Sal Liriano. "Isomorphism Classes and Derived Series of Almost-Free Groups," Experimental Mathematics 3 (1994) p.255-258.



The Manual

HTML versions of the Fermat manual for Mac OS9, OS X, L/Unix and Windows are available, and of the FFermat manual (FFermat, the float version, is Mac OS9 only). They were produced from the TeX versions by the program TTH (TeX to HTML), http://hutchinson.belmont.ma.us/tth/. You can find the


Linux/Unix/OSX/Windows manual here. the Mac OS9 manual here, and the FFermat Manual is here.


PS, PDF: In addition there are ps or pdf versions of the manuals:


Download the Linux/Unix/OSX/Windows ps manual.
Download the Mac OS 9 pdf manual.
Download the FFermat pdf manual.

Update History

August 6, 2008. Documented set of functions to run the Dixon-EDF method. This is the method of choice for solving symbolic systems of multivariate polynomial equations over most ground rings. See here.

May 6, 2008. Version 3.9.7 for OSX and Linux. See here.

February 10, 2008. Version 3.9.2 for OSX and Linux. See here.

October 29, 2007. Version 3.8.8 for Mac OSX and Linux; 3.6.9 for Windows; 3.7.0 for Mac 0S9. See here.

September 26, 2007. Version 3.8.7 for Mac OSX and Linux. See here.

March 13, 2007. Version 3.8.1 for Mac OSX and Linux. 3.6.8 for Windows. See here.

August 10, 2006. Version 3.7.9 for Mac OSX and Linux. See here.

June 26, 2006. Version 3.7.8 for Mac OSX and Linux, 3.6.7 for Windows. See here.

June 11, 2006. Version 3.7.7 for Mac OSX and Linux. See here.

April 13, 2006. Version 3.7.5 for Mac OSX and Linux. See here.

March 10, 2006. Version 3.7.4 for Mac OSX and Linux. 3.6.6 for Windows. See here.

February 18, 2006. Version 3.7.2 for Mac OSX, Linux. Bug fixes and speed ups. See here.

December 30, 2005. Version 3.6.9 for Mac OSX, Linux, Unix; 3.6.5 for OS 9 and Windows. Bug fixes. See here.

November 30, 2005. Version 3.6.7 for Mac OSX. Bug fix. See here.

November 22, 2005. Version 3.6.4 for Windows. Bug fix. See here.

October 19, 2005. Version 3.6.7 for Linux. Various new features. See here.

July 27, 2005. Version 3.6.4 for all versions but Unix. Fixed a small memory leak. See here.

July 18, 2005. Version 3.6.3 for Mac OSX/Linux. Fixed a bug in the speedups of 3.5.8 - 3.6.0. See here.

June 27, 2005. Version 3.6.0 for Mac OSX/Linux/Windows/OS9. Additional speedup in multivariate polynomial gcd. See the Fermat Tests Page. Fixed memory leak.

June 10, 2005. Version 3.5.0 - 3.5.8 for Mac OSX/Linux/Unix. Significant speedups in multivariate polynomial gcd, 14% - 60% in various tests. See the Fermat Tests Page. Various bug fixes. Updated the manual for these versions, html and ps. See here.

January 1, 2005: Version 3.4.9 now available for Windows and Mac OS 9. This is a fix of a stupid trivial bug that got introduced in 3.4.8 in these two versions. It affected GCD of polynomials. Also see here.

November 3, 2004: Version 3.4.8 now available for Linux, Mac OS X, and Mac OS 9. Details are here.

October 20, 2004: New version 3.4.8 for Windows. Details are here.

September 1, 2004: New versions 3.4.7 for Linux and Mac OSX. Details are here.

July 12, 2004: Bug fix all versions but Unix. Multiplying a rational number by a rational function was flawed in some cases.

July 6, 2004: Version 3.4.6 for Linux. Minor bug fix. Details are here.

July 2, 2004: Version 3.4.5 for Linux. Minor bug fixes. Details are here.

June 27, 2004: Version 3.4.4 for Linux. Bug fix. Details are here.

June 9, 2004: Version 3.4.2 for Unix. See June 6 below. Details are here.

June 6, 2004: Versions 3.3.2 and 3.4.2. Quite a few bug fixes, especially in Linux. A few new features in Linux, bringing it up to the other versions. Details are here.

May 1, 2004: New version 3.3 for Mac and Windows. This version provides remarkable speedups in basic one variable polynomial arithmetic and g.c.d. Read the April 30, 2004 description below. This is not version 3.4 because the larger moduli have not been implemented here. The Mac MPW version has not been updated, and will no longer be maintained unless I hear from someone who cares about it.

April 30, 2004: New version 3.4 for Linux. This version provides remarkable speedups in basic one variable polynomial arithmetic and g.c.d. Since that is used by other parts of Fermat, the speedup propagates throughout the system to some extent. Problem 3 on the Fermat Tests Page saves 30%. Some computations over Z/p save 60%. Also a new implementation of LaGrange interpolation for determinant of sparse matrices, and Z/p is implemented for primes < 2^31. See the second ReadMe.

December 9, 2003: Added a version for Linux compiled with the -static option for gcc. This creates a very large application (5.5 meg), but fixes the problem some people had because they have old Linux libraries.

December 5, 2003: Version 3.3 for Linux and Windows; bug fix. A user reported a bug, which sometimes occurs when a rational number is multiplied or divided by a quolynomial. The same bug occurs in Mac and Unix, but those fixes are not ready yet.

September 25, 2003: Added Linux and Unix versions.

September 3, 2003: Version 3.2. Bug fixes. For Windows, Fermat would sometimes hang after hitting return. See the second ReadMe.

April 23, 2003: Version 3.1. Implemented builtin functions to return the number of polyvars and the nth prime. Content now has a second [optional] parameter, the name or ordinal of a poly var. The major improvement is yet another large gain in the speed of multivariate GCD. The time for one benchmark is cut by more than 64%. See the second ReadMe.

September 12, 2002: Version 3.0. Significant speedups in the Windows version due to a small change in storage allocation. Several bug fixes and new features. See the second ReadMe.

January 20, 2002: Version 2.6.4. Significant speedups in polynomial gcd, up to 50% saved on some benchmark problems. The Macintosh MPW version, which works under MPW on 68K Macs, no longer requires an fpu or a math coprocessor chip. This also makes it easy to use on PPC Macs (any Mac bought in the last five years or so), as a software fpu is no longer required. See the second ReadMe.

October 15, 2001: Version 2.6.3. Modularmode (i.e., ground ring Z/p) is now implemented for modulus up to 2^24 - 1 = 16,777,215. The limit had been 2^16 - 1 = 65535. Improved the speed of polygcd by handling content better. Removed a few annoying features of saving data to a file. See the second ReadMe.

July 13, 2001: New improved Viewer for Windows95/98/whatever. Full support for change of fonts, non-ASCII characters. The Windows interface is now better than the Mac interface, thanks to Bill Hastings of Fordham University.

Legal Matters


This Software is provided on an "AS IS" basis, without warranty of any kind, including without limitation the warranties of merchantability, fitness for a particular purpose and non-infringement. The entire risk as to the quality and performance of the Software is borne by you. Should the Software prove defective, you assume the entire cost of any service and repair. This disclaimer of warranty constitutes an essential part of the agreement. Even if parts of the disclaimer violates legal rights of some jurisdiction, only these parts are invalid, but not the entire disclaimer.


Do not use Fermat for any application that may entail the loss of life or property.


Please note that although the author plans to release enhanced versions of this software, the author cannot guarantee indefinite software support.


Copyright Statement:


This website contains a variety of copyrighted material. Some of this is the intellectual property of individuals as named, some is owned and copyrighted by Robert Lewis, and some is in the public domain. Except for material which is unambiguously and unarguably in the public domain, no material anywhere on this website, graphical or text, may be copied or further disseminated without the express and written permission of the legal holder of that copyright. This implies that no permission is granted for any commercial use or sale of any of this material, nor for transmission to electronic information and retrieval systems, nor for publication in any journal or compilation, written or electronic, without the written permission of the author.


The information on this Web Page is for personal use only and is provided in good faith without any express or implied warranty. There is no guarantee given as to the accuracy or currency of any individual item on any page. Persons accessing the page who require such confirmation should refer enquires to Robert Lewis, who may or may not be able to provide such. Neither Robert Lewis nor Fordham University accepts any responsibility for any loss or damage occasioned by use of the information contained on the page nor from any access to the page. All access and use is at the risk of the user. This site provides hypertext links to a number of other websites as a service to users. This service does not mean that Fordham University or Robert Lewis endorses those sites or material on them in any way. Neither Fordham University nor Robert Lewis is responsible for the use of any hypertext link which may incur a commercial charge. Individual users are responsible for any charges that result from their use. Any user linking directly to pages of this website bypassing this disclaimer assumes complete responsibility for any resulting inappropriate use of those pages by any or all parties.

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(Linked toruses: another graphic created with Fermat)




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More Comments, Questions, etc.

You can also send comments or questions to me at the following address, which a human will have no trouble understanding:
rlewis at fordham dot edu


The address of this web page is http://www.bway.net/lewis/


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Richard P. Feynman, Nobel Prize Banquet Address 1965:


Your Majesty,


The work I have done has, already, been adequately rewarded and recognized. Imagination reaches out repeatedly trying to achieve some higher level of understanding, until suddenly I find myself momentarily alone before one new corner of nature's pattern of beauty and true majesty revealed. That was my reward. Then, having fashioned tools to make access easier to the new level, I see these tools used by other men straining their imaginations against further mysteries beyond. There, are my votes of recognition.