The latest versions of Fermat (September 20, 2025) can now run on Windows, thanks to the efforts of Bogdan Radu. Windows 10 introduced a new feature, named WSL (or the Windows Subsystem for Linux), which allows a user to natively run Linux applications within a Windows environment. See the instructions available here written by Bogdan Radu. May 2021.

Release of Fermat Source Code. The source code for version 6.21 was released under GNU GPL in August 2018. If you are interested in having a copy, and have not done so recently, email me.

Update: The source code for version 6.37 was released under GNU GPL in June 2020.

The manual for the 64 bit Linux and Mac versions has been updated. June 24, 2024; September 27, 2023.

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, multivariate polynomials, 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 are versions for OS X and old obsolete versions for OS 9. There are 64 bit versions. There is an old "float" version for graphics (no longer usable) and some new float versions (no graphics). All versions are available here.

New 64 bit Fermat 7.8, with improved multivariate polynomial factoring. September 20, 2025; June 25, 2024; February 28, 2024; February 11, 2024; January 11, 2024; November 29, 2023.

Quick introduction to Fermat: First Steps

64 bit Fermat 6.5, with vastly improved multivariate polynomial gcd ever since 6.0. June 25, 2021.

Fermat Linux as a C Library. revised September 5, 2023

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

See for example:

Compare Your Computer Algebra System. Take the Fermat Tests! September 2015.

There are now five tests. One test involves evaluation of rational functions, the second involves Smith Normal Form, the third resultants, the fourth rational function arithmetic, the fifth a multivariate determinant.

   Fermat is freeware. Those who have grant money are requested to contribute $70 per installed machine.

   64 bit Versions of Fermat, Version 5.25 July 5, 2016

   Mac OS X Intel Version of Fermat Available, 32 bit. 3.9.9x, revised July 5, 2016

   Float Versions of Fermat Now Available. April 22, 2015

   Linux Version of Fermat. 3.9.9x, revised July 5, 2016

   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.

   First Steps with Fermat:

Short introduction to the basics of using Fermat. September 2024

   PDF Version of the Manual:

Revised PDF manual for Linux, Windows, and OS X. June 2024; September 2023, small revisions April and June 2016, March 2020, March 30, 2021.

The old Windows manual (for the now obsolete Windows version) is from June 10, 2005. There are HTML versions of the Fermat manual for old Windows, U/Linux, and Mac, and of the FFermat manual (FFermat, the old float version, was Mac only). In addition, there are revised 2011 ps and pdf versions of the manuals. See below.

   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 [now basically obsolete] click here and here.

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

   Old Mac OS X G3-G5 Version of Fermat Available. 3.9.9x, revised October 8, 2010, for old G3-G5 Macs.

   "New" Windows Version of Fermat. 3.9.9x, March 28, 2013

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

   Old Obsolete Windows Version of Fermat. 3.6.9, revised October 29, 2007

   Mac OS 9 (Obsolete) Version of Fermat Available. 3.7.0, revised October 29, 2007

   Source Code for Old Windows Version Available. April 17, 2010

   Source Code for FFermat Now Available. January 19, 2009

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 Appendices 4 - 6 of the Manual. (This is the revised manual of July 2011 for Mac OS X and Linux. Some features are not implemented in Windows.)



Fermat in Actual Use

This is the method of choice for solving most systems of polynomial equations with fewer than, say, 20 variables, but any number of parameters.


Data From 2004: Fermat is Overall Best in the World at Polynomial GCD

According to two independent researchers.

Some research projects that used Fermat -- and needed it (100+ papers):

M. Egner, M. Fael, K. Schonwald, M. Steinhauser. Nonleptonic B-meson decays to next-to-next-to-leading order. arXiv:2406.19456v2 (2024)


Matteo Fael, Fabian Lange, Kay Schonwald, Matthias Steinhauser. Massive three-loop form factors: Anomaly contribution. PHYSICAL REVIEW D 107, 094017 (2023)

F. Buccioni, P. Alexander Kreer, X. Liu, L. Tancredi. One loop QCD corrections to g ->tt-H at O(epsilon^2). arXiv:2312.10015 [hep-ph] (2023)


Matteo Fael, Fabian Lange, Kay Schonwald and Matthias Steinhauser. Three-loop b -> sg vertex with current-current operators. ArXiv ePrint: 2309.14706 (2023)


J. Rada, M. Zamboj. 3-D Shadows of 4-D Algebraic Hypersurfaces in a 4-D Perspective. arXiv:2307.12986 [math.GM] (2023)


Schonwald. Massive form factors at O(alpha^3_s). arXiv:2207.06705v1 [hep-ph] (2023)


D. Blackman, S. Vigna. Scrambled Linear Pseudorandom Number Generators. arXiv:1805.01407v3 [cs.DS] 28 Mar 2022.


M. Fael, F. Lange, K. Schonwald, M. Steinhauser. Massive vector form factors to three loops. arXiv:2202.05276v1 [hep-ph] (2022)


V. Shtabovenko. Towards two- and three-loop QCD corrections to the width difference in Bs − Bs mixing. SciPost Phys. Proc. 7, 025 (2022)


M. Gerlach, U. Nierste, V. Shtabovenko, M. Steinhauser. The width difference in B--B mixing at order alpha_s and beyond. Journal of High Energy Physics V. 2022, article number 6, (2022).


V. Shtabovenko. NNLO QCD corrections to B-meson mixing. arXiv:2110.12044v1 [hep-ph] (2021)


F. Lange, P. Maierhofer, J. Usovitsch. Developments since Kira 2.0. arXiv:2111.01045 [hep-ph] (2021)


B. Agarwal, S. P. Jones, and A. von Manteuffel. Two-loop helicity amplitudes for gg -> ZZ with full top-quark mass effects. J. High Energ. Phys. 2021, 256 (2021)


J. Klappert, F. Lange, P. Maierhofer, J. Usovitsch. Integral Reduction with Kira 2.0 and Finite Field Methods. arXiv:2008.06494v2 [hep-ph] (2020)


T. Ahmed, A. H. Ajjath, Long Chen, P. K. Dhani, P. Mukherjee, V. Ravindran. Polarised Amplitudes and Soft-Virtual Cross Sections for bb' -> ZH at NNLO in QCD. arXiv:1910.06347v2 [hep-ph] (2020)


Fahimeh Mokhtari, Jan A. Sanders. Versal normal form for nonsemisimple singularities. Journal of Differential Equations, Volume 267, Issue 5, 15 August 2019, Pages 3083-3113.


H. Chawdhry, M. Lim, A. Mitov. Two-loop five-point massless QCD amplitudes within the integration-by-parts approach. PHYSICAL REVIEW D99,076011 (2019)


A. Neumann, Z. Sullivan. Off-shell single-top-quark production in the Standard Model Effective Field Theory. Journal of High Energy Physics volume 2019, Article number: 22 (2019)


J. Bohm, A. Georgoudis, K. J. Larsen, H. Schonemann, and Y. Zhang. Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections. Journal of High Energy Physics 2018. ArXiv ePrint: 1805.01873


Samuel Abreu, Ben Page and Mao Zeng. Differential equations from unitarity cuts: nonplanar hexa-box integrals. Journal of High Energy Physics 2019. DOI: 10.1007/JHEP01(2019)006.


Jonas Klappert and Fabian Lange. Reconstructing Rational Functions with FireFly. arXiv:1904.00009v2 [cs.SC] Oct. 2019.


A. Behring, J. Blumlein, A. De Freitas, A. Goedicke, S. Klein, A. von Manteuffel, C. Schneider, K. Schonwald. The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements. arXiv:1908.03779 [hep-ph] Aug. 2019.


V. Hirschi, S. Lionetti, and A. Schweitzer. One-loop weak corrections to Higgs production. arXiv:1902.10167 [hep-ph] Feb. 2019.


Patrick Dukes and Joe Rusinko. Commutation Classes of Double Wiring Diagrams. arXiv:1006.1076v1.pdf. June 2010. Also see: Involve: A Journal of Mathematics, V. 5, 2 (2012), 207 - 218.


R. Angeles-Mart́inez, M. Czakon and S. Sapeta. NNLO soft function for top quark pair production at small transverse momentum. arxiv.org/pdf/1809.01459.pdf. Nov. 2018.


L. Almeida, C. Sturm. Two-loop matching factors for light quark masses and three-loop mass anomalous dimensions in the RI/SMOM schemes. arXiv:1004.4613v2 [hep-ph] 2011.


L. Almeida, C. Sturm. Two-loop matching factors for light quark masses and three-loop mass anomalous dimensions in the regularization invariant symmetric momentum-subtraction schemes. PHYSICAL REVIEW D 82, 054017 (2010)


K. Kudashkina, K. Melnikova, C. Wever. Two-loop amplitudes for processes gg → Hg, qg → Hq and qq → Hg at large Higgs transverse momentum. arXiv:1712.06549 [hep-ph]. Dec. 2017.


J. Ablinger, A. Behring, J. Bluemlein, et al. Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering: Recent Results. arXiv:1711.07957. Nov. 2017.


T. Luthe, A. Maier, P. Marquard, Y. Schreoder. Complete renormalization of QCD at five loops. Journal of high energy physics. 2017 (3). 020


Philipp Maierhoefer, Johann Usovitsch, Peter Uwer. Kira - A Feynman Integral Reduction Program. arXiv:1705.05610 [hep-ph]. May 2017.


J. Ablinger, A. Behring, J. Blumlein, A. De Freitas, A. von Manteuffel, C. Schneider. The Three-Loop Splitting Functions P(2)qg and P(2,NF)gg. arXiv:1705.01508 [hep-ph]. May 2017.


Andreas von Manteuffel, Lorenzo Tancredi. A non-planar two-loop three-point function beyond multiple polylogarithms. arXiv:1701.05905v1. Jan. 2017.


Kirill Melnikov, Lorenzo Tancredi, Christopher Wever. Two-loop amplitudes for qg → Hq and qq¯→ Hg mediated by a nearly massless quark. arXiv:1702.00426 [hep-ph]. Feb 2017.


J. Ablinger, A. Behring, J. Blumlein, G. Falcioni, A. De Freitas, A. Hasselhuhn, et al. New Results on Massive 3-Loop Wilson Coefficients in Deep-Inelastic scattering. arXiv:1609.03397v2 [hep-ph] 2016.


T. Luthe, Y. Schroder. Five-loop massive tadpoles. arXiv:1609.06786v1 [hep-ph] 2016.


T. Luthe and Y. Schröder. Fun with higher-loop Feynman diagrams. 2016 J. Phys.: Conf. Ser. 762 012066.


T. Luthe, A. Maier, P. Marquard and Y. Schroder. Towards the five-loop Beta function for a general gauge group. JHEP 07 (2016) 127.


Sebastiano Vigna. An Experimental Exploration of Marsaglia's xorshift Generators, Scrambled. ACM Transactions on Mathematical Software, June 2016 Article No. 30.


A. Behring, J. Blumlein, A. De Freitas, A. von Manteuffel, and C. Schneider. The 3-Loop Non-Singlet Heavy Flavor Contributions to the Structure Function g_1(x,Q^2) at Large Momentum Transfer. ArXiv ID: 1504.08217, DOI: 10.1016/j.nuclphysb.2015.06.007. June 2015.


A. Behring, J. Blumlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, and C. Schneider. "The O(alpha^3) Heavy Flavor Contributions to the Charged Current Structure Function xF3(x,Q^2) at Large Momentum Transfer." Physical Review D 92(114005). November 2015.


A. von Manteuffel, E. Panzer, and R. Schabinger. "On the Computation of Form Factors in Massless QCD with Finite Master Integrals," http://arxiv.org/abs/1510.06758, October 2015.


T. Gehrmann. A. von Manteuffel, L. Tancredi. "The two-loop helicity amplitudes for qq → V1V2 →four leptons," arxiv:1503.04812v2, September 2015.


L. Tancredi, T. Gehrmann. A. von Manteuffel. "Two-loop QCD corrections to vector boson pair production at the LHC," 12th International Symposium on Radiative Corrections (Radcor 2015) and LoopFest XIV (Radiative Corrections for the LHC and Future Colliders), 15-19 June 2015. UCLA Department of Physics & Astronomy Los Angeles, CA, USA.


J. Ablinger, A. Behring, J. Blumlein, A. De Freitas, A. von Manteuffel, C. Schneider. "Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra," arXiv:1509.08324, September 2015.


Li, von Manteuffel, Schabinger, Zhu, "Soft-virtual corrections to Higgs production at N^3LO". arXiv:1412.2771, December 2014.


A. De Freitas, J. Ablinger, A. Behring, J. Bluemlein, A. Hasselhuhn, A. von Manteuffel, C.G. Raab, M. Round, C. Schneider, F. Wissbrock. Recent progress on the calculation of three-loop heavy flavor Wilson coefficients in Deep-Inelastic Scattering. Proceedings of Science. DOI: https://doi.org/10.22323/1.211.0041. August 2014.


J. Ablinger, A. Behring, J. Blumlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, et al. The 3-Loop Non-Singlet Heavy Flavor Contributions and Anomalous Dimensions for the Structure Function F2(x,Q2) and Transversity. arXiv:1406.4654v1 [hep-ph] June 2014.


J. Ablinger, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider. The O(α_s^3 T_F^2) Contributions to the Gluonic Operator Matrix Element. arXiv:1405.4259v1 [hep-ph] May 2014.


C. Sturm. Higher order QCD results for the fermionic contributions of the Higgs-boson decay into two photons and the decoupling function for the MS renormalized fine-structure constant. The European Physical Journal C volume 74, Article number: 2978 (2014)


J. Ablinger, J. Blumlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, F. Wissbrock. 3-Loop Heavy Flavor Corrections in Deep-Inelastic Scattering with Two Heavy Quark Lines. arXiv:1407.2821 [hep-ph] 2014.


Ablinger, Blumlein, DeFreitas, Hasselhuhn, von Manteuffel, Round, Schneider, Wissbrock, "The transition matrix element Agq(N) of the variable flavor number scheme at O(alpha s^3)". Nuclear Physics B, February 2014.


Sebastiano Vigna, "An experimental exploration of Marsaglia's xorshift generators, scrambled". Universita degli Studi di Milano, Italy. http://vigna.di.unimi.it/ftp/papers/xorshift.pdf. May 2014.


Maierhofer, P., Marquard, P., Complete three-loop QCD corrections to the decay H+gamma. Zbl 1307.81067. Phys. Lett., B 721, No. 1-3, 131-135 (2013).


Sturm, Christian. Leptonic contributions to the effective electromagnetic coupling at four-loop order in QED. Zbl 1282.81201. Nucl. Phys., B 874, No. 3, 698-719 (2013).


Schneider, Carsten (ed.) et al., Computer algebra in quantum field theory. Integration, summation and special functions. Wien: Springer (ISBN 978-3-7091-1615-9/hbk; 978-1-4614-8523-0/ebook). Texts and Monographs in Symbolic Computation, 361-379 (2013).


P.A. Baikov, K.G. Chetyrkin, J.H. Kuhn, C. Sturm. The relation between the QED charge renormalized in MSbar and on-shell schemes at four loops, the QED on-shell beta-function at five loops and asymptotic contributions to the muon anomaly at five and six loops. Nucl.Phys.B 867 (2013) 182-202.


Lloyd, Noel G.; Pearson, Jane Margaret. A cubic differential system with nine limit cycles. Zbl 1304.34062. J. Appl. Anal. Comput. 2, No. 3, 293-304 (2012).


David Yuen, "The Utility of Computations" and "The Satake Compactification of the Paramodular Group". First EU-US Conference on Automorphic Forms and Related Topics, Aachen University, August 2012.


William B. Kilgore and Christian Sturm. Two-loop virtual corrections to Drell-Yan production at order alpha_s alpha^3. Phys. Rev. D 85, 033005, Feb 2012.


Czakon, M. Double-real radiation in hadronic top quark pair production as a proof of a certain concept. Zbl 1215.81117. Nucl. Phys., B 849, No. 2, 250-295 (2011).


Shaun Ault. Symmetric Homology of Algebras. arXiv:0902.1274v3 [math.AT] 2011.


Grozin, A. G. Integration by parts: An introduction. International Journal of Modern Physics A 26(17) (2011).


J. M. Pearson and N. G. Lloyd, "Kukles revisited: Advances in computing techniques." Computers and Mathematics with Applications, v. 60, issue 10, 2010, pp. 2797 - 2805.


B. Palancz, J. Awange, P. Zaletnyik, "Computational Mathematics: Theory, Methods and Applications." Peter G. Chareton, ed. Nova Science Publishers, New York, January 2010.


Bekavac, S.; Grozin, A.G.; Seidel, D.; Smirnov, V.A. Three-loop on-shell Feynman integrals with two masses. Zbl 1194.81252. Nucl. Phys., B 819, No. 1-2, 183-200 (2009).


Zhao, ShiZhong; Fu, HongGuang. Three kinds of extraneous factors in Dixon resultants. Zbl 1179.13024. Sci. China, Ser. A 52, No. 1, 160-172 (2009).


A. V. Smirnov, "Algorithm FIRE: Feynman Integral Reduction." July 2008. http://arxiv.org/pdf/0807.3243v3
See also Smirnov's software that links to Fermat, FIRE


M. Oura, C. Poor, and D. S. Yuen, "Towards the Siegel Ring in Genus Four." International Journal of Number Theory, August 2008.


P. A. Baikov, K.G.Chetyrkin, and C. Sturm. New Results in Four and Five Loop QED calculations. Nuclear Physics B - Proceedings Supplements Volume 183, October 2008, Pages 8-13.


J. H. Kuhn, M. Steinhauser and M. Tentyukov, "Massless Four-Loop Integrals and the Total Cross Section in e+ e- Annihilation." High Performance Computing in Science and Engineering '07; Transactions of the High Performance Computing Center, Stuttgart (HLRS) 2008.


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.


A. Maier, P. Maierhofer and P. Marquard. Higher Moments of Heavy Quark Correlators in the Low Energy Limit at O(alpha^2_s). Nucl. Phys.B Proc.Suppl. 183 (2008) 209-214.


P. Marquard, L. Mihaila, J.H. Piclum and M. Steinhauser, Relation between the pole and the minimally subtracted mass in dimensional regularization and dimensional reduction to three-loop order. Nuclear Physics B Volume 773, Issues 1–2, 25 June 2007, Pages 1-18.


A.G. Grozin, P. Marquard, J.H. Piclum, and M. Steinhauser. Three-Loop Chromomagnetic Interaction in HQET. arXiv:0707.1388v1 [hep-ph] 2007.


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


S. Bekavac, D. Seidel1, M. Steinhauser and A. Grozin, Light quark mass effects in the on-shell renormalization constants. Journal of High Energy Physics, Volume 2007, JHEP10(2007)


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


K. G. Chetyrkin, M. Faisst, J. H. Kuhn, P. Maierhofer, and C. Sturm. Four-Loop QCD Corrections to the Electroweak rho Parameter. Phys. Rev. Lett. 97, 102003 (2006).


Fiala, Nick C.; Agre, Keith M. Searching for shortest single axioms for groups of exponent 6. Zbl 1107.68097 J. Autom. Reasoning 36, No. 3, 241-257 (2006).


K. Chetyrkin, J. Kuhn, and C. Sturm. Four-loop moments of the heavy quark vacuum polarization function in perturbative QCD. Eur. Phys. J. C 48, 107–110 (2006). https://doi.org/10.1140/epjc/s2006-02610-y


Fotiou, I.A.; Rostalski, P.; Parrilo, P.A.; Morari, M. Parametric optimization and optimal control using algebraic geometry methods. Zbl 1133.93337 Int. J. Control 79, No. 11, 1340-1358 (2006).


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


P. Marquard, J.H. Piclum, Dirk Seidel, M. Steinhauser. Fermionic corrections to the three-loop matching coefficient of the non-relativistic vector current. Nuclear Physics B 758(1): 144 - 160. August 2006.


K.G. Chetyrkin, J.H. Kuhn, C. Sturm. QCD decoupling at four loops. Nuclear Physics B 744 (2006) 121–135.


K. G. Chetyrkina, C. Sturm. Recent Results on Four-loop Tadpoles. NUCL PHYS B-PROC SUPPL, vol. 160, pp. 230-234, 2006.


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.


Zhao, Shizhong; Fu, Hongguang. An extended fast algorithm for constructing the Dixon resultant matrix. Zbl 1122.14302 Sci. China, Ser. A 48, No. 1, 131-143 (2005).


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.


K. G. Chetyrkin, J. H. Kuhn, P. Mastrolia, C. Sturm. Heavy-quark vacuum polarization: first two moments of the O(alphas^3 nf^2) contribution. http://arxiv.org/abs/hep-ph/0412055


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


M. Awramik, M. Czakon, A. Freitas, and G. Weiglein. Complete Two-Loop Electroweak Fermionic Corrections to the Effective Leptonic Weak Mixing Angle and Indirect Determination of the Higgs Boson Mass. Phys. Rev. Lett. 93, 201805 (2004)


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.


Some more papers that reference Fermat. In January 2024, this search found 250. No doubt there is overlap with the above list.

List of 250 papers.

Revised Manual for Linux and OSX. June 25, 2021

A revised and updated manual for OS X, Linux, and Windows is now available as a pdf (March 30, 2021) here. This has small revisions made in April and June 2016 though 2017 and 2019-2021. There is also an old html version, but the pdf is much more readable and attractive.


Older HTML versions of the Fermat manual for Mac OS9, Unix and old Windows are still available, and of the FFermat manual (FFermat, the old float version, was 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


OS X, Windows, and Linux manual here. You can find the Unix and old obsolete Windows manual here. the Mac OS9 manual here, and the FFermat Manual is here.


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


Download the revised Linux, Windows, and OSX pdf manual.
Download the Unix and old obsolete Windows ps manual.
Download the Mac OS 9 pdf manual. OS 9 is obsolete. The OS 9 version of Fermat is obsolete.
Download the FFermat pdf manual.

Jenks Prize Nomination Form

The 2008 nomination form for Fermat for the Jenks Prize can be read here.





Older Update History

June 7, 2016. Features and bug fixes to 64 bit versions. Bug fix only to 32 bit versions, now considered obsolete. Small revisions to manual.

November 25, 2015. 64 bit version 5.21: Significant revision of some heuristics for multivariate polynomial GCD. New interface features upon invoking Fermat.

October 20, 2014: 64 bit version 5.17. Bug fix in both Mac and Linux 64 bit versions. There was another small bug in monomial multiply involving constants. Also, the imperative command &(_o = ...) now works.

June 24, 2014: 5.15 or 5.16. Bug fix in both Mac and Linux 64 bit versions. A small bug introduced last August when monomial multiply was implemented could crash Fermat.

November 9, 2013. Minor improvements in the 64 bit versions; version 5.1.

October 25, 2013. New fast monomial-oriented multiplication for multivariate polynomials in the 64 bit versions; version 5.0. See here.

March 25, 2013. Bug fix, all versions, involving large two-variable polynomial multiplication. Refined heuristics for multivariable polynomial gcd.

November 6, 2012. Bug fix, all versions, involving Pseudet.

June 12, 2012. Bug fix, all versions.

January 3, 2012. Three float versions created. See here. Also, minor revisions of Mac OS X, Linux, and Windows.

November 10, 2011. New Windows version, created with Cygwin. The old Windows version is obsolete and very inferior.

November 1, 2011. New versions for Linux and OS X, 32 bit and 64 bit.

July 25, 2011. New revised manuals for Linux and OS X.

July 23, 2011. 3.9.99 Several bug fixes, new functions, faster multivariate polynomial gcd, 64 bit versions. See here.

October 7, 8, 2010. Version 3.9.9x for OSX and Linux. New 64 bit versions 4.08. See above and here.

October 19, 2009. Version 3.9.9i for OSX and Linux. See here.

January 19, 2009. Source code for FFermat, float version, now available. See here.

January 8, 2009. Version 3.9.8f for OSX and Linux. See here.

August 6, 2008. Uploaded a 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.

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.


THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNERS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.


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

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(Linked toruses: another graphic created with Fermat)




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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.