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A019590 Fermat's Last Theorem: a(n) = 1 if x^n + y^n = z^n has a nontrivial solution in integers, otherwise a(n) = 0. 107


%S 1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Fermat's Last Theorem: a(n) = 1 if x^n + y^n = z^n has a nontrivial solution in integers, otherwise a(n) = 0.

%C a(n) is the Hankel transform of A000045(n), n>=1 (Fibonacci numbers). See A055879 for the definition of Hankel transform. - _Wolfdieter Lang_, Jan 23 2007

%C 1, -1, 0, 0, 0, ... is the convolutional inverse of the all-ones sequence. - _Tanya Khovanova_, Jun 29 2007

%C Also parity of the Euler totient function A000010. - _Omar E. Pol_, Jan 15 2012

%C a(n-1) gives the row sums of A048994. - _Wolfdieter Lang_, May 09 2017

%D A. D. Aczel, Fermat's Last Theorem, Four Walls Eight Windows NY 1996.

%D A. C. Clarke, The Last Theorem, Gollancz SF 2004.

%D B. Cipra, What's Happening in the Mathematical Sciences 1994 Vol. 2, "A Truly Remarkable Proof" pp. 3-8 AMS Providence RI.

%D B. Cipra, What's Happening in the Mathematical Sciences 1995-6 Vol. 3, 'Fermat's Theorem-At Last' pp. 2-13 AMS Providence RI.

%D J. Coates and S.-T. Yau (Eds), Elliptic Curves, Modular Forms and Fermat's last Theorem, International Press Boston MA 1998.

%D G. Cornell, J. H. Silverman and G. Stevens (Eds), Modular Forms and Fermat's last Theorem, Springer NY 2000.

%D K. J. Devlin, Mathematics: The New Golden Age, Chapter 10, Columbia Univ. Press NY 1999.

%D L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 731.

%D H. M. Edwards, Fermat's Last Theorem, Springer, 1977.

%D G. Giorello & C. Sinigaglia, "Fermat: De defis en conjectures", Les genies de la science No. 32 Aug-Oct 2007, pp. 82-100, Pour la Science, Paris.

%D C. Goldstein, "Le Theoreme de Fermat", La Recherche, Vol. Mar 25 1994, pp. 268-275 Paris.

%D C. Goldstein, "Le Theoreme De Fermat Enfin Demontre", Chapter IX pp. 111-129 in 'Histoire Des Nombres', La Recherche, Tallandier, Paris 2007.

%D Y. Hellegouarch, "Fermat Vaincu", Quadrature No. 22 pp. 37-55 Editions du choix Argenteuil (France) 1995.

%D Y. Hellegouarch, "Fermat enfin demontre", Pour la Science, No. 220, 1996 pp. 92-97 Paris.

%D Y. Hellegouarch, Invitation aux mathematiques de Fermat-Wiles, Dunod Paris 2001.

%D Y. Hellegouarch, "L'Enigme du Theoreme de Fermat" pp. 31-41 in 'Qu'est-ce que l'Univers?", Universite de tous les savoirs, Vol. 4 (Edit. Y. Michaud), Odile Jacob Paris 2001.

%D Y. Hellegouarch, Invitation to the Mathematics of Fermat-Wiles, Academic Press NY 2001.

%D P. Hoffman, The Man Who Loved Only Numbers, pp. 183-200 Hyperion NY 1998

%D W. Lindsay, Fermat's Last Theorem, A Perfect Proof, Lulu Press, Morrisville NC 2005.

%D L. J. Mordell, Three lectures on Fermat's last theorem, Cambr. Univ. Press 1921 (Reprinted by The Scholarly Pub. Office, Univ. of Michigan Library 2005).

%D C. J. Mozzochi, The Fermat Diary, AMS Providence RI 2000.

%D C. J. Mozzochi, The Fermat Proof, New Bern NC 2004.

%D V. K. Murty, Seminar on Fermat's Last Theorem, Amer. Math. Soc. Providence RI 1995.

%D P. Odifreddi, The Mathematical Century, Chapter 2.14 "Number Theory: Wiles' Proof of Fermat's Last Theorem (1995)" p. 82 Princeton Univ. Press NJ 2004.

%D I. Peterson, The Mathematical Tourist, pp. 234-238 W. H. Freeman/Owl Book NY 2001.

%D I. Peterson, "A Marginal Note" in Islands of Truths, pp. 280-5 W. H. Freeman NY 1990.

%D A. van der Poorten, Notes on Fermat's Last Theorem, Wiley NY 1996

%D J. Propp, Who Proved Fermat's Last Theorem? Princeton Univ. Press NJ 2005.

%D P. Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, 1979.

%D P. Ribenboim, Fermat's Last Theorem for Amateurs, Springer Verlag NY 1999.

%D R. Schoof, "Wiles' proof of the Taniyama-Weil conjecture for semi-stable elliptic curves over Q", Chap. 14 in 'Ou En Sont Les Mathematiques ?' Soc. Math. de France (SMF), Vuibert, Paris 2002.

%D S. Singh, Fermat's Enigma, Walker and Co. NY 1997.

%D I. Stewart, "Fermat's Last Time-Trip", in Scientific American Nov. 1993 pp. 85.

%D I. Stewart, From Here To Infinity, Chapter 3 "Marginal Interest" pp. 25-48 OUP Oxford 1996.

%D I. Stewart and D. Tall, Algebraic Number Theory and Fermat's Last Theorem, A. K. Peters Natick MA 2001.

%D G. R. Talbott, Fermat's Last Theorem, Lotus Press WI 1991.

%D R. Van Vo, Fermat's Last Theorem, AuthorHouse, Bloomington IN 2002.

%D J. Vigouroux et al., Une aventure mathematique, le theoreme de Fermat, BT2 series No. 6, PEMF Mouans-Sartoux(France) 1998.

%H Amer. Math. Soc., <a href="http://www.ams.org/new-in-math/fermat.html">Fermat's Last Theorem</a>

%H Anonymous, <a href="http://www.bath.ac.uk/~ma0kvb">Fermat's Last Theorem</a>

%H A. J. Bailey, <a href="http://www.adrianbailey.co.uk/fermat/flt.html">Fermat's Last Theorem</a>

%H T. Ballard, <a href="http://www.fermatproof.com">A Short Form Proof Of Fermat's Last Theorem</a>

%H N. Barker, <a href="http://people.bath.ac.uk/ma1nlb/MATH0126.html">Fermat's Last Theorem</a>

%H K. Belabas and C. Goldstein, <a href="http://www.math.u-bordeaux.fr/~belabas/Orsay-info/fermat.html">Fermat et son Theoreme</a>

%H N. Boston, <a href="http://www.math.wisc.edu/~boston/869.pdf">The Proof Of Fermat's Last Theorem</a>

%H K. Buzzard, <a href="http://www.ams.org/bull/1999-36-02/S0273-0979-99-00778-8/S0273-0979-99-00778-8.pdf">Review of "Modular Forms and Fermat's Last Theorem" edited by G. Cornel, J. H. Silverman and G. Stevens</a>

%H C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=FermatsLastTheorem">Fermat's last theorem</a>

%H A. Camus College Team, <a href="http://www.col-camus-soufflenheim.ac-strasbourg.fr/Page.php?IDP=340">La conjecture de Fermat</a>

%H C. C. Chang, <a href="http://www.cs.wichita.edu/~chang/fermat.html">BBC's Horizon Program: "Fermat's Last Theorem"</a>

%H C. S. Chinea, <a href="http://personales.ya.com/casanchi/mat/conjeturafermat.pdf">Sobre La Conjetura de Fermat(Text in Italian)</a>

%H K. Choi, <a href="http://www.public.iastate.edu/~kchoi/fermat.htm">A Note on Fermat's Last Theorem-Was it a Right Question?</a>

%H K. Choi, <a href="http://www.public.iastate.edu/~kchoi/time.htm">Timeline of FLT</a>

%H K. Conrad, <a href="http://www.math.uconn.edu/~kconrad/blurbs/fltreg.pdf">Fermat's Last Theorem For Regular Primes</a>

%H J. Constant, <a href="http://www.coolissues.com/mathematics/Fermat/fermat">Proof of Fermat's Last Theorem</a>

%H P. Daley, <a href="http://people.bath.ac.uk/pid20/fermat.html">Fermat's Last Theorem</a>

%H C. Daney, <a href="http://aux.planetmath.org/files/lec/4/flt.pdf">The Proof of Fermat's Last Theorem</a>

%H H. Darmon, <a href="http://www.ams.org/notices/199911/comm-darmon.pdf">A Proof of the Full Shimura-Taniyama-Weil Conjecture Is Announced</a>

%H H. Darmon, <a href="http://citeseer.ist.psu.edu/correct/474804">The Shimura-Taniyama conjecture (d'apres Wiles)</a>

%H R. Ellman, <a href="http://www.the-origin.org/fermat.pdf">A Concise And Direct Proof of "Fermat's Last Theorem"</a>

%H K. M. Evans, <a href="https://web.archive.org/web/20091027025450/http://geocities.com/kerryme47714">In Defense of Mr. Fermat</a>

%H G. Faltings, <a href="http://www.ams.org/notices/199507/faltings.pdf">The Proof of Fermat's Last Theorem by R.Taylor and A.Wiles</a>

%H Larry Freeman's Blog Spot, <a href="http://fermatslasttheorem.blogspot.com/2005_04_24_archive.html">Fermat's Last Theorem</a>

%H Daniele A. Gewurz and Francesca Merola, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors of oligomorphic permutation groups</a>, J. Integer Seqs., Vol. 6, 2003.

%H C. Gorin, <a href="http://people.bath.ac.uk/ma3clg/Fermat&#39;s%20Last%20Theorem.html">Fermat's Last Theorem</a>

%H F. Q. Gouvea, <a href="http://www.americanscientist.org/template/BookReviewTypeDetail/assetid/14359?&amp;print=yes">Wiles's Proof, 1993-1995: Review of "The Fermat Diary" by C. J. Mozzochi</a>

%H A. Granville, <a href="http://www.ams.org/notices/199701/comm-granville.pdf">Review of BBC's Horizon Program:"Fermat's Last Theorem"</a>

%H D. Graves, <a href="http://www.maa.org/reviews/fltamat.html">Review of "Fermat's Last Theorem for Amateurs" by P.Ribenboim</a>

%H W. F. Hammond, <a href="http://www.albany.edu/~hammond/gellmu/examples/f356g.pdf">Fermat's Last Theorem after 356 years</a>

%H W. F. Hammond, <a href="http://math.albany.edu:8010/g/Math/topics/fermat">Wiles,Ribet,Shimura-Taniyama-Weil and Fermat's Last Theorem</a>

%H T. Hausberger, <a href="http://www.math.univ-montp2.fr/~hausberg/ens/licence/fermat/index.html">Autour du dernier Theoreme de Fermat</a>

%H S. Horsler, <a href="http://people.bath.ac.uk/ma3sah">Home Page on FLT</a>

%H A. Jackson, <a href="http://www.ams.org/notices/199710/comm-fermat.pdf">Review of "Fermat's Enigma" by S.Singh</a>

%H M. Jaiclin, <a href="http://www.brockport.edu/math/mathclub/fermat.talk.pdf">What Makes Fermat's Last Theorem So Hard, Anyway?</a>

%H N. de Jong, <a href="https://web.archive.org/web/20091026234130/http://www.geocities.com/elementaryfermat">Fermat's Last Theorem. Was Wiles's proof really first?</a>

%H D. Kaminski, <a href="http://www.cs.uleth.ca/~kaminski/esferm03.html">The extraordinary story of Fermat's Last Theorem</a>

%H A. Krowne, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/FermatsLastTheorem.html">Fermat's last theorem</a>

%H S. Lang, <a href="http://www.ams.org/notices/199511/forum.pdf">Some History of the Shimura-Taniyama Conjecture</a>

%H W. Lang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

%H L. Lemaire, <a href="http://www.ulb.ac.be/assoc/bms/info/rechmath.pdf">Le theoreme de Fermat-Wiles (pages 5-11/35)</a>

%H R. Lisker, <a href="http://rendezvous.com/tangledweb/conferences/fermat">Fermat's Last Theorem (Conference Report)</a>

%H A. Lopez-Ortiz, <a href="http://www.math.uncc.edu/flt.php">What We Know About Fermat's Last Theorem</a>

%H E. R. Lucier, <a href="https://arxiv.org/abs/math/0502245">Fermat's Last Theorem,Further Study</a>, arXiv:math/0502245 [math.GM], 2005-2012.

%H L. Lusk, <a href="http://mathematics.gulfcoast.edu/mgf1107/newpage11.htm">A Brief History of Fermat's Last Theorem</a>

%H John Lynch (writer/producer) and Simon Singh (director), <a href="https://www.youtube.com/watch?v=BQG9McMC59U">Fermat's Last Theorem.</a> Horizon. BBC TV/WGBH Boston, Jan 15 1996.

%H M. Marcionelli, <a href="http://www.math.ethz.ch/~michele/AZ/fr/ind_Fermat.html">La conjecture de Fermat</a>

%H Mathematical Database, Poster, <a href="http://mathdb.org/gallery/poster/description/e_poster_02.htm">Fermat's Last Theorem</a>

%H Mathforum, <a href="http://mathforum.org/dr.math/faq/faq.fermat.html">Fermat's Last Theorem</a>

%H G. Mazzarino, <a href="http://www.economics.ox.ac.uk/Members/giuseppe.mazzarino/Fermat_March_2003_abstract.htm">Fermat's Last Theorem</a>

%H G. Mazzarino, <a href="http://www.economics.ox.ac.uk/Members/giuseppe.mazzarino/Fermat_Revisited_April_2003_abstract.htm">Fermat's Last Theorem Revisited</a>

%H P. L. Mills, <a href="http://members.lycos.co.uk/comms1/fidn1/fermat.html">Fermat's Last Theorem-A New Proof</a>

%H J. S. Milne, <a href="http://www.jmilne.org/math/CourseNotes/math679.html">Elliptic Curves</a>

%H New York Times, <a href="http://partners.nytimes.com/books/first/s/singh-fermat.html">"Fermat's Enigma" by S. Singh; Chapter 1</a>

%H J. J. O'Connor and E. F. Robertson, <a href="http://www.math.uga.edu/~schang/math/fermat.pdf">Fermat's Last Theorem</a>

%H R. Osserman et al., <a href="http://www.msri.org/publications/ln/msri/1993/outreach/fermat/1/main.html">Fermat's Last Theorem: A Supplement to the Video</a>

%H I. Peterson, <a href="http://www.maa.org/mathland/mathtrek_11_22_99.html">Curving Beyond Fermat</a>

%H J. Propp, <a href="http://www.math.wisc.edu/~propp/courses/491/articles">Bibliography of articles on FLT</a>

%H S. Rees, <a href="http://www.bath.ac.uk/~ma1sr/project.html">Fermat's Last Theorem</a>

%H D. Rideout, <a href="http://www.math.uncc.edu/~hbreiter/m6105/pythag.pdf">Pythagorean Triples, the Unit Circle and Fermat's Last Theorem</a>

%H K. Rubin, <a href="http://www.math.ohio-state.edu/history/flt/old.fltslides">The Solving of Fermat's Last Theorem (Slides)</a>

%H K. Rubin and A. Silverberg, <a href="http://citeseer.ist.psu.edu/correct/346535">A Report on Wiles' Cambridge Lectures</a>

%H K. Rubin and A. Silverberg, <a href="https://arxiv.org/abs/math/9407220">A report on Wiles' Cambridge lectures</a>, arXiv:math/9407220 [math.NT], 1994.

%H K. Rubin and A. Silverberg, <a href="http://www.ams.org/bull/pre-1996-data/199431-1/rubin.pdf">A Report On Wiles' Cambridge Lectures</a>

%H D. Rusin, The Mathematical Atlas, <a href="http://www.math.niu.edu/~rusin/known-math/index/11D41.html">Higher degree equations; Fermat's equation</a> [Broken link]

%H D. Rusin, The Mathematical Atlas, <a href="/A019590/a019590.html">Higher degree equations; Fermat's equation</a> [Cached copy, but just of the top page, so none of the internal links will work]

%H School of Mathematics and Statistics, University of St Andrews, <a href="http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Fermat&#39;s_last_theorem.html">Fermat's last theorem</a>.

%H P. Schorer, <a href="http://www.occampress.com/fermat.pdf">Is There a "Simple" Proof of Fermat's Last Theorem</a>

%H J. L. Selfridge, C. A. Nicol and H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/41/11/970.pdf">Proof Of Fermat's Last Theorem For All Prime Exponents Less Than 4002</a>

%H D. Shay, <a href="http://fermat.workjoke.com">Fermat's Last Theorem</a>

%H Kaida Shi, <a href="https://arxiv.org/abs/math/0309005">The n-dimensional Cube--A New Way to Prove the Fermat's Last Theorem</a>, arXiv:math/0309005 [math.GM], 2003-2009.

%H S. Singh, <a href="http://www.simonsingh.com/Fermat_Corner.html">Fermat Corner</a>

%H S. Singh, <a href="http://www.prometheus.demon.co.uk/01/01fermat.htm">Fermat's Last Theorem</a>

%H S. Somani, <a href="http://web.bii.a-star.edu.sg/slides/journal/2002/Fermat_Last_Theorem.pdf">The Saga of Fermat's Last Theorem</a>

%H K. Spiliopoulos, <a href="http://users.forthnet.gr/ath/kimon/FLTmilestones.htm">The Final Milestones in the Proof of Fermat's Last Theorem</a>

%H S. Stedman, <a href="http://students.bath.ac.uk/ma0sjbs/fermat.html">Fermat's Last Theorem</a>

%H W. A. Stein, <a href="http://modular.math.washington.edu/edu/Fall2001/124/lectures/lecture32/lecture32">Fermat's Last Theorem and Modularity of Elliptic Curves</a>

%H D. Surendran, <a href="http://www.uz.ac.zw/science/maths/zimaths/flt.htm">Fermat's Last Theorem</a>

%H Think Quest, <a href="http://library.thinkquest.org/28049/final_stages_in_proving_fermat.htm">The final stages in proving Fermats Last Theorem</a>

%H C. Thornhill, <a href="http://www.missouri.edu/~cst398/fermat/fermat1.htm">Fermat's Last Theorem</a>

%H A. van der Poorten, <a href="http://www.maths.mq.edu.au/~alf/NotesonFLT.html">"Notes on Fermat's Last Theorem": Table of Contents</a>

%H A. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/alfpapers/a142e.pdf">Fermat's Last Theorem</a>

%H D. L. Vestal, <a href="http://www.maa.org/reviews/fermatwiles.html">Review of "Invitation to the Mathematics of Fermat-Wiles" by Y. Hellegouarch</a>

%H G. Villemin's Almanach of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Addition/ThFermat.htm">Theoreme de Fermat-Wiles</a>

%H A. H. Warren, <a href="http://www.memagazine.org/contents/current/webonly/fermfinal310.pdf">Proof Of The Fermat's Last Theorem</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fermat&#39;s_Last_Theorem">Fermat's Last Theorem</a>

%H A. J. Wiles, <a href="http://modular.math.washington.edu/edu/Spring2004/129/references/flt/flt.pdf">Modular elliptic curves and Fermat's last theorem</a>, Ann. Math. 141 (1995), 443-551.

%H A. J. Wiles, <a href="http://modular.math.washington.edu/edu/Spring2003/21n/papers/Wiles,Modular_Elliptic_Curves_and_Fermats_Last_Theorem.pdf">Modular elliptic curves and Fermat's Last Theorem</a>

%H T. Yee, <a href="http://ihome.ust.hk/~tonyee/flt.htm">The Fermat's Last Theorem(FLT)</a>

%H Yan X Zhang, <a href="https://arxiv.org/abs/1508.00318">Four Variations on Graded Posets</a>, arXiv preprint arXiv:1508.00318 [math.CO], 2015.

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%F a(n) = (-1)^n*sum{k=0..floor(n/2), (-1)^A010060(n-2k)mod(C(n, 2k), 2)}. - _Paul Barry_, Jan 03 2005

%F a(n) = 1-[(n+2) mod (n+1)]+[n!^2 mod (n+1)]*[(n+1)!^2 mod (n+2)]. - _Paolo P. Lava_, Aug 29 2007

%F a(n) = [(n-1)! mod 2], with n>=1. - _Paolo P. Lava_, Feb 15 2008

%F a(n+1) = (1/2)*[1+(-1)^n]*a(n), with a(0)=1. - _Paolo P. Lava_, Apr 16 2008

%F Euler transform of length 2 sequence [1, -1]. - _Michael Somos_, Jul 05 2009

%F a(n) is multiplicative with a(2) = 1, a(2^e) = 0 if e>1, a(p^e) = 0^e if p>2. - _Michael Somos_, Jul 05 2009

%F G.f.: x + x^2 = x * (1 - x^2) / (1 - x). - _Michael Somos_, Jul 05 2009

%F Dirichlet g.f.: 1 + 2^(-s). - _Michael Somos_, Jul 05 2009

%F a(n) = A000035(A000010(n)). - _Omar E. Pol_, Oct 28 2013

%o (PARI) {a(n) = (n==1) + (n==2)}; /* _Michael Somos_, Jul 05 2009 */

%Y Cf. A000004, A000007, A010051, A012450.

%Y INVERT transform gives Fibonacci numbers, A000045.

%Y Convolution inverse of A062157. Dirichlet convolution inverse of A154269.

%Y Cf. A229382, A229383 (near-miss counterexamples to FLT).

%Y Cf. A048994 (row sums).

%K nonn,nice,easy,mult

%O 1,1

%A _N. J. A. Sloane_

%E Additional links provided by _Lekraj Beedassy_, Nov 20 2003

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