%I A019590
%S A019590 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,
%T A019590 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 A019590 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
%N 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.
%C A019590 a(n) is the Hankel transform of A000045(n),n>=1 (Fibonacci numbers).
See A055879 for the definition of Hankel transform. W. Lang Jan 23
2007.
%C A019590 1, -1, 0, 0, 0, ... is the convolutional inverse of the all-ones sequence.
- Tanya Khovanova (tanyakh(AT)yahoo.com), Jun 29 2007
%C A019590 a(n) = n - (sum of all natural numbers less than n), if the result is
positive, 0 otherwise. Up to signs and a shift this sequence is the
convolutional inverse of all the one sequence: 1, -1, 0, 0, ... -
Tanya Khovanova (tanyakh(AT)yahoo.com), Jul 14 2007
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%D A019590 B. Cipra, What's Happening in the Mathematical Sciences 1995-6 Vol. 3,
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%D A019590 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public.
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%D A019590 Y. Hellegouarch, "Fermat Vaincu", Quadrature No. 22 pp. 37-55 Editions
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%D A019590 Y. Hellegouarch, "Fermat enfin demontre", Pour la Science, No. 220, 1996
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%D A019590 Y. Hellegouarch, Invitation to the Mathematics of Fermat-Wiles, Academic
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%D A019590 P. Hoffman, The Man Who Loved Only Numbers, pp. 183-200 Hyperion NY 1998
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%D A019590 C. J. Mozzochi, The Fermat Proof, New Bern NC 2004.
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RI 1995.
%D A019590 P. Odifreddi, The Mathematical Century, Chapter 2.14 " Number Theory
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%D A019590 I. Peterson, The Mathematical Tourist, pp. 234-8 W.H.Freeman/Owl Book
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%D A019590 I. Peterson, "A Marginal Note" in Islands of Truths, pp. 280-5 W.H.Freeman
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%D A019590 A. van der Poorten, Notes on Fermat's Last Theorem, Wiley NY 1996
%D A019590 J. Propp, Who Proved Fermat's Last Theorem? Princeton Univ. Press NJ
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%D A019590 P. Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, 1979.
%D A019590 P. Ribenbiom, Fermat's Last Theorem for Amateurs, Springer Verlag NY
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%D A019590 S. Singh, Fermat's Enigma, Walker and Co. NY 1997
%D A019590 I. Stewart, "Fermat's Last Time-Trip", in Scientific American Nov.1993
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%D A019590 I. Stewart, From Here To Infinity, Chapter 3 "Marginal Interest" pp.
25-48 OUP Oxford 1996
%D A019590 I. Stewart and D. Tall, Algebraic Number Theory and Fermat's Last Theorem,
A. K. Peters Natick MA 2001.
%D A019590 G. R. Talbott, Fermat's Last Theorem, Lotus Press WI 1991.
%D A019590 R. Van Vo, Fermat's Last Theorem, AuthorHouse, Bloomington IN 2002.
%D A019590 J. Vigouroux et al., Une aventure mathematique, le theoreme de Fermat,
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%D A019590 C. Goldstein, "Le Theoreme de Fermat", La Recherche, Vol. 25 March 1994,
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%D A019590 C. Goldstein, "Le Theoreme De Fermat Enfin Demontre", Chapter IX pp.
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%D A019590 R. Schoof, "Wiles' proof of the Taniyama-Weil conjecture for semi-stable
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%H A019590 Amer. Math. Soc., <a href="http://www.ams.org/new-in-math/fermat.html">
Fermat's Last Theorem</a>
%H A019590 Anonymous, <a href="http://www.bath.ac.uk/~ma0kvb">Fermat's Last Theorem</
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%H A019590 A. J. Bailey, <a href="http://www.adrianbailey.co.uk/fermat/flt.html">
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%H A019590 T. Ballard, <a href="http://www.fermatproof.com">A Short Form Proof Of
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%H A019590 N. Barker, <a href="http://people.bath.ac.uk/ma1nlb/MATH0126.html">Fermat's
Last Theorem</a>
%H A019590 K. Belabas and C. Goldstein, <a href="http://www.math.u-bordeaux.fr/~belabas/
Orsay-info/fermat.html">Fermat et son Theoreme</a>
%H A019590 N. Boston, <a href="http://www.math.wisc.edu/~boston/869.pdf">The Proof
Of Fermat's Last Theorem</a>
%H A019590 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
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%H A019590 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
page.php?sort=FermatsLastTheorem">Fermat's last theorem</a>
%H A019590 A. Camus College Team, <a href="http://www.col-camus-soufflenheim.ac-strasbourg.fr/
Page.php?IDP=340">La conjecture de Fermat</a>
%H A019590 C. C. Chang, <a href="http://www.cs.wichita.edu/~chang/fermat.html">BBC's
Horizon Program: "Fermat's Last Theorem"</a>
%H A019590 C. S. Chinea, <a href="http://personales.ya.com/casanchi/mat/conjeturafermat.pdf">
Sobre La Conjetura de Fermat(Text in Italian)</a>
%H A019590 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 A019590 K. Choi, <a href="http://www.public.iastate.edu/~kchoi/time.htm">Timeline
of FLT</a>
%H A019590 K. Conrad, <a href="http://www.math.uconn.edu/~kconrad/blurbs/fltreg.pdf">
Fermat's Last Theorem For Regular Primes</a>
%H A019590 J. Constant, <a href="http://www.coolissues.com/mathematics/Fermat/fermat">
Proof of Fermat's Last Theorem</a>
%H A019590 P. Daley, <a href="http://people.bath.ac.uk/pid20/fermat.html">Fermat's
Last Theorem</a>
%H A019590 C. Daney, <a href="http://aux.planetmath.org/files/lec/4/flt.pdf">The
Proof of Fermat's Last Theorem</a>
%H A019590 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 A019590 H. Darmon, <a href="http://citeseer.ist.psu.edu/correct/474804">The Shimura-Taniyama
conjecture (d'apres Wiles)</a>
%H A019590 R. Ellman, <a href="http://www.the-origin.org/fermat.pdf">A Concise And
Direct Proof of "Fermat's Last Theorem"</a>
%H A019590 K. M. Evans, <a href="http://geocities.com/kerryme47714">In Defense of
Mr. Fermat</a>
%H A019590 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 A019590 Larry Freeman's Blog Spot, <a href="http://fermatslasttheorem.blogspot.com/
2005_04_24_archive.html">Fermat's Last Theorem</a>
%H A019590 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">Sequences realized as Parker vectors ...</
a>, J. Integer Seqs., Vol. 6, 2003.
%H A019590 C. Gorin, <a href="http://people.bath.ac.uk/ma3clg/Fermat's%20Last%20Theorem.html">
Fermat's Last Theorem</a>
%H A019590 F. Q. Gouvea, <a href="http://www.americanscientist.org/template/BookReviewTypeDetail/
assetid/14359?&print=yes">Wiles's Proof, 1993-1995 : Review of "The
Fermat Diary" by C. J. Mozzochi</a>
%H A019590 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 A019590 D. Graves, <a href="http://www.maa.org/reviews/fltamat.html">Review of
"Fermat's Last Theorem for Amateurs" by P.Ribenboim</a>
%H A019590 W. F. Hammond, <a href="http://www.albany.edu/~hammond/gellmu/examples/
f356g.pdf">Fermat's Last Theorem after 356 years</a>
%H A019590 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 A019590 T. Hausberger, <a href="http://www.math.univ-montp2.fr/~hausberg/ens/
licence/fermat/index.html">Autour du dernier Theoreme de Fermat</
a>
%H A019590 S. Horsler, <a href="http://people.bath.ac.uk/ma3sah">Home Page on FLT</
a>
%H A019590 A. Jackson, <a href="http://www.ams.org/notices/199710/comm-fermat.pdf">
Review of "Fermat's Enigma" by S.Singh</a>
%H A019590 M. Jaiclin, <a href="http://www.brockport.edu/math/mathclub/fermat.talk.pdf">
What Makes Fermat's Last Theorem So Hard, Anyway?</a>
%H A019590 N. de Jong, <a href="http://www.geocities.com/elementaryfermat">Fermat's
Last Theorem. Was Wiles's proof really first?</a>
%H A019590 D. Kaminski, <a href="http://www.cs.uleth.ca/~kaminski/esferm03.html">
The extraordinary story of Fermat's Last Theorem</a>
%H A019590 A. Krowne, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
FermatsLastTheorem.html">Fermat's last theorem</a>
%H A019590 S. Lang, <a href="http://www.ams.org/notices/199511/forum.pdf">Some History
of the Shimura-Taniyama Conjecture</a>
%H A019590 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
On generalizations of Stirling number triangles</a>, J. Integer Seqs.,
Vol. 3 (2000), #00.2.4.
%H A019590 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 A019590 R. Lisker, <a href="http://rendezvous.com/tangledweb/conferences/fermat">
Fermat's Last Theorem (Conference Report)</a>
%H A019590 A. Lopez-Ortiz, <a href="http://www.math.uncc.edu/flt.php">What We Know
About Fermat's Last Theorem</a>
%H A019590 E. R. Lucier, <a href="http://arXiv.org/abs/math/0502245">Fermat's Last
Theorem,Further Study</a>
%H A019590 L. Lusk, <a href="http://mathematics.gulfcoast.edu/mgf1107/newpage11.htm">
A Brief History of Fermat's Last Theorem</a>
%H A019590 M. Marcionelli, <a href="http://www.math.ethz.ch/~michele/AZ/fr/ind_Fermat.html">
La conjecture de Fermat</a>
%H A019590 Mathematical Database, Poster, <a href="http://mathdb.org/gallery/poster/
description/e_poster_02.htm">Fermat's Last Theorem</a>
%H A019590 Mathforum, <a href="http://mathforum.org/dr.math/faq/faq.fermat.html">
Fermat's Last Theorem</a>
%H A019590 G. Mazzarino, <a href="http://www.economics.ox.ac.uk/Members/giuseppe.mazzarino/
Fermat_March_2003_abstract.htm">Fermat's Last Theorem</a>
%H A019590 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 A019590 P. L. Mills, <a href="http://members.lycos.co.uk/comms1/fidn1/fermat.html">
Fermat's Last Theorem-A New Proof</a>
%H A019590 J. S. Milne, <a href="http://www.jmilne.org/math/CourseNotes/math679.html">
Elliptic Curves</a>
%H A019590 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 A019590 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 A019590 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 A019590 I. Peterson, <a href="http://www.maa.org/mathland/mathtrek_11_22_99.html">
Curving Beyond Fermat</a>
%H A019590 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 A019590 A. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/alfpapers/
a142e.pdf">Fermat's Last Theorem</a>
%H A019590 J. Propp, <a href="http://www.math.wisc.edu/~propp/courses/491/articles">
Bibliography of articles on FLT</a>
%H A019590 S. Rees, <a href="http://www.bath.ac.uk/~ma1sr/project.html">Fermat's
Last Theorem</a>
%H A019590 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 A019590 K. Rubin, <a href="http://www.math.ohio-state.edu/history/flt/old.fltslides">
The Solving of Fermat's Last Theorem (Slides)</a>
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346535">A Report on Wiles' Cambridge Lectures</a>
%H A019590 K. Rubin and A. Silverberg, <a href="http://arXiv.org/abs/math.NT/9407220">
A report on Wiles' Cambridge lectures</a>
%H A019590 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 A019590 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>
%H A019590 School of Mathematics and Statistics, University of St Andrews, <a href="http:/
/www-gap.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html">
Fermat's last theorem</a>.
%H A019590 P. Schorer, <a href="http://www.occampress.com/fermat.pdf">Is There a
"Simple" Proof of Fermat's Last Theorem</a>
%H A019590 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 A019590 D. Shay, <a href="http://fermat.workjoke.com">Fermat's Last Theorem</
a>
%H A019590 Kaida Shi, <a href="http://uk.arXiv.org/ftp/math/papers/0309/0309005.pdf">
The n-dimensional Cube--A New Way to Prove the Fermat's Last Theorem</
a>
%H A019590 S. Singh, <a href="http://www.simonsingh.com/Fermat_Corner.html">Fermat
Corner</a>
%H A019590 S. Singh, <a href="http://www.prometheus.demon.co.uk/01/01fermat.htm">
Fermat's Last Theorem</a>
%H A019590 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 A019590 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 A019590 S. Stedman, <a href="http://students.bath.ac.uk/ma0sjbs/fermat.html">
Fermat's Last Theorem</a>
%H A019590 W. A. Stein, <a href="http://modular.fas.harvard.edu/edu/Fall2001/124/
lectures/lecture32/lecture32">Fermat's Last Theorem and Modularity
of Elliptic Curves</a>
%H A019590 D. Surendran, <a href="http://www.uz.ac.zw/science/maths/zimaths/flt.htm">
Fermat's Last Theorem</a>
%H A019590 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 A019590 C. Thornhill, <a href="http://www.missouri.edu/~cst398/fermat/fermat1.htm">
Fermat's Last Theorem</a>
%H A019590 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 A019590 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/
Addition/ThFermat.htm">Theoreme de Fermat-Wiles</a>
%H A019590 A. H. Warren, <a href="http://www.memagazine.org/contents/current/webonly/
fermfinal310.pdf">Proof Of The Fermat's Last Theorem</a>
%H A019590 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fermat's_last_theorem">
Fermat's Last Theorem</a>
%H A019590 A. J. Wiles, <a href="http://modular.fas.harvard.edu/edu/Spring2004/129/
references/flt/flt.pdf">Modular elliptic curves and Fermat's last
theorem</a>, Ann. Math. 141 (1995), 443-551.
%H A019590 A. J. Wiles, <a href="http://modular.fas.harvard.edu/edu/Spring2003/21n/
papers/Wiles,Modular_Elliptic_Curves_and_Fermats_Last_Theorem.pdf">
Modular elliptic curves and Fermat's Last Theorem</a>
%H A019590 T. Yee, <a href="http://ihome.ust.hk/~tonyee/flt.htm">The Fermat's Last
Theorem(FLT)</a>
%H A019590 YouTube, <a href="http://www.youtube.com/results?search_query=Fermat%27s+Last+Theorem">
BBC's Horizon Program : "Fermat's Last Theorem" in 5 parts</a>
%F A019590 a(n)=(-1)^n*sum{k=0..floor(n/2), (-1)^A010060(n-2k)mod(C(n, 2k), 2)}
- Paul Barry (pbarry(AT)wit.ie), Jan 03 2005
%F A019590 a(n)=1-[(n+2) mod (n+1)]+[n!^2 mod (n+1)]*[(n+1)!^2 mod (n+2)]; - Paolo
P. Lava (ppl(AT)spl.at), Aug 29 2007
%F A019590 a(n)=[(n-1)! mod 2], with n>=1 - Paolo P. Lava (ppl(AT)spl.at), Feb 15
2008
%F A019590 a(n+1)=(1/2)*[1+(-1)^n]*a(n), with a(0)=1 - Paolo P. Lava (ppl(AT)spl.at),
Apr 16 2008
%F A019590 Euler transform of length 2 sequence [ 1, -1]. - Michael Somos Jul 05
2009
%F A019590 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 A019590 G.f.: x + x^2 = x * (1 - x^2) / (1 - x). - Michael Somos Jul 05 2009
%F A019590 Dirichlet g.f.: 1 + 2^(-s). - Michael Somos Jul 05 2009
%o A019590 (PARI) {a(n) = (n==1) + (n==2)} /* Michael Somos Jul 05 2009 */
%Y A019590 Cf. A000004, A000007, A010051, A012450.
%Y A019590 INVERT transform gives Fibonacci numbers, A000045.
%Y A019590 Convolution inverse of A062157. Dirichlet convolution inverse of A154269.
%Y A019590 Sequence in context: A105812 A134323 A060576 this_sequence A154955 A014040
A014071
%Y A019590 Adjacent sequences: A019587 A019588 A019589 this_sequence A019591 A019592
A019593
%K A019590 nonn,nice,easy,mult
%O A019590 1,1
%A A019590 N. J. A. Sloane (njas(AT)research.att.com).
%E A019590 Additional links provided by Lekraj Beedassy (blekraj(AT)yahoo.com),
Nov 20 2003
|
