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A Look at the ABC Conjecture Via Elliptic Curves
A Look at the ABC Conjecture via Elliptic Curves Nicole Cleary Brittany DiPietro Alexander Hill Gerard D.Koffi Beihua Yan Abstract We study the connection between elliptic curves and ABC triples. Two important results are proved. The first gives a method for finding new ABC triples. The second result states conditions under which the power of the new ABC triple increases or decreases. Finally, we present two algorithms stemming from these two results. 1 Introduction The ABC conjecture is a central open problem in number theory. It was formu- lated in 1985 by Joseph Oesterl´eand David Masser, who worked separately but eventually proposed equivalent conjectures. Like many other problems in number theory, the ABC conjecture can be stated in relatively simple, understandable terms. However, there are several profound implications of the ABC conjecture. Fermat's Last Theorem is one such implication which we will explore in the second section of this paper. 1.1 Statement of The ABC Conjecture Before stating the ABC conjecture, we define a few terms that are used frequently throughout this paper. Definition 1.1.1 The radical of a positive integer n, denoted rad(n), is defined as the product of the distinct prime factors of n. That is, rad(n) = Q p where p is prime. pjn 1 Example 1.1.2 Let n = 72 = 23 · 32: Then rad(n) = rad(72) = rad(23 · 32) = 2 · 3 = 6: Definition 1.1.3 Let A; B; C 2 Z. A triple (A; B; C) is called an ABC triple if A + B = C and gcd(A; B; C) = 1. -
Remembrance of Alan Baker
Hardy-Ramanujan Journal 42 (2019), 4-11Hardy-Ramanujan 4-11 Remembrance of Alan Baker Gisbert W¨ustholz REMEMBRANCE OF ALAN BAKER It was at the conference on Diophantische Approximationen organized by Theodor Schneider by which took place at Oberwolfach Juli 14 - Juli 20 1974 when I first met Alan Baker. At the time I was together with Hans Peter SchlickeweiGISBERT a graduate WUSTHOLZ¨ student of Schneider at the Albert Ludwigs University at Freiburg i. Br.. We both were invited to the conference to help Schneider organize the conference in practical matters. In particular we were determined to make the speakers write down an abstract of their talks in the Vortagsbuch which was considered to be an important historical document for the future. Indeed if you open the webpage of the Mathematisches Forschungsinstitut Oberwolfach you find a link to the Oberwolfach Digital Archive where you can find all these Documents It was at the conference on Diophantische Approximationen organized by Theodor Schneider which took place at fromOberwofach 1944-2008. Juli 14 - Juli 20 1974 when I first met Alan Baker. At the time I was together with Hans Peter Schlickewei a graduateAt this student conference of Schneider I was at verythe Albert-Ludwig-University much impressed and at Freiburg touched i. Br.. by We seeing both were the invitedFields to Medallist the conference Baker whoseto help work Schneider I had to studied organize the in conference the past in year practical to matters.do the Inp-adic particular analogue we were determined of Baker's to make recent the versions speakers A write down an abstract of their talks in the Vortagsbuch which was considered to be an important historical document for sharpeningthe future. -
Program of the Sessions San Diego, California, January 9–12, 2013
Program of the Sessions San Diego, California, January 9–12, 2013 AMS Short Course on Random Matrices, Part Monday, January 7 I MAA Short Course on Conceptual Climate Models, Part I 9:00 AM –3:45PM Room 4, Upper Level, San Diego Convention Center 8:30 AM –5:30PM Room 5B, Upper Level, San Diego Convention Center Organizer: Van Vu,YaleUniversity Organizers: Esther Widiasih,University of Arizona 8:00AM Registration outside Room 5A, SDCC Mary Lou Zeeman,Bowdoin upper level. College 9:00AM Random Matrices: The Universality James Walsh, Oberlin (5) phenomenon for Wigner ensemble. College Preliminary report. 7:30AM Registration outside Room 5A, SDCC Terence Tao, University of California Los upper level. Angles 8:30AM Zero-dimensional energy balance models. 10:45AM Universality of random matrices and (1) Hans Kaper, Georgetown University (6) Dyson Brownian Motion. Preliminary 10:30AM Hands-on Session: Dynamics of energy report. (2) balance models, I. Laszlo Erdos, LMU, Munich Anna Barry*, Institute for Math and Its Applications, and Samantha 2:30PM Free probability and Random matrices. Oestreicher*, University of Minnesota (7) Preliminary report. Alice Guionnet, Massachusetts Institute 2:00PM One-dimensional energy balance models. of Technology (3) Hans Kaper, Georgetown University 4:00PM Hands-on Session: Dynamics of energy NSF-EHR Grant Proposal Writing Workshop (4) balance models, II. Anna Barry*, Institute for Math and Its Applications, and Samantha 3:00 PM –6:00PM Marina Ballroom Oestreicher*, University of Minnesota F, 3rd Floor, Marriott The time limit for each AMS contributed paper in the sessions meeting will be found in Volume 34, Issue 1 of Abstracts is ten minutes. -
The Abc Conjecture
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 8-2002 The abc conjecture Jeffrey Paul Wheeler University of Tennessee Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Recommended Citation Wheeler, Jeffrey Paul, "The abc conjecture. " Master's Thesis, University of Tennessee, 2002. https://trace.tennessee.edu/utk_gradthes/6013 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Jeffrey Paul Wheeler entitled "The abc conjecture." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Science, with a major in Mathematics. Pavlos Tzermias, Major Professor We have read this thesis and recommend its acceptance: Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a thesis written by Jeffrey Paul Wheeler entitled "The abc Conjecture." I have examined the finalpaper copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Mathematics. Pavlas Tzermias, Major Professor We have read this thesis and recommend its acceptance: 6.<&, ML 7 Acceptance for the Council: The abc Conjecture A Thesis Presented for the Master of Science Degree The University of Tenne�ee, Knoxville Jeffrey Paul Wheeler August 2002 '"' 1he ,) \ � �00� . -
Elliptic Curves and the Abc Conjecture
Elliptic Curves and the abc Conjecture Anton Hilado University of Vermont October 16, 2018 Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 1 / 37 Overview 1 The abc conjecture 2 Elliptic Curves 3 Reduction of Elliptic Curves and Important Quantities Associated to Elliptic Curves 4 Szpiro's Conjecture Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 2 / 37 The Radical Definition The radical rad(N) of an integer N is the product of all distinct primes dividing N Y rad(N) = p: pjN Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 3 / 37 The Radical - An Example rad(100) = rad(22 · 52) = 2 · 5 = 10 Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 4 / 37 The abc Conjecture Conjecture (Oesterle-Masser) Let > 0 be a positive real number. Then there is a constant C() such that, for any triple a; b; c of coprime positive integers with a + b = c, the inequality c ≤ C() rad(abc)1+ holds. Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 5 / 37 The abc Conjecture - An Example 210 + 310 = 13 · 4621 Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 6 / 37 Fermat's Last Theorem There are no integers satisfying xn + y n = zn and xyz 6= 0 for n > 2. Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 7 / 37 Fermat's Last Theorem - History n = 4 by Fermat (1670) n = 3 by Euler (1770 - gap in the proof), Kausler (1802), Legendre (1823) n = 5 by Dirichlet (1825) Full proof proceeded in several stages: Taniyama-Shimura-Weil (1955) Hellegouarch (1976) Frey (1984) Serre (1987) Ribet (1986/1990) Wiles (1994) Wiles-Taylor (1995) Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 8 / 37 The abc Conjecture Implies (Asymptotic) Fermat's Last Theorem Assume the abc conjecture is true and suppose x, y, and z are three coprime positive integers satisfying xn + y n = zn: Let a = xn, b = y n, c = zn, and take = 1. -
Fermat's Last Theorem
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have an infinite number of solutions.[1] The proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica; Fermat added that he had a proof that was too large to fit in the margin. However, there were first doubts about it since the publication was done by his son without his consent, after Fermat's death.[2] After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995; it was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016.[3] It also proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.[4] Contents The 1670 edition of Diophantus's Arithmetica includes Fermat's Overview commentary, referred to as his "Last Pythagorean origins Theorem" (Observatio Domini Petri Subsequent developments and solution de Fermat), posthumously published Equivalent statements of the theorem by his son. -
Fall 2008 [Pdf]
Le Bulletin du CRM • crm.math.ca • Automne/Fall 2008 | Volume 14 – No 2 | Le Centre de recherches mathématiques The Fall 2008 Aisenstadt Chairs Four Aisenstadt Chair lecturers will visit the CRM during the 2008-2009 thematic year “Probabilistic Methods in Mathemat- ical Physics.” We report here on the series of lectures of Wendelin Werner (Université Paris-Sud 11) and Andrei Okounkov (Princeton University), both of whom are Fields Medalists, who visited the CRM in August and September 2008 respectively. The other Aisenstadt Chairs will be held by Svante Janson (Uppsala University) and Craig Tracy (University of California at Davis). Wendelin Werner Andrei Okounkov de Yvan Saint-Aubin (Université de Montréal) by John Harnad (Concordia University) Wendelin Werner est un A metaphor from an ancient fragment by Archilochus “The Fox spécialiste de la théo- knows many things but the Hedgehog knows one big thing” rie des probabilités. Il a was used by Isaiah Berlin as title and theme of his essay on Tol- obtenu son doctorat en stoy’s view of history (“. by nature a fox, but believed in be- 1993 sous la direction de ing a hedgehog”) [1]. It is also very suitably applied to styles in Jean-François Le Gall. Il science. In his presentation of the work of Andrei Okounkov est professeur au labo- when he was awarded the Fields medal at the 2006 Interna- ratoire de mathématiques tional Congress of Mathematicians in Madrid, Giovanni Felder à l’Université Paris-Sud said: “Andrei Okounkov’s initial area of research was group XI à Orsay depuis 1997, representation theory, with particular emphasis on combinato- ainsi qu’à l’École nor- rial and asymptotic aspects. -
On a Problem Related to the ABC Conjecture
On a Problem Related to the ABC Conjecture Daniel M. Kane Department of Mathematics / Department of Computer Science and Engineering University of California, San Diego [email protected] October 18th, 2014 D. Kane (UCSD) ABC Problem October 2014 1 / 27 The ABC Conjecture For integer n 6= 0, let the radical of n be Y Rad(n) := p: pjn For example, Rad(12) = 2 · 3 = 6: The ABC-Conjecture of Masser and Oesterl´estates that Conjecture For any > 0, there are only finitely many triples of relatively prime integers A; B; C so that A + B + C = 0 and max(jAj; jBj; jCj) > Rad(ABC)1+: D. Kane (UCSD) ABC Problem October 2014 2 / 27 Applications The ABC Conjecture unifies several important results and conjectures in number theory Provides new proof of Roth's Theorem (with effective bounds if ABC can be made effective) Proves Fermat's Last Theorem for all sufficiently large exponents Implies that for every irreducible, integer polynomial f , f (n) is square-free for a constant fraction of n unless for some prime p, p2jf (n) for all n A uniform version of ABC over number fields implies that the Dirichlet L-functions do not have Siegel zeroes. D. Kane (UCSD) ABC Problem October 2014 3 / 27 Problem History The function field version of the ABC conjecture follows from elementary algebraic geometry. In particular, if f ; g; h 2 F[t] are relatively prime and f + g + h = 0 then max(deg(f ); deg(g); deg(h)) ≥ deg(Rad(fgh)) − 1: The version for number fields though seems to be much more difficult. -
The Geometry Center Reaches
THE NEWSLETTER OF THE MA THEMA TICAL ASSOCIATION OF AMERICA The Geometry Center Reaches Out Volume IS, Number 1 A major mission of the NSF-sponsored University of Minnesota Geometry Center is to support, develop, and promote the communication of mathematics at all levels. Last year, the center increased its efforts to reach and to educate diffe rent and dive rse groups ofpeople about the beauty and utility of mathematics. Center members Harvey Keynes and Frederick J. Wicklin describe In this Issue some recent efforts to reach the general public, professional mathematicians, high school teach ers, talented youth, and underrepresented groups in mathematics. 3 MAA President's Column Museum Mathematics Just a few years ago, a trip to the local 7 Mathematics science museum resembled a visit to Awareness Week a taxidermy shop. The halls of the science museum displayed birds of 8 Highlights from prey, bears, cougars, and moose-all stiff, stuffed, mounted on pedestals, the Joint and accompanied by "Don't Touch" Mathematics signs. The exhibits conveyed to all Meetings visitors that science was rigid, bor ing, and hardly accessible to the gen 14 NewGRE eral public. Mathematical Fortunately times have changed. To Reasoning Test day even small science museums lit erally snap, crackle, and pop with The graphical interface to a museum exhibit that allows visitors to interactive demonstrations of the explore regular polyhedra and symmetries. 20 Letters to the physics of electricity, light, and sound. Editor Visitors are encouraged to pedal, pump, and very young children to adults, so it is accessible push their way through the exhibit hall. -
The Paradox of the Proof | Project Wordsworth
The Paradox of the Proof | Project Wordsworth Home | About the Authors If you enjoy this story, we ask that you consider paying for it. Please see the payment section below. The Paradox of the Proof By Caroline Chen MAY 9, 2013 n August 31, 2012, Japanese mathematician Shinichi Mochizuki posted four O papers on the Internet. The titles were inscrutable. The volume was daunting: 512 pages in total. The claim was audacious: he said he had proved the ABC Conjecture, a famed, beguilingly simple number theory problem that had stumped mathematicians for decades. Then Mochizuki walked away. He did not send his work to the Annals of Mathematics. Nor did he leave a message on any of the online forums frequented by mathematicians around the world. He just posted the papers, and waited. Two days later, Jordan Ellenberg, a math professor at the University of Wisconsin- Madison, received an email alert from Google Scholar, a service which scans the Internet looking for articles on topics he has specified. On September 2, Google Scholar sent him Mochizuki’s papers: You might be interested in this. http://projectwordsworth.com/the-paradox-of-the-proof/[10/03/2014 12:29:17] The Paradox of the Proof | Project Wordsworth “I was like, ‘Yes, Google, I am kind of interested in that!’” Ellenberg recalls. “I posted it on Facebook and on my blog, saying, ‘By the way, it seems like Mochizuki solved the ABC Conjecture.’” The Internet exploded. Within days, even the mainstream media had picked up on the story. “World’s Most Complex Mathematical Theory Cracked,” announced the Telegraph. -
NEWSLETTER No
NEWSLETTER No. 458 May 2016 NEXT DIRECTOR OF THE ISAAC NEWTON INSTITUTE In October 2016 David Abrahams will succeed John Toland as Director of the Isaac Newton Institute for Mathematical Sciences and NM Rothschild and Sons Professor of Mathematics in Cambridge. David, who is a Royal Society Wolfson Research Merit Award holder, has been Beyer Professor of Applied Mathematics at the University of Man- chester since 1998. From 2014-16 he was Scientific Director of the International Centre for Math- ematical Sciences in Edinburgh and was President of the Institute of Mathematics and its Applica- tions from 2007-2009. David’s research has been in the broad area of applied mathematics, mainly focused on the theoretical understanding of wave processes including scattering, diffraction, localisation and homogenisation. In recent years his research has broadened somewhat, to now cover topics as diverse as mathematical finance, nonlinear vis- He has also been involved in a range of public coelasticity and glaciology. He has close links with engagement activities over the years. He a number of industrial partners. regularly offers mathematics talks of interest David plays an active role within the internation- to school students and the general public, al mathematics community, having served on over and ran the annual Meet the Mathematicians 30 national and international working parties, outreach events for sixth form students with panels and committees over the past decade. Chris Howls (Southampton). With Chris Budd This has included as a Member of the Applied (Bath) he has organised a training conference Mathematics sub-panel for the 2008 Research As- in 2010 on How to Talk Maths in Public, and in sessment Exercise and Deputy Chair for the Math- 2014 co-chaired the inaugural Festival of Math- ematics sub-panel in the 2014 Research Excellence ematics and its Applications. -
Multiplicative Number Theory: the Pretentious Approach Andrew
Multiplicative number theory: The pretentious approach Andrew Granville K. Soundararajan To Marci and Waheeda c Andrew Granville, K. Soundararajan, 2014 3 Preface AG to work on: sort out / finalize? part 1. Sort out what we discuss about Halasz once the paper has been written. Ch3.3, 3.10 (Small gaps)and then all the Linnik stuff to be cleaned up; i.e. all of chapter 4. Sort out 5.6, 5.7 and chapter 6 ! Riemann's seminal 1860 memoir showed how questions on the distribution of prime numbers are more-or-less equivalent to questions on the distribution of zeros of the Riemann zeta function. This was the starting point for the beautiful theory which is at the heart of analytic number theory. Until now there has been no other coherent approach that was capable of addressing all of the central issues of analytic number theory. In this book we present the pretentious view of analytic number theory; allowing us to recover the basic results of prime number theory without use of zeros of the Riemann zeta-function and related L-functions, and to improve various results in the literature. This approach is certainly more flexible than the classical approach since it allows one to work on many questions for which L-function methods are not suited. However there is no beautiful explicit formula that promises to obtain the strongest believable results (which is the sort of thing one obtains from the Riemann zeta-function). So why pretentious? • It is an intellectual challenge to see how much of the classical theory one can reprove without recourse to the more subtle L-function methodology (For a long time, top experts had believed that it is impossible is prove the prime number theorem without an analysis of zeros of analytic continuations.