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Fermat's Last Theorem (FLT) Pierre De Fermat​(1601-1665, France) In Fermat’s Last Theorem (FLT) Pierre de Fermat (1601-1665, France) ​ In 1623, Fermat studied law at the University of Orleans and graduated in 1626. He moved to the city of Bordeaux where he became an attorney in the high courts. Mathematics was his hobby where eventually he founded modern number theory and made advancements in areas such as probability theory, infinitesimal calculus, analytic geometry, and optics. His contributions include Fermat numbers and Fermat primes, Fermat's principle, Fermat's Little Theorem, and Fermat's Last Theorem. Fermat’s Last Theorem n n n Fermat observed , if x ​ +y ​ = z where​ n>2 has any whole number solutions. He ​ ​ ​ believed that there were no such solutions and also believed that he found proof that showed that there were no whole number solutions. He hinted inside a book he was reading that he found the proof. His son, Clement-Samuel re-discovered this book and published a new version of it with all of Fermat’s little notes printed in the text, but there was no sufficient proof for his conjecture that there is no solution for FLT . The mathematics required to prove FLT did not exist until the second half of the twentieth century. Fermat might have had another way of proving it but, it seems unlikely. We have many different ways of proving that there is infinitely many whole number solutions for 2 2 2 2 2 x ​ +y ​ = z where​ the positive integers x,y and z are exactly those of the form: x=r(s ​ - t ),​ y=2rst ​ ​ 2​ 2 ​ ​ and z=r(s ​ + t ).​ If we attempt to repeat the same strategy as proving the case for n=2, for n>2 we ​ 2 ​ 2 2 find that y ​ = z -​ x ​ does not split into linear factors over the integers (we need also a primitive nth ​ ​ ​ root of unity). In the proof for n=2 we use some facts from number theory over the integers (Fundamental Theorem of Arithmetic, concepts of "common factors" and "primes") which might not translate to nth root of unity in the factorization for n>2, and this problem motivates the study of number theory in such a ring. Subsequent Developments and Solutions Fermat proved the case n=4 which is sufficient to establish that if the theorem is false for some exponent n that is not a prime number, it must also be false for some smaller n. Hence, only prime values of n need further investigation. In 1637-1839 the conjecture was proved for primes: 3, 5, and 7. During this time, Sophie Gemain proved an approach relevant to an entire class of primes. Later in mid 19th century, Ernst Kummer proved the theorem for all regular primes, leaving irregular primes to be analyzed individually and around 1955, Goro Shimura and Yutaka Taniyama suspected linkage between elliptic curves and modular forms, known as the Tainyama- Shimura conjecture. At the time, it was unsolved and had no connection to FLT. Later in 1984 Gernard Frey noticed a link between these two unsolved and unrelated problems. In 1986 Ken Ribet accomplished a full proof that these two problems were closely linked by building on a partial proof of Jean-Pierre Serre. 1993: Andrew Wiles, professor at Cambridge University succeeded in proving enough of the conjecture to prove FLT. However, a flaw was discovered and further investigation was needed. At last in 1995 The final proof accompanied by a smaller joint paper showing the fixed steps were valid. Andrew Wiles honoured by 2016 Abel Prize What was Fermat’s Last Theorem useful for? This opened up a whole new area of mathematics and number theory. Also it is allowing us to further understand new equations. The world of Cryptography depends on the mathematics discovered in the journey to solve FLC. References “Andrew Wiles.” Wikipedia, Wikimedia Foundation, 17 Dec. 2019, en.wikipedia.org/wiki/Andrew_Wiles. ​ ​ Boyer, Carl B. “Pierre De Fermat.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., 23 Jan. 2020, ​ ​ www.britannica.com/biography/Pierre-de-Fermat. ​ En.wikipedia.org. 2020. Pierre De Fermat. [online] Available at: ​ ​ <https://en.wikipedia.org/wiki/Pierre_de_Fermat#/media/File:Pierre_de_Fermat.jpg> [Accessed 10 March 2020]. “Fermat's Last Theorem.” Wikipedia, Wikimedia Foundation, 22 Feb. 2020, ​ ​ en.wikipedia.org/wiki/Fermat's_Last_Theorem. “Fermat's Last Theorem Proof Secures Mathematics' Top Prize for Sir Andrew Wiles.” University of Oxford, ​ ​ www.ox.ac.uk/news/2016-03-15-fermats-last-theorem-proof-secures-mathematics-top-prize-sir-andrew-wiles. ​ Fermat Portraits, mathshistory.st-andrews.ac.uk/PictDisplay/Fermat.html. ​ “Home.” Famous Scientists, www.famousscientists.org/pierre-de-fermat/. ​ ​ ​ ​ Klinger, Gary. “Fermat’s Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem.” The Mathematics teacher. 90.6 (1997): n. pag. Web. “Pierre De Fermat.” Pierre Fermat (1601 - 1665), mathshistory.st-andrews.ac.uk/Biographies/Fermat.html. ​ ​ Poorten, Alf van der. Notes on Fermat's Last Theorem. Wiley, 1996. ​ ​ Radford, Tim. “Fermat's Last Theorem by Simon Singh – Review | Tim Radford.” The Guardian, Guardian News and ​ ​ Media, 2 Aug. 2013, www.theguardian.com/science/2013/aug/02/fermats-last-theorem-simon-singh-review. Ribenboim, Paulo. Fermat's Last Theorem for Amateurs. Springer, 2000. ​ ​ Study.com. 2020. Pierre De Fermat: Contributions To Math & Accomplishments. [online] Available at: ​ ​ <https://study.com/academy/lesson/pierre-de-fermat-contributions-to-math-accomplishments.html> [Accessed 10 March 2020]. YouTube. 2020. Fermat’s Last Theorem | A Mathematical Mystery. [online] Available at: ​ ​ <https://www.youtube.com/watch?v=KDpf70xguCM> [Accessed 10 March 2020]. YouTube. 2020. Fermat’s Last Theorem Numberphile. [online] Available at: ​ ​ <https://www.youtube.com/watch?v=qiNcEguuFSA> [Accessed 10 March 2020]. YouTube. 2020. What Is Fermat's Last Theorem?. [online] Available at: ​ ​ <https://www.youtube.com/watch?v=1BSFyEIY2BY> [Accessed 10 March 2020]. Lunar Libration (Group 6) March 10th, 2020 Aristotelian Views (Wrong): • Planets orbited in perfect circles • Only 50% of the Moon is visible What is Lunar Libration • “Waggling” of the Moon (the combination of the moon nodding “yes” and “no” to the Earth throughout the course of a lunar cycle) • Allows us to see 59% of her William Gilbert (1540 - 1603): • English Physicist and Natural Philosopher • Drew out various maps of the moon in Seleniography, 1600 Galileo Galilei (1564 - 1642): • Credited as the first person to mention the movement of the moon • This was contradictory to what Aristotle said since Aristotle was a clown Joseph-Louis Lagrange (1736 - 1813): • French Mathematician Born in Turin, Italy • Won The Académie des Sciences in Paris in 1764 for explaining lunar libration • Also won for explaining the moons of Jupiter and creating the new Lagrange Points 1774, 1778 Three Causes of Lunar Libration • Change in Longitude (180 degrees becomes 188 degrees) o Rotation is ahead of orbit when moon is moving farther o Opposite when moon is moving closer • Change in Latitude (7% more) o Moon’s orbit around Earth is not parallel to Earth’s orbit around the sun o We can see past the north pole when the moon is “lower” than Earth, and past the south pole when the moon is “higher” than Earth • Parallax based on position on the Earth (Almost no addition) Lagrange Points • Points at which a third, small body amongst two larger bodies, would remain almost stable between the gravitational pull of the two large bodies • First three points (L1, L2, L3) discovered by Euler o Collinear with two large bodies • Points L4 and L5 discovered by Lagrange o Each point forms a separate equilateral triangle with each body, including either L4 or L5, at the vertices • Lagrange wrote “Essay on the three-body problem” in 1772 and won the Académie des Sciences again because of it (shared with Euler) “Joseph-Louis Lagrange.” Joseph-Louis Lagrange (1736 - 1813), JOC/EFR, mathshistory.st- andrews.ac.uk/Biographies/Lagrange.html. Galilei, Galileo. Dialogue Concerning the Two Chief World Systems - Ptolemaic & Copernican. Translated by Stillman Drake, Forward by Albert Einstein University of California Press, 1967. Gilbert, William. “On the Load Stone and Magnetic Bodies.” Internet Archive, [New York, J. Wiley & Sons, archive.org/details/williamgilbertof00gilbrich/page/xx/mode/2up. Lagrange, Joseph-Louis. “Recherche Sur La Libration De La Lune.” Oeuvres De Lagrange, 1873, pp. 8–65. Galica, https://gallica.bnf.fr/ark:/12148/bpt6k229225j/f10.image. Ptolemaeus, Claudius, et al. Ptolemys Almagest. Translated by G. J. Toomer Springer-Verlag, 1984. Pumfrey, Stephen. “The Seleographia of William Gilbert: His Pre-Telescopic Map of the Moon And His Discovery of Lunar Libration.” SagePub, 2011, pp. 193–203., https://journals-sagepub- com.ezproxy.library.yorku.ca/doi/pdf/10.1177/002182861104200205. Young, Charles A. Manual of Astronomy. The Anthenaem Press, 1902. Group #7: Baker & Venkatesh Alan Baker ● Born August 19 1939 in London England ● Died February 4 2018 in Cambridge England ● Fellow at Trinity College from 1964 until his death ● 1970 Fields Medal at the age of 31 ○ Solved Hilbert’s 7th problem → generalized the Gelfond–Schneider theorem ​ ​ ■ If a =/ 0, 1 and b = i then ab is a transcendental number ■ Baker’s Theorem is an extension to what Gelfond-Schneider did ■ Construction of the function and finding it’s zeroes ● 1972 Adam’s Prize ● 1974 Professor of pure math at Cambridge until 2006 when he became emeritus ● 2012 became a fellow at the American Mathematical Society Akshay Venkatesh ● Born: November 21, 1981 in Delhi, India ● Australian mathematician → moved to Australia at age 2 ● Started
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