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: ​ ​ [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: ​ ​ [Accessed

10 March 2020].

YouTube. 2020. Fermat’s Last Theorem | A Mathematical Mystery. [online] Available at: ​ ​ [Accessed 10 March 2020].

YouTube. 2020. Fermat’s Last Theorem Numberphile. [online] Available at: ​ ​ [Accessed 10 March 2020].

YouTube. 2020. What Is Fermat's Last Theorem?. [online] Available at: ​ ​ [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 his studies at University of Western Australia at age 13 ○ Went straight into his second year & youngest ever student ○ Completed the 4 year program in just 3 years (graduated at age 16) ○ Youngest person to earn First Class Honours in pure math at UWA ● PhD at Princeton University (1998-2000)(finished it at age 20) ● Had a postdoc position at the Massachusetts Institute of Technology ● Became a full professor at Stanford University on September 1, 2008 ● Currently: permanent faculty member at Institute for Advanced Study (IAS) ● Many prizes/accomplishments such as: Salem Prize (2007), Packard Fellowship (2007), speaker at International Congress of Mathematicians (2010), Infosys Prize in Mathematical Sciences (2016), Ostrowski Prize (2017), Fields Medal (2018) ● Made significant contributions to many areas such as: analytic number theory, homogeneous dynamics, topology, representation theory

Evolution of Mathematics: Impacts on Mathematical “Discoveries” & Medal Winning ● “Discoveries” have went from being general to specific ● In the beginning: humans attempted to explain the world around them through basic math ● Math became represented by symbols, given structure and explained with precise definitions ● As mathematics has evolved, we see less “discoveries” and more “contributions” to math as well as its applications. This means: ➢ Medallists are not “reinventing the wheel” , but making enhancements to mathematics that already exists (e.g. Baker generalized Hilbert’s 7th problem) ➢ Medallists are applying their work to a combination of fields in mathematics and science. (e.g. Shing-Tung Yau’s Fields Medal (1982) involved his work on partial differential equations, physics (general relativity) etc. )

References

A Mathematical Chronology. (n.d.). Retrieved from http://mathshistory.st-andrews.ac.uk/Chronology/full.html

Alan Baker. (n.d.). Retrieved from http://mathshistory.st-andrews.ac.uk/Biographies/Baker_Alan.html

Aravind, I. (2018, August 4). Meet Akshay Venkatesh: The Delhi-born who got the Nobel Prize in mathematics. Retrieved from https://economictimes.indiatimes.com/news/science/akshay-venkatesh-a-new-star-is-born/a rticleshow/65273660.cms?from=mdr

Baker, A. (1975). Transcendental Number Theory (Cambridge Mathematical Library). ​ ​ Cambridge: Cambridge University Press. doi:10.1017/CBO9780511565977

Conrad, K. (n.d.). THE LOCAL-GLOBAL PRINCIPLE. Retrieved from https://kconrad.math.uconn.edu/blurbs/gradnumthy/localglobal.pdf

Ellenberg, J., & Venkatesh, A. (2008). Local-global principles for representations of quadratic forms. Inventiones Mathematicae, 171(2), 257–279. https://doi.org/10.1007/s00222-007-0077-7

InternationalMathematicalUnion. (n.d.). Retrieved from https://www.mathunion.org/imu-awards/fields-medal

The Development of Mathematics. (n.d.). Retrieved from https://mathscitech.org/articles/development-of-mathematics

Zhou, N. (2018, August 1). Australian Akshay Venkatesh wins Fields medal – the 'Nobel for maths'. Retrieved from https://www.theguardian.com/science/2018/aug/02/australian-akshay-venkatesh-wins-field s-medal-the-nobel-for-maths

GROUP 3 -- PRIME NUMBER THEOREM

The Prime Number Theorem (PNT) was conjectured in 1896 by Legendre, although Gauss claimed to be working on a similar idea around the same time: it says (x) ∼ x/log x , or (x) lim x = 1 for some counting function 휋(x) = the number of primes less than or equal to any x→∞ log x number x. These were approximations for how many primes existed between 1 and x. The first to prove the PNT were Jacques Hadamard and Charles-Jean de le Vallée Poussin, using the findings of other mathematicians (like Gauss, Chebyshev, and Dirichlet, to name a few).

n ⋍ dx Gauss himself found a better approximation: (x) ∫ log x ; the right side is li(x), or the 0 logarithmic integral of x, giving error estimates that de la Vallée Poussin explored. Earlier proofs of the PNT involved complex analysis, including the famed Riemann zeta function ∞ 1 (s) = ∑ ns that was introduced by Euler. Euler also proved, through unique factorization of n=1 positive integers, that (s) = ∏(1 − p−s)−1 as a product as well as a sum. Dirichlet extended p Gauss’ discoveries of prime factorization into a series of zeta function terms multiplied by characters, creating Dirichlet L functions. Chebyshev estimated the bounds of ​ ​ (x) .921 x/log x ≤ (x) ≤ 1.106 x/log x , and showed that lim x =1. Riemann had considered x→∞ log x both 휁 as a complex-valued function and 휋(x) as a complex integral involving 휁. Along with proving that 휁 was analytic on the complex plane with one singularity and expressing a complex number as s = 휎 + 휄t, Riemann also conjectured one of the most important unsolved problems in mathematics, the Riemann hypothesis: all non-trivial zeroes of 휁(s) lie on the line 1 Re(s)= 2 ; Hadamard and de la Vallée Poussin assumed this as true for their proofs. Both mathematicians would keep these above discoveries in mind for their proofs.

Hadamard first showed that 휁 was nonzero using a different variation of the zeta function log zeta, log (s) =− ∑log(1 − 1/ps) . In his 1896 paper “Arithmetic Consequences, p Hadamard used a cleaner form of the Mellin inversion formula, corresponding to Chebyshev’s work and one of Dirichlet’s series that satisfied the zeta function. de la Vallee Poussin also showed that 휁 had no zeroes with real part 1, and used a variant of a smoother Mellin inversion formula. He also published an article explaining his methods for solving the PNT with an error estimate (x) − li(x) = O(x exp(− c logx)) , which would be improved over time. D.J. Newman’s proof is currently one of the shortest proofs of the PNT, where the zeta function, Chebyshev’s formula, and holomorphicity were proven to be related to what he ∞ proposed as the ‘analytic theorem’ (involving the integral g(z) = ∫ f(t)eztdt with Re(z)>0). 0 Newman showed that the prime number theorem would follow.

Earlier proofs of the PNT were analytic, but there was an uprising of elementary proofs (did not require complex analysis and focused on arithmetic) in recent years. In 1949, both Paul Erdӧs and Atle Selberg both published an elementary proof of the PNT. Interestingly, this was not a joint effort and they both had independent publications. This led to a controversial dispute between the two mathematicians, the accounts of which have become unclear; despite this Selberg and Erdos were both recognized for their work and received the Fields Medal and the Cole Prize respectively. The results of PNT are two-fold: prime number theory became more popular as a mathematical strand, in the sense that more mathematicians were working on the subject. At the same time, it highlighted that collaboration between mathematicians could either be beneficial (without teamwork, the PNT would not have been solved) or detrimental (eg. the Selberg and Erdös controversy). GROUP 3 -- PRIME NUMBER THEOREM

Works Cited

Avigad, Jeremy, et al. “A Formally Verified Proof of the Prime Number Theorem.” ArXiv.org, 6 Apr. 2006, arxiv.org/abs/cs/0509025. ​ Bateman, Paul T., and Harold G. Diamond. “A Hundred Years of Prime Numbers.” The ​ American Mathematical Monthly, vol. 103, no. 9, 1996, pp. 729–741. JSTOR, ​ ​ ​ www.jstor.org/stable/2974443. Accessed 2 Mar. 2020. Bruckman, Paul S. “A Generalization of the Prime Number Theorem.” International Journal ​ of Mathematical Education in Science and Technology, vol. 39, no. 5, July 2008, pp. ​ 631–635, doi:10.1080/00207390701867455. Cartwright, Mary L. “Jacques Hadamard. 1865-1963.” Biographical Memoirs of Fellows of ​ the Royal Society, vol. 11, 1965, pp. 75–99. JSTOR, www.jstor.org/stable/769262. ​ ​ ​ Accessed 1 Mar. 2020. Goldfeld, Dorian. “The Elementary Proof of the Prime Number Theorem: An Historical Perspective.” SpringerLink, Springer, New York, NY, 2004, ​ ​ https://doi.org/10.1007/978-1-4419-9060-0_10, https://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf Goldstein, L. J. “A History of the Prime Number Theorem.” The American Mathematical ​ Monthly, vol. 80, no. 6, 1973, pp. 599–615. JSTOR, www.jstor.org/stable/2319162. ​ ​ ​ Accessed 1 Mar. 2020. Gowers, Timothy, et al., editors. “Branches of Mathematics.” The Princeton Companion to ​ Mathematics, Princeton University Press, Princeton; Oxford, 2008, pp. 315–680. ​ JSTOR, www.jstor.org/stable/j.ctt7sd01.8. Accessed 28 Feb. 2020. ​ Gowers, Timothy, et al., editors. “Mathematicians.” The Princeton Companion to ​ Mathematics, Princeton University Press, Princeton; Oxford, 2008, pp. 733–826. ​ JSTOR, www.jstor.org/stable/j.ctt7sd01.10. Accessed 28 Feb. 2020. ​ Gowers, Timothy, et al., editors. “Theorems and Problems.” The Princeton Companion to ​ Mathematics, Princeton University Press, Princeton; Oxford, 2008, pp. 681–732. ​ JSTOR, www.jstor.org/stable/j.ctt7sd01.9. Accessed 28 Feb. 2020. ​ Jameson, G. J. O. (Graham James Oscar). The Prime Number Theorem. Cambridge University Press, 2003. Richter, Florian K. “A New Elementary Proof of the Prime Number Theorem.” ArXiv.org, 9 ​ ​ Feb. 2020, arxiv.org/abs/2002.03255. Spencer, Joel, and Ronald Graham. “The Elementary Proof of the Prime Number Theorem.” The Mathematical Intelligencer, Springer-Verlag, 25 June 2009, link.springer.com/article/10.1007/s00283-009-9063-9. Trigiante, G., and D. Trigiante. “A Discrete Approach to the Prime Number Theorem.” Journal of Difference Equations and Applications, vol. 8, no. 1, Jan. 2002, pp. ​ 93–100, doi:10.1080/10236190211939. Zagier, D. “Newman's Short Proof of the Prime Number Theorem.” The American ​ Mathematical Monthly, vol. 104, no. 8, 1997, pp. 705–708. JSTOR, ​ ​ ​ www.jstor.org/stable/2975232. Accessed 29 Feb. 2020. Kepler Conjecture – Group 2

In 1611, Johannes Kepler came up with the theorem: no packing of balls of the same radius in three dimensions has a density greater than the face-centered cubic packing. This theorem was named “Kepler Conjecture”. This theorem arose when Sir Walter Raleigh asked about the best way of stacking cannonballs on the decks. Although Kepler came up with a solution, he did not provide with a proof. The first person who came up with a proof was Gauss, in the year of 1831, but he only proved that the conjecture is true if the spheres are arranged in a regular lattice. In 1900, Hilbert included this problem in his twenty-three problems but did not provide a proof, either. In 1947, Rankin came up with a proof that showed that the density of the arrangement can be as small as 0.828, which compared with the density associated 휋 with Kepler’s way of arranging: = 0.7405, was close, but not enough. Using a √18 similar method, Fejes Toth in 1953 proved that all arrangements can be reduced to a finite but a very large number of calculations. This indicated that a proof by exhaustion was, in principle, possible. In 1990, Wu-Yi Hsiang claimed to prove the Kepler conjecture using geometric methods, although, in some other mathematicians' opinions, it was more like a generalization instead of a formal proof. Thomas Hales was the first person to provide a proof of the Kepler conjecture. His first publication began in 1988, and in 2017 his formal proof was finally accepted. His proof relied on computer calculations to verify. Hales and his colleague Ferguson began a research program to systematically apply linear programming methods to find lower bounds of a function to verify the Kepler conjecture. They needed to find the lower bound of a function, which meant solving 100 000 linear programming problems. They created a program called Flyspeck. The program they created is an AI system capable of providing a wide range of mathematical conjectures automatically. In 2014, the team’s software verified the proof to be correct, and published A formal proof of the Kepler conjecture in 2015. The formal proof was then verified in 2017 and accepted by the mathematical community. Flyspeck incorporated machine learning methods. It is a significant milestone in the AI field, as it not only connects mathematics with AI methods, but also provides an initial dataset for machine learning in regards to mathematical proofs..

Kepler Conjecture – Group 2

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