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Students can participate in this class as a transition to higher mathematics, working to learn the details, relationships, proofs, and counterexamples that form the core of advanced mathematical thinking. Studnet can also participate in this course as an introduction to higher-level mathematics without being mathematically prepared to work through all the proofs, and will still learn the interesting facets of the number system and useful techniques for solving number theoretical problems without being saddled with the expectations being placed on the "high flyer" mathematics students.

We will not formally be using a textbook this semester.

Number Theory and Cryptology

Many textbooks available in the IRC, in bookstores, and even some complete texts online cover the main topics that we will discuss. Students are encouraged to use these as references and sources for extra practice problems. Students will be expected to keep complete and detailed class notes in lieu of a textbook. Reference Text: Rosen, Kenneth H.


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Elementary Number Theory and its Applications. Addison Wesley. This course is an introduction to the fundamentals of number theory, with a survey of additional topics that are accessible once the basics are in place. The course is intended to introduce concepts of number theory from a rigorous point of view, and then show students some of the applications of the material they have just learned. Not as deep, but generally broader than a one-semester course for math majors at universities.

The First part of the course is core material: axioms for the number system, divisibility and the GCD, factorization, Euclidean algorithm, linear diophantine equations, the multiplicative functions sum and number of divisors. Congruences, systems and the Chinese Remainder Theorem, reduced residue systems and Euler's totient function. Euler and Fermat theorems, primality testing, order and primitive roots.


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Representations of numbers in base b , periodic expansions, irrational numbers and cardinality. This core material comprises about two-thirds of the term. Second part of course consists of optional units selected by the students. Units in recent past have included continued fractions, cryptography, quadratic reciprocity, and combinatorial game theory.

Carl Pomerance

Nowakowski, and C. Analytic number theory in honor of Helmut Maier's 60th birthday , M. Rassias and C.

Basic Number Theory Lecture #1 Rising Star Scademy

Fields talk , Counting Fields. Multiplicative order talk , The multiplicative order mod n , on average. Version of March, , Brigham Young University. Version of April, , University of Rochester. Long version of March, , Brigham Young University undergraduate colloquium.

Combinatorial Number Theory

The first dynamical system? Counting in number theory, the Rademacher Lectures, U Penn, September, Elementary number theory , Finite cyclic groups , Fibonacci integers , Counting fields Two problems in combinatorial number theory , Number theory and its applications, Debrecen, Hungary, Oct. Version of November, Version of December, Vermont, December 2, UC Irvine colloquium, February 9, Sums and products Dartmouth Mathematics Society, May 16, Sums and products U.

Instructor

Sums and products , West Chester U. Sums and products , Arizona State U. Sums and products , Providence College, April 2, What we still don't know about addition and multiplication , Richard and Louise Guy Lecture, University of Calgary, September 12, What we still don't know about addition and multiplication , Woods Lecture Series, Butler University, December 3, What we still don't know about addition and multiplication , Christie Lecture, Bowdoin College, April 7, Hawaii, Honolulu, March, Akron, Akron, OH, October, The set of values of an arithmetic function, Mathematical Congress of the Americas, Guanajuato, Mexico, August 9, Ram Murty, University of Montreal, October 15—17, The first function , U.

Georgia colloquium, December 3, The first function , Middlebury College Seminar, April 21, The first function , Dartmouth Colloquium, May 26, The first function , Western Michigan U. The ranges of some familiar functions , December 8, The first function , December 11, Amicable numbers , December 12, Here are some sample letters that are discussed near the end of the talk.

The first dynamical system with a short feature , Summer school on fractal geometry and complex dimensions, Cal Poly San Luis Obispo, June 27, Vermont, September , West Georgia, October 6, The ranges of some familiar arithmetic functions , Michigan State U. Colloquium, October 13, The ranges of some familiar arithmetic functions , Max Planck Institute for Mathematics, November 2, The first dynamical system , Charles U. Prague Seminar, November 8, Random number theory , Charles U. Prague Colloquium, November 8, Georgia talks, March , Random number theory , Euclidean prime generators , What we still don't know about addition and multiplication , The first dynamical system.

What we still don't know about addition and multiplication , Leonard C. See draft of paper on this topic with some updated results. New Brunswick, June 4, What we still don't know about addition and multiplication , Trjitzinsky Lecture 1, U. Illinois Urbana-Champaign, November 27, Random number theory , Trjitzinsky Lecture 2, U.

Illinois Urbana-Champaign, November 28, Primality testing: then and now , Trjitzinsky Lecture 3, U. Illinois Urbana-Champaign, November 29, Primality testing: then and now , Boise State University, February 20, Papers Odd perfect numbers are divisible by at least seven distinct primes , C.

Pomerance, Acta Arith. On Carmichael's conjecture , C. Pomerance, Proc. A search for elliptic curves with large rank , D. Penney and C. Pomerance, Math. Nelson, D.

Art of Problem Solving

Penney, and C. Pomerance, J. Three elliptic curves with rank at least seven , D. The second largest prime factor of an odd perfect number , C. On an interesting property of , J. Hunsucker and C. Pomerance, Fibonacci Quarterly 13 , — There are no odd super perfect numbers less than 7 x 10 24 , J. Pomerance, Indian J. Some new results on odd perfect numbers , G. Dandapat, J.

M3P14: Elementary Number Theory

Hunsucker, and C. Pomerance, Pacific J. On multiply perfect numbers with a special property , C. Multiply perfect numbers, Mersenne primes and effective computability , C.

On a tiling problem of R. Eggleton , C. Pomerance, Discrete Math. On the distribution of amicable numbers , C. Pomerance, Aequationes Math. On a class of relatively prime sequences , P. Number Theory 10 , — The prime number graph , C. On a problem of Evelyn—Linfoot and Page in additive number theory , C. Pomerance and D. Suryanarayana, Publ.

Debrecen 26 , — Nearly parallel vectors , H. Diamond and C. Pomerance, Mathematika 26 , — Some number theoretic matching problems , C. Ribenboim, ed. Combinatorial Theory A 28 , — A note on the least prime in an arithmetic progression , C. Number Theory 12 , — The pseudoprimes to 25 x 10 9 , C. Selfridge, and S. Wagstaff, Jr.

Matching the natural numbers up to n with distinct multiples in another interval , P. Pomerance, Nederl.


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  6. A 83 , — Proof of D. Newman's coprime mapping conjecture , C. Pomerance and J. Selfridge, Mathematika 27 , 69— Popular values of Euler's function , C.