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SF2724, Topics in mathematics IV: Prime numbers, 7.5 hp, Spring 2009
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Instructor: Anders Karlsson, akarl@kth.se,
room 3538, Lindstedsv. 25, 790 71
73.
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Course secretary: Rose-Marie Jansson, jansson@math.kth.se,
room 3527, Lindstedsv. 25, 790 72 01
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Time and place: Thursdays 10.15-12.00, room 3733,
Department of Mathematics, KTH (This is preliminary.) The first meeting will be Thursday
January 22. The course will probably be given in English.
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Literature:
The most suitable book might be Tennenbaum and
Mendes France:
The Prime Numbers and Their Distribution.
A few papers will be handed out.
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Prerequisites: Calculus, and ideally some complex analysis.
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Topics:
The course will cover some classic results related to prime numbers and we will discuss some of the
outstanding unsolved problems. More precise content includes: The fundamental theorem of
arithmetic. Euclid's and Euler's proof of the infinitude of primes. Theta inversion. The Riemann
zeta function, including the Euler product, functional equation and special values. The
prime number theorem. The Riemann hypothesis. Quadratic reciprocity. RSA cryptography.
Goldbach's conjecture. The ABC-conjecture, with an analogue for polynomials and the relation
to Fermat's last theorem. Theorems of Dirichlet and Green-Tao concerning primes and
arithmetic progressions.
The hope is that during this walk through landmark theorems and conjectures on primes,
we will get a glimpse of the beauty and unity of mathematics, in particular through the use
of analytic methods for the solution of problems apparently "discrete" at first sight.
Although the course is primarily intended for advanced undergraduates, anyone interested
is welcome to attend.
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Examination: The examination will be in the form of homework problems and a written exam. The homeworks, which are due every week, count for 50% of the grade and the exam for the remaining 50%. Several of the exam problems will be taken from the homework exercises. The exam is scheduled for Tuesday, May 19, 14:00-18:00 in room E53.
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Oppen forelasning pa KVA: Primtal, 2008-12-17
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Lecturenotes: Lectures 1-.
Optional reading for lectures 3-6.
No homework due for February 12.
Additional exercises. Sample exam, Exam., Tenta losningar.
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