BSM, Fall 2018, Conjecture and Proof C&P

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Course syllabus

General info

Problem sheets: 1. ; 2.

Short summaries: 1. Algebraic numbers ; 2. Diophantine approximation


Homeworks are assigned on Thursday and are due on next Thursday. Each problem is one point worth (partial credit is possible), the weekly assignment contains 3 problems from the problem sheets.

HW1. Due on September 20: (i) Any three of the altogether five parts 1/b,c,f,g,h; (ii) Any three of the altogether six parts of Problems 2 and 7; (iii) Problem 8. Problem 10 can replace one of (i), (ii), and (iii).

HW2. Due on September 27: (i) Any three parts of 5; (ii) Any three parts of 6a,b,d,e,f; (iii) One of 11, 13, and 17.

Extra problems


You can hand in the solutions of these problems without time limit.

X1. Ad 14: Prove the statement for squares working with differences instead of quotients used in class.

What happened in class?

September 10: We "tasted" Problem 9: we gave some upper (see also the first page of Section 8 in the textbook) and lower bounds for the maximal size of a Sidon set in the interval [1,n]. We shall return to better estimates later. Then we gave 5 proofs for 1a (see also Section 1 in the Textbook).

September 13: We solved 1e first (see also Section 1 in the textbook), then after summarizing the basic facts about complex numbers, we solved 1d. We introduced algebraic numbers, minimal polynomial, and degree, and discussed some of their properties (see Handout 1, you are encouraged to prove (a), (b), and (c) on the sheet, but do not try to prove (d), we shall return to it later).

September 17: We proved properties (a), (b), and (c) in Handout 1, solved Problems 4, 3, and 6c, stated the Gelfond-Schneider theorem about powers of algebraic numbers with a non-rational algebraic exponent, and applied it to prove that log base 10 of 3 is transcendental. During the office hour we tried to handle rationally some problems of irrationality.

September 20: Diophantine approximation (see Handout 2): The different order of approximation of rational and irrational numbers, the fractional parts of the multiples of an irrational number are dense in interval [0,1], Problem 14 as an application, square root 2 cannot be approximated "too well" (Problem 12). You are encouraged to show that cube root of 2 cannot be approximated too well and to make a conjecture about the poor approximation of algebraic numbers in general depending on their degress - to be discussed next week. This will be the key to construct a transcendental number.