My email address is freud@cs.elte.hu.
Final is on June 8, 10-12, in Room 3-715, office hour is on June 7 at 9, in Room 4-202. Sample final
Midterm is on March 28, as scheduled. We shall discuss the problems of the sample midterm on March 27, Monday, at 2 PM.
Problem sheets: 1. ; 2. ; 3. ; 4. ; 5. ; 6.
Handouts: Algebraic numbers ; Sums of powers ; All sums are distinct ; Mersenne and Fermat primes ; Sidon sets .
Homeworks are due at 4 PM next Tuesday. Each problem is one point worth (partial credit is possible), the weekly assignment contains 3 problems from the problem sheets.
HW1. Due on February 21: 2b, 3, any two parts of 1.
HW2. Due on February 28: (i) Prove that 341 is not a pseudprime with base 3, i.e. 3^{340}-1 is not divisible by 341; (ii)-(iii) Any two problems of 2c, 4a, 5, and 8.
HW3. Due on March 7: (i) Problem 6; (ii)-(iii) Any two problems of 9, 10, 11, and 12.
HW4. Due on March 14: Any three of the following five problems: 13a, 14a, 15, 16, and 17.
HW5. Due on March 21: 18, 19.
HW6. Due on April 4: 20, 21, 22.
HW7. Due on April 11: Any four of the following six problems: 24a,b; 24c,d; 25, 26, 27, 28.
HW8. Due on May 2: 31, 33, 35. Extended till May 9.
HW9. Due on May 9: (i) One of the following three probles: 36, 37a, 37b; (ii) Any three parts of 40; (iii) Any three parts of 41a,b,d,e,f; (iv) Any four parts of 42.
HW10. Due on May 16: Any four of the following nine problems: 45a1, 45a2, 45b, 46, 47a, 47b, 48, 49, and 50. Extended till May 24.
February 14: We proved that there are infinitely many primes both of form 4k-1 and 4k+1, and stated Dirichlet's theorem for the number of primes in arithmetic progressions. We showed some facts concerning Mersenne and Fermat primes, solving also 2a.
February 21: We sketched the contrast between the difficulties of primality testing and prime factorization. We mentioned some famous unsolved problems concerning prime: twin primes, Goldbach conjecture. There can be arbitrarily big gaps between consecutive primes, whereas there is always a prime between n and 2n (Chebyshev's Theorem). We stated also the Prime Number Theorem.
February 28: We discussed some consequences of the Prime Number Theorem and solved Problems 1b, 5, and 7.
March 7: We discussed Problems 6 and 11, then showed, using the Prime Number Theorem, that the sum of the reciprocals of the primes is divergent and the interval [n, (1+c)n] always contains a prime if n is large enough.
March 14: We discussed Problems 9, 10, and 12, then we sketched Euler's formula for the series of primes, and we established the characterization of the Pythagorean triples.
March 21: We solved Problem 13, and stated the general results for Pell's equation (partly without proof). As a preparation to the two-squares-theorem, we introduced the Gaussian integers, defined the norm, determined the units, and proved the division with remainder. This implies the unique factorization theorem as we reviewed it through the examples of the integers and the polynomials over a field.
March 28: Midterm.
April 4: We determined all Gaussian primes, solved Problems 21c and 23, and stated the two-square-theorem, the proof will follow next week.
April 11: After proving the two-squares-theorem we showed that the numbers a+bV-5 (with integer a and b) do not satisfy UFT, e.g. 6 has two distinct decompositions as the product of irreducible elements. We stated the three- and four-squares-theorems, then generalized the question for higher powers. We stated some results concerning Waring's problem, and gave a lower bound for g(k). We also atated G(6)>8, to be proven in next class. You are encouraged to establish some lower boound for G(6) using the residues of the 6th powers mod 7, 8, or 9.
May 2: We proved G(6)>8. Then we got acquainted with the basic notions and theorems about algebraic and transcendental numbers, and solved Problems 38, 39, and 41c.
May 9: We showed that the cardinality of algebraic numbers is countable, whereas the reals are power of continuum, which proves the existence of transcendental numbers. We gave lower and upper bounds for the size of a maximal Sidon set in [1,n] using greedy algorithm and counting arguments.
May 16: Besides discussing several HW problems, we proved that if all subset sums are distinct, then the sum of reciprocals of the elements is less than 2.