For any infinite subset X of the rationals and a subset F ⊆ X which has no isolated points in X we construct a function f : X→X such that f(f(x)) = x for each x∈X and F is the set of discontinuity points of f.

The Bolzano-Weierstrass principle of choice is the oldest method of the set theory, traditionally used in mathematical analysis. We are extending it towards transfinite sequences of steps indexed by ordinals. We are introducing the notions: hiker's tracks, hiker's maps and principles P_n(X,Y,m); which are used similarly in finite, countable and uncountable cases. New proofs of Ramsey's theorem and Erdős-Rado theorem are presented as some applications

We define several types of Lucas and Frobenius pseudoprimes and prove some theorems on these pseudoprimes. In particular: There exist infinitely many arithmetic progressions formed by three different Frobenius-Fibonacci pseudoprimes.

We will deal with numbers given by the relation
H_n(k) = [(k+1)ⁿ-\binom{n}{2}k²-nk-1]/k³,
where k is equal to 1, 2 or 3. These numbers arise from a generalization Bernoulli's inequality. In this paper some results about divisibility and primality of the numbers H_n(l), H_n(2) and H_n(3) are found. For example any positive integer n>1 does not divide H_n(2) and n ≡ 2mod4 is the necessary condition for divisibility H_n(l) and H_n(3) by n>2. In addition certain properties of their divisibility are used for finding primes among these numbers.

In this paper Kummer's elements in the Stickelberger ideal of the group ring of the Galois group of the extension ℚ(ζ)/ℚ over the ring of rational integers are studied. A special linear operator is constructed for better understanding to these elements, and Kummer's elements are mapped to Kummer's vectors. Then one computational method is presented and the coherence between its results and the property of being essential for Kummer's vectors is shown.

Report of Meeting. The Third Katowice-Debrecen Winter Seminar on Functional Equations and Inequalities January 29 - February 1, 2003, Będlewo, Poland.