The well-known Carlitz theorem for the Bernoulli numbers B_n, (see [3]) is extended to the case of values of the Bernoulli polynomials B_n(y) at rational points a/b, where (b,n!) = 1.

Let K be a field. Let us define K̅ as a formal set K∪{∞} (which can be identified with P¹(K)). For convenience we put that the degree of a zero polynomial is zero (not -∞). For relatively prime polynomials f,g∈K[X] of degrees n, m and leading coefficients a_n, b_m, respectively, we define rational function ϕ(X) = f(X)/g(X) as a mapping ϕ: K̅ ↦ K̅ as follows:
         f(ξ)/g(ξ) for ξ∈K, g(ξ)≠0
                   ∞ for ξ∈K, g(ξ)=0
ϕ(ξ)= an/bm for ξ=∞, n=m
                   ∞ for ξ=∞, n>m
                    0 for ξ=∞, n<m,
where we put 1/0 as ∞.
More generally for (f,g) ≠ (0,0) and ϕ = f/g we put ϕ = f₁/g₁, where f₁ = f/d, g₁ = g/d, d = gcd(f,g).
A k-tuple x₀, ...,x_k-1 of distinct elements of K̅ is called a cycle of ϕ of length k if
ϕ(x_i) = x_i+1 for i = 0, 1, ..., k-2  and  ϕ(x_k-1) = x₀.
The set of all positive integers which are not lengths of a cycle for ϕ will be denoted by Exc(ϕ).
The case of a rational mapping not of the form (aX + b)/(cX + d) over algebraically closed field of characteristic zero was solved by I.N. Baker [2], who showed that Exc(ϕ) is always finite and gave all possible examples of Exc(ϕ). However for positive characteristic the situation differs, in fact it was shown in [3] that for some polynomials ϕ the set Exc(ϕ) is infinite.
The aim of this paper is to prove that for a large class of rational ϕ over algebraically closed field of positive characteristic the set Exc(ϕ) is "lacunar", i.e. either Exc(ϕ) is finite or Exc(ϕ) = {a₁ < a₂ < ...} with a_i+1/a_i > λ > 1 for all i.

In this note we exhibit a connection between theory of associative rings and number theory by an example of necessary and sufficient conditions under which the integral group ring of a finite group is reduced.

The following theorem answers a question asked by I. Korec at the Second Czech & Polish Conference on Number Theory.

Theorem. If m∈ℕ, ord₂m is even, x₀,y₀,z₀∈ℤ and
(1)      x₀²+y₀² ≡ z₀²(modm),
then there exist x,y,z∈ℤ such that
x²+y² = z²,  x² ≡ x₀²,  y² ≡ y₀²,  z² ≡ z₀²(modm).

Functions F: ℕ→ℝ which satisfy the condition
∀x,y,u,v (ax²+cy² = au²+cv² ⇒ a·F(x)+c·F(y) = a·F(u)+c·F(v))
are considered. For some small positive integers a, c they are completely characterized. E.g., for a = c = 1 they form a σ-dimensional real vector space with a base consisting of F₀(x) = x² and five periodical functions with periods 1,..., 5. Further, functions F: ℕ→ℝ which satisfy the condition
∀x,y,z (x² = y²+z² ⇒ F(x) = F(y)+F(z))
are studied; 17 linearly independent examples of such functions are presented, including periodical ones with the periods 16, 9, 25, 13.

In [CPS] we have observed that each class of Witt equivalent quadratic number fields, except for the singleton class containing only ℚ(√-l), contains a field whose class group has 2-rank as large as we wish.
Here we generalize this observation from the case of quadratic number fields to fields of arbitrary even degree n. We prove that each class of Witt equivalent number fields of even degree n > 2 contains a field K with the 2-rank of class group as large as we wish. In fact, we prove a stronger result saying that the field in question has large 2-rank of S-class group for a finite set S of primes of K containing all infinite and all dyadic primes of the field.
We combine here an interpretation of the parity of S-class numbers in terms of a localization map (Proposition 6) with a valuation-theoretic result of Endler on the existence of fields with prescribed completions. The latter has been used in [Sz] to construct fields with prescribed Witt equivalence invariants. Here we discuss this technique again to make clear its applicability in constructing, in a given Witt class, number fields with special properties.

In this paper we discuss pertinent questions closely related to well known RSA cryptosystem [5]. From practical point of view it is reasonable to use as a public exponent an integer s = 2^k+1, i.e., so called short exponent, with the lowest possible binary weight. The most common are for k = 1 and k = 2⁴, the two Fermat primes. In this paper we prove two theorems which give a percentage of acceptable public exponents s = 2k+1, 1 ≤ k ≤ 1023 to two randomly selected primes of 512 bits each. In fact, our results are valid for arbitrary set of exponents s. We also present results of our experiments. In our simulation, for all such acceptable public exponents, the corresponding secret exponent t had a weight within the range of 451-567. Thus, although it is recommended in [8] not to use short public exponents, by our observation to use the attack based on continuos fractions is infeasible.

In this paper the results of [4] and [5] on the congruence of Ankeny-Artin-Chowla type modulo p² for real subfields of the Q(ζ_p) of a prime degree l > 2 are simplified to explicit forms (2) and (3) of the Theorem 1. The congruence is then used to calculations of class numbers in special quintic fields and to some calculations in cubic fields.

Forms of degree higher than 2 behave in a quite different way than quadratic forms. Jordan [J] proved finiteness of orthogonal groups of nonsingular forms of degree ≥ 3, whereas it is known that quadratic forms, even if nonsingular, provide us mainly with infinite orthogonal groups. In this paper we describe the orthogonal groups of separable forms of degree at least 3 and for any Clifford-Littlewood-Eckmann group G we construct a form over the rational number field ℚ with the orthogonal group isomorphic to G.

For a number field K let ????_K be the ring of algebraic integers of K. A basic result on the Witt ring W????_K of symmetric bilinear forms over the ring ????_K was established in [MH]. The structure of the Witt group W????_K, in terms of arithmetical invariants of K, was determined in [Sh]. Here we state precisely this description. We find generators of cyclic direct summands in the decomposition of the group W????_K into direct sum of cyclic groups. We will also describe products of these generators. This completely determines the structure of the ring W????_K. As an illustration of these results we determine the structure of Witt rings W????_K for all quadratic, and some cubic and some biquadratic fields K. The results of this paper allow us to find arithmetical conditions for the existence of an isomorphism of Witt rings W????_K → W????_L (for details see [Cz2]).

We are concerned with the equation of the title, where n is a fixed positive integer, h(n) is a given integer-valued arithmetic function and the unknowns take positive integral values 1 ≤ x₁ ≤ x₂ ≤ ... ≤ x_n. We estimate the number m for which x_i = 1 (i = 1, 2,...,m) in every solution. Next we give an upper bound for the number of coordinates of a solution which can be greater than 1. Further we estimate the number of all solution of the equation, and the paper concludes with a list of open problems.

In this paper we discuss the problem to carry over the well-known Minkowski-Hasse local-global principle to the context of an infinite algebraic extension of the rationals or the rational function fields ????_q(x) over finite fields. Applying this result we give a new proof of the elementary type conjecture for Witt rings of infinite algebraic extensions of global fields. This generalizes a result of I. Efrat [Ef] who proved, using Galois cohomology methods, a similar fact for algebraic extensions of the rationals.

At the Thirteen Czech-Slovak International Number Theory Conference in Ostravice in 1997 and at JA in Limoges in 1997 A. Schinzel proposed the following problem.

PROBLEM 1. Disprove the following statement.
There exists such a prime number p₀, that for all prime numbers p > p₀ and all n∈ℕ the following condition holds
2ⁿ ≡ 3 mod p ⇔ 3ⁿ ≡ 2 mod p.

We can reformulate Problem 1 in the following way.

Prove that for every prime number p₀ there is a prime number p > p₀ and there is an n∈ℕ such that either
2ⁿ ≡ 3 mod p and 3ⁿ ≢ 2 mod p
or
2ⁿ ≢ 3 mod p and 3ⁿ ≡ 2 mod p.

We solve Problem 1 by proving the following theorem.

THEOREM 1.
(a) For every prime number p₀ there is a prime number p > p₀ and there is an n∈ℕ such that
2ⁿ ≡ 3 mod p and 3ⁿ ≢ 2 mod p.
(b) For every prime number p₀ there is a prime number p > p₀ and there is an n∈ℕ such that
3ⁿ ≡ 2 mod p and 2ⁿ ≢ 3 mod p.

The notion of a regular system of divisors of a number underlying a Narkiewicz's idea of the generalization of the familiar Dirichlet and unitary convolution of arithmetical functions is used to extend some combinatorial results about systems of subsets of divisors of a generalized integer.

I. Korec [Ko] introduced the following definition of a P-additive function.

DEFINITION. A function F: N→R is said to be Pythagorean-additive (P-additive, for short) if for all x,y,z∈N
x² = y²+z² ⇒ F(x) = F(y)+F(z).

The aim of this paper is to determine all prime numbers that are periods of P-additive functions. The main result is the following theorem.

THEOREM. Let the prime number p be a period of a P-additive function. Then p∈{2,3,5,13}.

A reciprocity equivalence between two number fields is a Hilbert symbol preserving pair of maps (t,T), in which t is a group isomorphism between the global square class groups of the two fields, and T is a bijection between the sets of primes. For two reciprocity equivalent number fields, it is proved that: Theorem A : The Dirichlet density of the wild set of any reciprocity equivalence is zero. Theorem B: There exists a reciprocity equivalence whose wild set is infinite. Theorem C: Given (t,T), the bijection T determines the global square class isomorphism t.

The 2nd Czech and Polish Conference on Number Theory Cieszyn 1998. Problem session.