Research during the last few years concerned problems in different branches of number theory which will be
reviewed in the order adopted by Mathematical Reviews.
In elementary number theory a problem (proposed by T. Cochrane and G. Meyerson) concerning
covering systems of congruences has been solved in [10] and another one (proposed by W. Narkiewicz) concerning
arithmetical functions in [14]. Various problems concerning generalized pseudoprimes have been solved in
[1]-[4] and a problem on this subject proposed by C. Pomerance has been solved by A. Rotkiewicz (as yet
unpublished).
A criterion for reducibility over the rationals of the non-cyclotomic kernel of a non-reciprocal lacunary
polynomial has been given by A. Schinzel (as yet unpublished).
In diophantine equations, [8] gives a solution to Problem D16 from the book of R. Guy "Unsolved
Problems in Number Theory" and [15] solves a problem on Pythagorean triangles proposed by I. Korec. [16]
deals with a Diophantine equation related to generalized Bernoulli numbers.
In the metric theory of algorithms a problem of M. Deleglise concerning continued fractions has been
solved in [13].
In multiplicative number theory the connection between
squarefree values of polynomials and the
abc-conjecture has been studied in [6], while some estimates for pseudo-squares have been given in [11].
In algebraic number theory exponential congruences have been studied in [5], [12] and [18], while a
certain problem of I. Korec concerning algebraic integers has been solved in [9].
J. Urbanowicz [17] together with G. J. Fox and K. S. Williams has obtained a divisibility property for
generalized Bernoulli numbers that generalizes the classical result of Gauss concerning divisibility by powers of
two of class numbers of quadratic forms. An application of generalized Bernoulli numbers to class number
formulae for imaginary quadratic fields has been found by A. Schinzel,
J. Urbanowicz and P. van Wamelen
(as yet unpublished).
Research papers published in 1996-1999 (June)
- A. Rotkiewicz, On Lucas pseudoprimes of the form
ax^{2}+bxy+cy^{3}, in: Applications of Fibonacci
Numbers, vol. 6, 409-421.
- A. Rotkiewicz, On the theorem of Wójcik, Glasgow
Math. J. 38 (1996), 157-162.
- A. Rotkiewicz, There are infinitely many
arithmetical progressions formed by three different
Fibonacci pseudoprimes, in: Applications of Fibonacci
Numbers, vol. 7, 327-332.
- A. Rotkiewicz, Arithmetical progressions formed by
Lucas pseudoprimes, in: Number Theory.
Diophantine, Computational and Algebraic Aspects, 465-472.
- A. Schinzel, O pokazatelnykh
sravneniyakh, in: Mat. Zapiski, vol. 2, 121-126.
- A. Schinzel (with J. Browkin, M. Filaseta, G.
Greaves), Squarefree values of polynomials and the
abc-conjecture, in: London Math. Soc. Lecture
Note Ser. 237, 65-85.
- A. Schinzel, On the Mahler measure of polynomials in
many variables, Acta
Arith. 79 (1997), 77-81.
- A. Schinzel, Triples of positive integers with the
same sum and the same
product, Serdica Math. J. 22 (1996), 587-588.
- A. Schinzel, A class of algebraic numbers, in:
Tatra Mountains Math. Publ.,
vol. 11, 35-42.
- A. Schinzel, On homogeneous covering
congruences, Rocky Mountain J. Math. 27 (1997), 335-342.
- A. Schinzel, On pseudosquares, in: New Trends in
Probability and Statistics, vol. 4, 213-220.
- A. Schinzel (with D. Barsky and J. P. Bézivin),
Une caractérisation arithmétique de suites
récurrentes linéaires, J. Reine Angew. Math. 494 (1998),
73-84.
- A. Schinzel (with I. Aliev and S.
Kanemitsu), On the metric theory of continued fractions,
Colloq. Math. 77 (1998), 141-146.
- A. Schinzel, A property of the unitary convolution,
Colloq. Math. 78 (1998), 93-96.
- A. Schinzel, On
Pythagorean triangles, Ann. Math. Siles. 12 (1998), 31-33.
- J. Urbanowicz, On diophantine equations
involving sums of powers with quadratic characters as
coefficients, II, Compositio Math. 102 (1996), 125-140.
- J. Urbanowicz (with G. J. Fox and K. S. Williams),
Gauss' congruence for Dirichlet's class number formula and
generalizations,
in: Number Theory in Progress, vol. 2, de Gruyter, 1999, 813-839.
- J. Wójcik, On a problem in algebraic number
theory, Math. Proc. Cambridge Philos. Soc. 119
(1996), 191-200.