Number Theory
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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been solved in [1]-[4] and a problem on this subject proposed by C. Pomerance has been solved by A. Rotkiewicz (as yet unpublished).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rationals of the non-cyclotomic kernel of a non-reciprocal lacunary polynomial has been given by A. Schinzel (as yet unpublished).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number formulae for imaginary quadratic fields has been found by A. Schinzel, J. Urbanowicz and P. van Wamelen (as yet unpublished).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)

  1. A. Rotkiewicz, On Lucas pseudoprimes of the form ax2+bxy+cy3, in: Applications of Fibonacci Numbers, vol. 6, 409-421.
  2. A. Rotkiewicz, On the theorem of Wójcik, Glasgow Math. J. 38 (1996), 157-162.
  3. A. Rotkiewicz, There are infinitely many arithmetical progressions formed by three different Fibonacci pseudoprimes, in: Applications of Fibonacci Numbers, vol. 7, 327-332.
  4. A. Rotkiewicz, Arithmetical progressions formed by Lucas pseudoprimes, in: Number Theory. Diophantine, Computational and Algebraic Aspects, 465-472.
  5. A. Schinzel, O pokazatelnykh sravneniyakh, in: Mat. Zapiski, vol. 2, 121-126.
  6. 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.
  7. A. Schinzel, On the Mahler measure of polynomials in many variables, Acta Arith. 79 (1997), 77-81.
  8. A. Schinzel, Triples of positive integers with the same sum and the same product, Serdica Math. J. 22 (1996), 587-588.
  9. A. Schinzel, A class of algebraic numbers, in: Tatra Mountains Math. Publ., vol. 11, 35-42.
  10. A. Schinzel, On homogeneous covering congruences, Rocky Mountain J. Math. 27 (1997), 335-342.
  11. A. Schinzel, On pseudosquares, in: New Trends in Probability and Statistics, vol. 4, 213-220.
  12. A. Schinzel (with D. Barsky and J. P. Bézivin), Une caractérisation arithmétique de suites récurrentes linéaires, J. Reine Angew. Math. 494 (1998), 73-84.
  13. A. Schinzel (with I. Aliev and S. Kanemitsu), On the metric theory of continued fractions, Colloq. Math. 77 (1998), 141-146.
  14. A. Schinzel, A property of the unitary convolution, Colloq. Math. 78 (1998), 93-96.
  15. A. Schinzel, On Pythagorean triangles, Ann. Math. Siles. 12 (1998), 31-33.
  16. J. Urbanowicz, On diophantine equations involving sums of powers with quadratic characters as coefficients, II, Compositio Math. 102 (1996), 125-140.
  17. 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.
  18. J. Wójcik, On a problem in algebraic number theory, Math. Proc. Cambridge Philos. Soc. 119 (1996), 191-200.