Negatively reduced ideals in orders of real quadratic fields: even discriminants
Nil-clean and unit-regular elements in certain subrings of
An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson’s lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl’s question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements that are...
Nonexistence of a geometric progression that contains four triangular numbers.
Nonextendibility of -triples of the form .
Norm form equations and continued fractions.
Normal bivariate Birkhoff interpolation schemes and Pell equation
Finding the normal Birkhoff interpolation schemes where the interpolation space and the set of derivatives both have a given regular “shape” often amounts to number-theoretic equations. In this paper we discuss the relevance of the Pell equation to the normality of bivariate schemes for different types of “shapes”. In particular, when looking at triangular shapes, we will see that the conjecture in Lorentz R.A., Multivariate Birkhoff Interpolation, Lecture Notes in Mathematics, 1516, Springer, Berlin-Heidelberg,...
Note on a Shannon's theorem concerning the Fibonacci numbers and Diophantine quadruples
Note on a theorem of Rockett and Szusz on a diophantine equation x² - dy² = N
Note sur la géométrie des quinconces
Note sur la résolution en nombres entiers et positifs du système des deux équations indéterminées
Note sur la résolution en nombres entiers positifs du système des trois équations
Note sur quelques équations indéterminées