Dvě poznámky k teorii číselné
Dvě věty arithmetické
Eine Klasse von Abzählproblemen.
Elementare Abschätzungen für das kleinste gemeinsame Vielfache.
End-symmetric continued fractions and quadratic congruences
We show that for a fixed integer n ≠ ±2, the congruence x² + nx ± 1 ≡ 0 (mod α) has the solution β with 0 < β < α if and only if α/β has a continued fraction expansion with sequence of quotients having one of a finite number of possible asymmetry types. This generalizes the old theorem that a rational number α/β > 1 in lowest terms has a symmetric continued fraction precisely when β² ≡ ±1(mod α ).
Étude bibliographique. Sur la théorie des nombres
Exponents of primes in generalized binomial coefficients.
Extended Euclidean Algorithm and CRT Algorithm
In this article we formalize some number theoretical algorithms, Euclidean Algorithm and Extended Euclidean Algorithm [9]. Besides the a gcd b, Extended Euclidean Algorithm can calculate a pair of two integers (x, y) that holds ax + by = a gcd b. In addition, we formalize an algorithm that can compute a solution of the Chinese remainder theorem by using Extended Euclidean Algorithm. Our aim is to support the implementation of number theoretic tools. Our formalization of those algorithms is based...
Extended GCD and Hermite normal form algorithms via lattice basis reduction.
Farey fractions and sums over coprime pairs
General -product function.
Généralisation de l'identité de MM. Tchébychew et De Polignac
Greatest common divisors in shifted Fibonacci sequences.
Ideals in the semiring N
Inequalities related to the unitary analogue of Lehmer problem.
Le déterminant de Smith et Mansion
Lower bounds for the largest eigenvalue of the gcd matrix on
Consider the matrix with ’th entry . Its largest eigenvalue and sum of entries satisfy . Because cannot be expressed algebraically as a function of , we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that for all . If is large enough, this follows from F. Balatoni (1969).
Maximal sets of integers with distinct divisors.
Méthodes élémentaires en vue du théorème de Sylvester
More on Divisibility Criteria for Selected Primes
This paper is a continuation of [19], where the divisibility criteria for initial prime numbers based on their representation in the decimal system were formalized. In the current paper we consider all primes up to 101 to demonstrate the method presented in [7].