An identity involving Ramanujan's sum.
An improvement of Euclid's algorithm
The paper introduces the calculation of a greatest common divisor of two univariate polynomials. Euclid’s algorithm can be easily simulated by the reduction of the Sylvester matrix to an upper triangular form. This is performed by using - transformation and -factorization methods. Both procedures are described and numerically compared. Computations are performed in the floating point environment.
An infinite product with bounded partial quotients
An inverse theorem mod p
An iteration problem involving the divisor function
An unconventional problem in the elementary theory of numbers
An upper bound for odd perfect numbers.
An upper bound for the norm of a gcd-related matrix.
An upper bound for the number of solutions of a system of congruences
Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan.
Another generalization of Euler's theorem
Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden
Aposyndesis in
We consider the Golomb and the Kirch topologies in the set of natural numbers. Among other results, we show that while with the Kirch topology every arithmetic progression is aposyndetic, in the Golomb topology only for those arithmetic progressions with the property that every prime number that divides also divides , it follows that being connected, being Brown, being totally Brown, and being aposyndetic are all equivalent. This characterizes the arithmetic progressions which are aposyndetic...
Appending digits to generate an infinite sequence of composite numbers.
Application of the circle method on multidimensional limit-periodic functions
Applications d'un corps fini dans lui-même
Applications of differential algebra to algebraic independence of arithmetic functions
We generalize and unify the proofs of several results on algebraic independence of arithmetic functions and Dirichlet series by using a theorem of Ax on the differential Schanuel conjecture. Along the way, we find counter-examples to some results in the literature.
Applications of Jacobi's symbol to Lehmer's numbers
Approximate polynomial GCD
The computation of polynomial greatest common divisor (GCD) ranks among basic algebraic problems with many applications, for example, in image processing and control theory. The problem of the GCD computing of two exact polynomials is well defined and can be solved symbolically, for example, by the oldest and commonly used Euclid’s algorithm. However, this is an ill-posed problem, particularly when some unknown noise is applied to the polynomial coefficients. Hence, new methods for the GCD computation...
Approximation by Nearest Integer Continued Fractions (II).