An application of Kloosterman sums
An approximation property of quadratic irrationals
Let be irrational. Several authors studied the numberswhere is a positive integer and denotes the set of all real numbers of the form with restricted integer coefficients . The value of was determined for many particular Pisot numbers and for the golden number. In this paper the value of is determined for irrational numbers , satisfying with a positive integer .
An arithmetic formula of Liouville
An elementary proof is given of an arithmetic formula, which was stated but not proved by Liouville. An application of this formula yields a formula for the number of representations of a positive integer as the sum of twelve triangular numbers.
An arithmetic function
An arithmetical mapping and applications to Ω-results for the Riemann zeta function
An Artin Character and Representations of Primes by Binary Quadratic Forms III.
An awful problem about integers in base four
An effective procedure for minimal bases of ideals in Z[x]
We give an effective procedure to find minimal bases for ideals of the ring of polynomials over the integers.
An Egyptian algorithm for polynomials.
An elementary proof of a congruence by Skula and Granville
Let be a prime, and let be the Fermat quotient of to base . The following curious congruence was conjectured by L. Skula and proved by A. Granville In this note we establish the above congruence by entirely elementary number theory arguments.
An exercise on Fibonacci representations
We give a partial answer to a question of Carlitz asking for a closed formula for the number of distinct representations of an integer in the Fibonacci base.
An extension of a theorem of Nielsen
An extension of Daykin's generalized Möbius function to unitary divisors.
An identity for a class of arithmetical functions of several variables.
An identity for a class of arithmetical functions of two variables.
An identity involving multiplicative orders.
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