Semihappy numbers.
Sequences and series involving the sequence of composite numbers.
Sequences in power residue classes.
Sequences of generalized happy numbers with small bases.
Sequences of reducible -polynomials modulo a prime.
Sequenzen der Länge 2 von Restklassencharakteren.
Séries formelles et théorie des automates.
Sets of integers composed of few prime numbers
Sets of -expansions and the Hausdorff measure of slices through fractals
We study natural measures on sets of -expansions and on slices through self similar sets. In the setting of -expansions, these allow us to better understand the measure of maximal entropy for the random -transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading...
Seven octonary quadratic forms
Several inequalities about the number of positive divisors of a natural number .
Sharp elementary estimates for the sequence of primes
Short character sums with Fermat quotients
Short remark on Fibonacci-Wieferich primes
This paper has been inspired by the endeavour of a large number of mathematicians to discover a Fibonacci-Wieferich prime. An exhaustive computer search has not been successful up to the present even though there exists a conjecture that there are infinitely many such primes. This conjecture is based on the assumption that the probability that a prime is Fibonacci-Wieferich is equal to . According to our computational results and some theoretical consideratons, another form of probability can...
Sign changes of certain arithmetical function at prime powers
We examine an arithmetical function defined by recursion relations on the sequence and obtain sufficient condition(s) for the sequence to change sign infinitely often. As an application we give criteria for infinitely many sign changes of Chebyshev polynomials and that of sequence formed by the Fourier coefficients of a cusp form.
Signed bits and fast exponentiation
An exact analysis is given of the benefits of using the non-adjacent form representation for integers (rather than the binary representation), when computing powers of elements in a group in which inverting is easy. By counting the number of multiplications for a random exponent requiring a given number of bits in its binary representation, we arrive at a precise version of the known asymptotic result that on average one in three signed bits in the non-adjacent form is non-zero. This shows that...
Simple groups of square order and interesting sequence of primes
Simple proofs of some generalizations of the Wilson’s theorem
In this paper a remarkable simple proof of the Gauss’s generalization of the Wilson’s theorem is given. The proof is based on properties of a subgroup generated by element of order 2 of a finite abelian group. Some conditions equivalent to the cyclicity of (Φ(n), ·n), where n > 2 is an integer are presented, in particular, a condition for the existence of the unique element of order 2 in such a group.
Sobre el cardinal de ciertos subconjuntos finitos del cuadrado unidad.
Solution des questions proposées par le P. Pepin