Poznámka o vzorci pro součet kladných a celistvých mocnin čísel přirozené řady
PPCM de suites de polynômes
Preface
Preservation of log-concavity on summation
We extend Hoggar's theorem that the sum of two independent discrete-valued log-concave random variables is itself log-concave. We introduce conditions under which the result still holds for dependent variables. We argue that these conditions are natural by giving some applications. Firstly, we use our main theorem to give simple proofs of the log-concavity of the Stirling numbers of the second kind and of the Eulerian numbers. Secondly, we prove results concerning the log-concavity of the sum of...
Primality tests using algebraic groups.
Prime constellations in triangles with binomial coefficient congruences
The primality of numbers, or of a number constellation, will be determined from residue solutions in the simultaneous congruence equations for binomial coefficients found in Pascal’s triangle. A prime constellation is a set of integers containing all prime numbers. By analyzing these congruences, we can verify the primality of any number. We present different arrangements of binomial coefficient elements for Pascal’s triangle, such as by the row shift method of Mann and Shanks and especially by...
Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields
Let be a linear integer recurrent sequence of order , and define as the set of primes that divide at least one term of . We give a heuristic approach to the problem whether has a natural density, and prove that part of our heuristics is correct. Under the assumption of a generalization of Artin’s primitive root conjecture, we find that has positive lower density for “generic” sequences . Some numerical examples are included.
Prime divisors of Lucas sequences
Prime divisors of some shifted products.
Prime divisors of the Lagarias sequence
We solve a 1985 challenge problem posed by Lagarias [5] by determining, under GRH, the density of the set of prime numbers that occur as divisor of some term of the sequence defined by the linear recurrence and the initial values and . This is the first example of a ænon-torsionÆ second order recurrent sequence with irreducible recurrence relation for which we can determine the associated density of prime divisors.
Prime factors of with an irreducible polynomial .
Prime factors of binomial coefficients and related problems
Prime numbers along Rudin–Shapiro sequences
For a large class of digital functions , we estimate the sums (and , where denotes the von Mangoldt function (and the Möbius function). We deduce from these estimates a Prime Number Theorem (and a Möbius randomness principle) for sequences of integers with digit properties including the Rudin-Shapiro sequence and some of its generalizations.
Prime numbers with Beatty sequences
A study of certain Hamiltonian systems has led Y. Long to conjecture the existence of infinitely many primes which are not of the form p = 2⌊αn⌋ + 1, where 1 < α < 2 is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately implies that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic...
Prime power divisors of binomial coefficients.
Prime power divisors of binomial coefficients: Reprise.
Primefree shifted Lucas sequences
We say a sequence is primefree if |sₙ| is not prime for all n ≥ 0, and to rule out trivial situations, we require that no single prime divides all terms of . In this article, we focus on the particular Lucas sequences of the first kind, , defined by u₀ = 0, u₁ = 1, and uₙ = aun-1 + un-2 for n≥2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k such that both of the shifted sequences are simultaneously primefree. This result extends...
Prime-like sequences
Primes in Fibonacci -step and Lucas -step sequences.
Primes in intervals