Chou, Hsu and Shiue gave some applications of Faà di Bruno's formula to characterize inverse relations. Our aim is to develop some inverse relations connected to the multipartitional type polynomials involving to binomial type sequences.
In this article we compute the th power values of the quadratic polynomials with negative squarefree discriminant such that is coprime to the class number of the splitting field of over . The theory of unique factorisation and that of primitive divisors of integer sequences is used to deduce a bound on the values of which is small enough to allow the remaining cases to be easily checked. The results are used to determine all perfect power terms of certain polynomially generated integer...
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...
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...
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.
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.