A generalized Lucas sequence and permutation binomials

In this paper, we search all repdigits which are products of arbitrary three terms of -generalized Lucas sequences. Thus, we find all nonnegative integer solutions of the Diophantine equation , where and for all and .

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes p.

New classes of permutation binomials and permutation trinomials over finite fields

Two generalizations of Lucas sequence

On generalized Lucas sequences

Some Generalized Lucas Sequences

Green-function and information–geometric correspondences between inverse eigenvalue loci of generalized Lucas sequences and the Mandelbrot set

On the Partition of Horadam's Generalized Sequences into Generalized Fibonacci and Generalized Lucas Sequences

There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. However, by extending ideas of Garland, Borel proved that these values are related to the higher K-theory of the ring of integers. Shortly afterwards, Bloch and Kato proposed a Tamagawa number-type conjecture for these values, and showed that it would follow from a result in motivic cohomology which was unknown at the time. This vital result from motivic cohomology was subsequently proven by Huber, Kings, and Wildeshaus. Bringing together key results from K-theory, motivic cohomology, and Iwasawa theory, this book is the first to give a complete proof, accessible to graduate students, of the Bloch–Kato conjecture for odd positive integers. It includes a new account of the results from motivic cohomology by Huber and Kings.

In his second notebook, Ramanujan discovered the following identity for the special values of ζ (s) at the odd positive integers where α and β are positive numbers such that αβ = π 2 and m is a positive integer. As shown by Berndt in the viewpoint of general transformation of analytic Eisenstein series, it is a natural companion of Euler’s famous formula for even zeta values. In this note, we prove an analogue of the above Ramanujan’s identity in the functions fields setting, which involves the Bernoulli-Carlitz numbers.

Let $\mathcal A$ denote a set of positive integers, and let $p(\mathcal A,n)$ denote the associated partition function. Let $\beta $ be an odd positive integer, and let $P(z)$ be a polynomial in $ \mathbb F_2[z]$ of order $\beta $ such that $P(0)=1$. J.-L

We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p p can be a proper divisor of N p = ( p p − 1 ) / ( p − 1 ) N_p = (p^p-1)/(p-1) . It is known that the period always divides N p N_p . The period is shown to equal N p N_p for most primes p p below 180. The investigation leads to interesting new results about the possible prime factors of N p N_p . For example, we show that if p p is an odd positive integer and m m is a positive integer and q = 4 m 2 p + 1 q=4m^2 p+1 is prime, then q q divides p m 2 p − 1 p^{m^2p}-1 . Then we explain how this theorem influences the probability that q q divides N p N_p .

The aim of this paper is to classify all monogenic ternary semigroups, up to isomorphism. We divide them to two groups: finite and infinite. We show that every infinite monogenic ternary semigroup is isomorphic to the ternary semigroup O, the odd positive integers with ordinary addition. Then we prove that all finite monogenic ternary semigroups with the same index and the same period are isomorphic. We also investigate structure of finite monogenic ternary semigroups and we prove that any finite monogenic ternary semigroup is isomorphic to a quotient ternary semigroup.

Two-colorings of a dense subgroup of [formula omitted]n that forbid many distances

Type 2 constacyclic codes over [formula omitted] of oddly even length