Abstract. Kurepa's conjecture states that there is no odd prime p that divides !p = 0!+1!+· · ·+(p−1)!. We search for a counterexample to this conjecture for all p < 2 34 . We introduce new optimization techniques and perform the computation using graphics processing units. Additionally, we consider the generalized Kurepa's left factorial given by ! k n = (0!) k +(1!) k +· · ·+((n−1)!) k , and show that for all integers 1 < k < 100 there exists an odd prime p such that p |! k p.
This paper is devoted to the study of the relation between Osserman algebraic curvature tensors and algebraic curvature tensors which satisfy the duality principle. We give a short overview of the duality principle in Osserman manifolds and extend this notion to null vectors. Here, it is proved that a Lorentzian totally Jacobi-dual curvature tensor is a real space form. Also, we find out that a Clifford curvature tensor is Jacobi-dual. We provide a few examples of Osserman manifolds which are totally Jacobidual and an example of an Osserman manifold which is not totally Jacobi-dual.
We investigate the connection between the duality principle and the Osserman condition in a pseudo-Riemannian setting. We prove that a connected pointwise two-leaves Osserman manifold of dimension n 5 is globally Osserman and investigate the relation between the special Osserman condition and the two-leaves Osserman one.
Fibonacci numbers have engaged the attention of mathematicians for several centuries, and whilst many of their properties are easy to establish by very simple methods, there are several unsolved problems connected to them. In this paper we review the history of the conjecture that the only perfect powers in Fibonacci sequence are 1, 8, and 144. Afterwards we consider more stronger conjecture and give the new characterization of closely related Wall-Sun-Sun primes.
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