Related papers: Kurepa's conjecture on the left factorial function…
We establish a connection between the subfactorial function S(n) and the left factorial function of Kurepa K(n). Some elementary properties and congruences of both functions are described. Finally, we give a calculated distribution of…
In the article [Petojevic 2006], A. Petojevi\' c verified useful properties of the $K_{i}(z)$ functions which generalize Kurepa's [Kurepa 1971] left factorial function. In this note, we present simplified proofs of two of these results and…
We present improved algorithms for computing the left factorial residues $!p=0!+1!+...+(p-1)! \!\mod p$. We use these algorithms for the calculation of the residues $!p\!\mod p$, for all primes $p$ up to $2^{40}$. Our results confirm that…
We show that the Jacobian conjecture of the two dimensional case is true.
We give a new proof of the Mordell-Lang conjecture in positive characteristic for finitely generated subgroups. We also make some progress towards the full Mordell-Lang conjecture in positive characteristic.
Kurepa's conjecture states that there is no odd prime $p$ that divides $!p=0!+1!+\cdots+(p-1)!$. We search for a counterexample to this conjecture for all $p<2^{34}$. We introduce new optimization techniques and perform the computation…
The article presents the proof of Casas-Alvero conjecture.
In the study of Zeilberger's conjecture on an integer sequence related to the Catalan numbers, Lassalle proposed the following conjecture. Let $(t)_n$ denote the rising factorial, and let $\Lambda_{\mathbb{R}}$ denote the algebra of…
We settle in the affirmative the Graham-Sloane conjecture.
In this paper we consider Kurepa's function $K(z)$ \cite{Kurepa_71}. We give some recurrent relations for Kurepa's function via appropriate sequences of rational functions and gamma function. Also, we give some inequalities for Kurepa's…
Kurepa's hypothesis asserts that for each integer $n\ge 2$ the greatest common divisor of $!n:=\sum_{k=0}^{n-1}k!$ and $n!$ is $2$. Motivated by an equivalent formulation of this hypothesis involving derangement numbers, here we give a…
We prove duality between the left and right fractional derivatives, independently on the type of fractional operator. Main result asserts that the right derivative of a function is the dual of the left derivative of the dual function or,…
We establish the exact overlaps conjecture for iterated functions systems on the real line with algebraic contractions and arbitrary translations.
By Probability theory, that is, by a kind of quasi-law of the iterated logarithm, we prove the title claim.
In this paper we consider alternating Kurepa's function $A(z)$ \cite{Petojevic_02}. We give some recurrent relations for alternating Kurepa's function via appropriate sequences of rational functions and gamma function. Also we give some…
It is shown that the question raised in Section 5.7 of [1] has an affirmative answer.
We introduce a new factorial function which agrees with the usual Euler gamma function at both the positive integers and at all half-integers, but which is also entire. We describe the basic features of this function.
We discuss some problems in number theory posed by Djuro Kurepa (1907-1993), including his classical left factorial hypothesis that an odd prime $p$ does not divide $0! + 1! + ... + (p-1)!$.
We state and prove a function field analogue of Furusho for multiple zeta values.
We give a number new examples analytically and numerically to confirm the Kohler conjecture. It turned out that for a rather large class of nonnegative functions the equality (A) hold.