# On Kurepa's problems in number Theory

@article{Ivic2003OnKP, title={On Kurepa's problems in number Theory}, author={A. Ivi'c and vZarko Mijajlovi'c}, journal={arXiv: Number Theory}, year={2003} }

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 $0! + 1! + ... + (p-1)!$.

#### 12 Citations

Some remarks on Kurepa's left factorial

- Mathematics
- 2004

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,… Expand

Some considerations in connection with alternating Kurepa's function

- Mathematics, Physics
- 2004

In this paper, we consider the functional equation for alternating factorial sum and some of its particular solutions alternating Kurepa's function A(z) and function A 1(z). We determine an extension… Expand

Searching for a counterexample to Kurepa's conjecture

- Computer Science, Mathematics
- Math. Comput.
- 2016

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… Expand

The function v M m ( s ; a , z ) and some well-known sequences

- 2002

In this paper we define the function vMm(s; a, z), and we study the special cases 1Mm(s; a, z) and nM−1(1; 1, n + 1). We prove some new equivalents of Kurepa’s hypothesis for the left factorial.… Expand

Bell Numbers Modulo a Prime Number, Traces and Trinomials

- Mathematics, Computer Science
- Electron. J. Comb.
- 2014

Several applications of the formula and of the condition are included, and equivalent forms of the conjecture of Kurepa that $B(p-1)$ is $\neq 1$ modulo $p$ are given. Expand

Generalized Factorial Functions, Numbers and Polynomials

- 2004

The generalized factorial functions and numbers and some classes of polynomials associated with them are considered. The recurrence relations, several representations, asymptotic and other properties… Expand

On Some Finite Sums with Factorials

- Mathematics, Physics
- 2004

The summation formula $$ \sum^{n-1}_{i=0}\epsilon^i i! (i^k+u_k) = v_k+\epsilon^{n-1} n! A_{k-1}(n) $$ $(\epsilon=\pm 1; k=1,2,...; u_k, v_k\in \msbm\hbox{Z}; A_{k-1}$ is a polynomial) is derived and… Expand

The Kurepa-Vandermonde matrices arising from Kurepa's left factorial hypothesis

- Mathematics
- 2015

Kurepa's (left factorial) hypothesis asserts that for each integer $n\ge 2$ the greatest common divisor of $!n:=\sum_{k=0}^{n-1}k!$ and $n!$ is $2$. It is known that Kurepa's hypothesis is equivalent… Expand

A pr 2 00 8 DIFFERENTIALLY TRANSCENDENTAL FUNCTIONS

- 2008

The aim of this article is to exhibit a method for proving that certain analytic functions are not solutions of algebraic differential equations. The method is based on model-theoretic properties of… Expand

3 1 Ja n 20 05 DIFFERENTIALLY TRANSCENDENTAL FUNCTIONS

- 2005

The aim of this paper is to exhibit a method for proving that certain analytic functions are not solutions of algebraic differential equations. The method is based on model-theoretic properties of… Expand

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