THERE ARE INFINITELY MANY SMARANDACHE DERIVATIONS, INTEGRATIONS AND LUCKY NUMBERS

A number is said to be a Smarandache Lucky Number if an incorrect calculation leads to a correct result. In general, a Smarandache Lucky Method or Algorithm is said to be any incorrect method or algorithm, which leads to a correct result. In this note we find an infinite sequence of distinct lucky fractions

On the prime power factorization of n!

In this paper we prove two results. The first theorem uses a paper of Kim \cite{K} to show that for fixed primes $p_1,...,p_k$, and for fixed integers $m_1,...,m_k$, with $p_i\not|m_i$, the numbers $(e_{p_1}(n),...,e_{p_k}(n))$ are uniformly distributed modulo $(m_1,...,m_k)$, where $e_p(n)$ is the order of the prime $p$ in the factorization of $n!$. That implies one of Sander's conjecture from \cite{S}, for any set of odd primes. Berend \cite{B} asks to find the fastest growing function $f(x)$ so that for large $x$ and any given finite sequence $\epsilon_i\in \{0,1\}, i\le f(x)$, there exists $n<x$ such that the congruences $e_{p_i}(n)\equiv \epsilon_i\pmod 2$ hold for all $i\le f(x)$. Here, $p_i$ is the $i$th prime number. In our second result, we are able to show that $f(x)$ can be taken to be at least $c_1 (\log x/(\log\log x)^6)^{1/9}$, with some absolute constant $c_1$, provided that only the first odd prime numbers are involved.Comment: 7 pages; accepted Journal of Number Theor

Landscape Boolean Functions

In this paper we define a class of Boolean and generalized Boolean functions defined on $\mathbb{F}_2^n$ with values in $\mathbb{Z}_q$ (mostly, we consider $q=2^k$), which we call landscape functions (whose class containing generalized bent, semibent, and plateaued) and find their complete characterization in terms of their components. In particular, we show that the previously published characterizations of generalized bent and plateaued Boolean functions are in fact particular cases of this more general setting. Furthermore, we provide an inductive construction of landscape functions, having any number of nonzero Walsh-Hadamard coefficients. We also completely characterize generalized plateaued functions in terms of the second derivatives and fourth moments.Comment: 19 page