Praeludium We will call prime numbers the integers n ≥ 2 which are divisible only by 1 and themselves. Euclid (fourth century B. C.) first showed that there exist infinitely many prime numbers. His proof is an excellent example of a mathematical argument: if 2, 3, 5, . . . , p were the only prime numbers, we could construct the number 2 · 3 · 5 · · · p + 1, which has the property of leaving the...

An overpartition of the nonnegative integer n is a non-increasing sequence of natural numbers whose sum is n in which the first occurrence of a number may be overlined. Let k ≥ 1 be an integer. An overpartition k-tuple of a positive integer n is a k-tuple of overpartitions wherein all listed parts sum to n. Let pk(n) be the number of overpartition k-tuples of n. In this paper, we will give a sh...

Forward We start with the set of natural numbers, N = {1, 2, 3, . . .} equipped with the familiar addition and multiplication and assume that it satisfies the induction axiom. It allows us to establish division with a residue and the Euclid’s algorithm that computes the greatest commond divisor of two natural numbers. It also leads to a proof of the fundamental theorem of arithmetic: Every natu...

It is well known that the Fibonacci number Fn can be a prime only If n 4 or n p, where p is an odd prime. Throughout this paper, p will denote a prime. In a very interesting paper, Drobot [2] proved that Fp is composite for certain primes p. In particular, he proved that if p > 7, p = 2 or 4 (mod 5), and 2p -1 is also a prime, then 2p -11 Fp and Fp > 2p . For example, 371 Fl9 = 4181-37-113. A s...

We study properties of operators defined on the space E+(p) of right half-infinite sequences with entries chosen from Zp where p is prime. The operators in question allow solution of the problem of finding predecessor states for certain cellular automata evolutions and they can be thought of as discrete integration with respect to sequence index. These operators are self-accumulating, not solip...

*Correspondence: [email protected] 2Department of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450046, P.R. China Full list of author information is available at the end of the article Abstract In this paper, we show that if λ1, λ2, λ3, λ4 are positive real numbers, at least one of the ratios λi/λj (1≤ i < j≤ 4) is irrational, then the inequality |λ1x2 1 +...

A nonempty subset A of {1, 2, . . . , n} is said to be relatively prime if gcd(A) = 1. Nathanson [4] defined f(n) to be the number of relatively prime subsets of {1, 2, . . . , n} and, for k ≥ 1, fk(n) to be the number of relatively prime subsets of {1, 2, . . . , n} of cardinality k. Nathanson [4] defined Φ(n) to be the number of nonempty subsets A of the set {1, 2, . . . , n} such that gcd(A)...

In this paper we classify fuzzy subgroups of a rank-3 abelian group $G = mathbb{Z}_{p^n} + mathbb{Z}_p + mathbb{Z}_p$ for any fixed prime $p$ and any positive integer $n$, using a natural equivalence relation given in cite{mur:01}. We present and prove explicit polynomial formulae for the number of (i) subgroups, (ii) maximal chains of subgroups, (iii) distinct fuzzy subgroups, (iv) non-isomorp...