Let f(m,n) denote the number of relatively prime subsets of {m + 1,m + 2, . . . , n}, and let Φ(m,n) denote the number of subsets A of {m+1,m+2, . . . , n} such that gcd(A) is relatively prime to n. Let fk(m,n) and Φk(m,n) be the analogous counting functions restricted to sets of cardinality k. Simple explicit formulas and asymptotic estimates are obtained for these four functions. A nonempty s...

Abstract Let R be a ring and let M an -module with $$S={\text {End}}_R(M)$$ S = End R ( M ) . The module is called quasi-dual Baer if for every fully invari...

Let $G$ be a group with identity $e$. $R$ commutative $G$-graded ring non-zero identity, $S\subseteq h(R)$ multiplicatively closed subset of and $M$ graded $R$-module. In this article, we introduce study the concept $S$-1-absorbing prime submodules. A submodule $N$ $(N:_{R}M)\cap S=\emptyset$ is said to prime, if there exists an $s_{g}\in S$ such that whenever $a_{h}b_{h'}m_{k}\in N$, then eith...

A graph G(V,E) with vertex set V is said to have a prime labeling if there exist a bijection f : V (G) → {1, 2, . . . , |V |} such that for each edge xy ∈ E(G), f(x) and f(y) are relatively prime. It becomes an interesting problem to investigate the maximum possible edges that can be constructed in a graph with n vertices having prime labeling property. We interpret this problem in number theor...

Let f (m, n) denote the number of relatively prime subsets of {m+ 1, m + 2,. .. , n}, and let Φ(m, n) denote the number of subsets A of {m + 1, m + 2,. .. , n} such that gcd(A) is relatively prime to n. Let f k (m, n) and Φ k (m, n) be the analogous counting functions restricted to sets of cardinality k. Simple explicit formulas and asymptotic estimates are obtained for these four functions. A ...

Let $$N=2n^2-1$$ or $$N=n^2+n-1$$ , for any $$n\ge 2$$ . $$M=\frac{N-1}{2}$$ We construct families of prime knots with Jones polynomials $$(-1)^M\sum _{k=-M}^{M} (-1)^kt^k$$ Such have Mahler measure equal to 1. If N is prime, these are cyclotomic $$\Phi _{2N}(t)$$ up some shift in the powers t. Otherwise, they products such polynomials, including In particular, all roots unity $$\zeta _{2N}$$ o...

Let $R$ be a commutative ring with identity. A proper submodule $N$ of an $R$-module $M$ is an n-submodule if $rmin N~(rin R, min M)$ with $rnotinsqrt{Ann_R(M)}$, then $min N$. A number of results concerning n-submodules are given. For example, we give other characterizations of n-submodules. Also various properties of n-submodules are considered.