(2) hn divides hm whenever n divides m. They have attracted number theoretical and combinatorial interest as some of the simplest nonlinear recurrence sequences (see [3] for references), but for us their interest lives in the underlying geometry: Ward demonstrates that an elliptic divisibility sequence arises from any choice of elliptic curve over Q and rational point on that curve. Theorem 1 (...

pα||n α. As usual p will denote primes, pα||n means that p divides n but p does not, and a function f(n) is additive if f(mn) = f(m)+f(n) whenever (m,n) = 1. From the pioneering works of Alladi and Erdős [1]-[2], P. Erdős’s perspicacity and insight have been one of the main driving forces in the research that brought on many results on summatory functions of large additive functions and P (n). ...

Let n be a positive integer n and let uj(n), 0 (n) 3 r (n) ? (n) and a(n) be the classical arithmetic functions of n. That is, cj(n), fi(n), and r(n) count the number of distinct prime divisors of n, the total number of prime divisors of n, and the number of divisors of n, respectively, while 4>(n) and a(n) are the Euler function of n and the sum of divisors function of n respectively. A lot...

let r be a commutative ring with identity and m be a unitary r-module. let : s(m) −! s(m) [ {;} be a function, where s(m) is the set of submodules ofm. suppose n  2 is a positive integer. a proper submodule p of m is called(n − 1, n) − -prime, if whenever a1, . . . , an−1 2 r and x 2 m and a1 . . . an−1x 2p(p), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x...

Given integers k ≥ 2 and n ≥ k, let c(n, k) denote the maximum possible number of edges in an n-vertex graph which has no k-connected subgraph. It is immediate that c(n, 2) = n − 1. Mader [2] conjectured that for every k ≥ 2, if n is sufficiently large then c(n, k) ≤ (1.5k − 2)(n− k + 1), where equality holds whenever k − 1 divides n. In this note we prove that when n is sufficiently large then...

For a graph G whose number of edges is divisible by k, let R(G,Zk) denote the minimum integer r such that for every function f : E(Kr) 7→ Zk there is a copy G′ of G in Kr so that ∑ e∈E(G′) f(e) = 0 (in Zk). We prove that for every integer k, R(Kn, Zk) ≤ n + O(k log k) provided n is sufficiently large as a function of k and k divides ( n 2 ) . If, in addition, k is an odd prime-power then R(Kn, ...

In the present paper, using result of Bennett [1] on characterization factorable matrices, we give necessary and sufficient conditions in order that Σλ_{n}x_{n} is summable |R,p_{n}|_{s} whenever Σμ_{n}x_{n} |C,0|_{k}, |C,0|_{s} |R,p_{n}|_{r},. where 1

Let ?: R0?R be a ring homomorphism and suppose that a and a0, respectively, are ideals of R and R0 such that is an Artinian ring. Let M and N be two finitely generated R-modules and suppose that (R0,m0) is a local ring. In this note we prove that the R-modules and are Artinian for all integers i and j, whenever and . Also we will show that if a is principal, then the R-modules and ...

For all positive integers $r\geq 3$ and $n$ such that $r^2-r$ divides an affine plane of order $r$ exists, we construct $r$-edge colored graph with minimum degree $(1-\frac{r-2}{r^2-r})n-2$ the largest monochromatic component has less than $\frac{n}{r-1}$. This generalizes example Guggiari Scott and, independently, Rahimi for $r=3$ thus disproves a conjecture Gyarfas Sarkozy exists.

In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if n divides ( n k ) , then the complete k-uniform hypergraph on n vertices has a decomposition into Hamilton Berge cycles. Here a Berge cycle consists of an alternating sequence v1, e1, v2, . . . , vn, en of distinct vertices vi and distinct edges ei so that each ei contains vi and vi+1. So the divisibility condition is clearly nec...