Let A be an elliptic curve over Q of square free conductor N . Suppose A has a rational torsion point of prime order r such that r does not divide 6N . We prove that then r divides the order of the cuspidal subgroup C of J0(N). If A is optimal, then viewing A as an abelian subvariety of J0(N), our proof shows more precisely that r divides the order of A ∩ C. Also, under the hypotheses above, we...

Fermat wrote in a letter to Frenicle, that whenever p is prime, p divides a p?1 ? 1 for all integers a not divisible by p, a result now known as Fermat's `little theorem'. An equivalent formulation is the assertion that p divides a p ? a for all integers a, whenever p is prime. The question naturally arose as to whether the primes are the only integers exceeding 1 that satisfy this criterion, b...

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For n ≥ 3, let r = r(n) ≥ 3 be an integer and let k = (k1, . . . , kn) be a vector of nonnegative integers, where each kj = kj(n) may depend on n. Let M = M(n) = ∑n j=1 kj for all n ≥ 3, and define the set I = {n ≥ 3 | r(n) divides ...

Let $R$ be a unitary ring with an endomorphism $σ$ and $F∪{0}$ be the free monoid generated by $U={u_1,…,u_t}$ with $0$ added, and $M$ be a factor of $F$ setting certain monomial in $U$ to $0$, enough so that, for some natural number $n$, $M^n=0$. In this paper, we give a sufficient condition for a ring $R$ such that the skew monoid ring $R*M$ is quasi-Armendariz (By Hirano a ring $R$ is called...

This note answers, and generalizes, a question of Kaisa Matomaki. We show that given two cuspidal automorphic representations $\pi_1$ $\pi_2$ $GL(n)$ over number field $F$ respective conductors $N_1$, $N_2$, every character $\chi$ such $\pi_1\otimes\chi\simeq\pi_2$ conductor $Q$, satisfies the bound: $Q^n\mid N_1N_2$. If at finite place $v$, $\pi_{1,v}$ is discrete series whenever it ramified, ...

‎In this work‎, ‎we introduce the concept of classical 2-absorbing secondary modules over a commutative ring as a generalization of secondary modules and investigate some basic properties of this class of modules‎. ‎Let $R$ be a commutative ring with‎ ‎identity‎. ‎We say that a non-zero submodule $N$ of an $R$-module $M$ is a‎ ‎emph{classical 2-absorbing secondary submodule} of $M$ ...

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 2 P...

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. Suppose that $phi:S(M)rightarrow S(M)cup lbraceemptysetrbrace$ be a function where $S(M)$ is the set of all submodules of $M$. A proper submodule $N$ of $M$ is called an $(n-1, n)$-$phi$-classical prime submodule, if whenever $r_{1},ldots,r_{n-1}in R$ and $min M$ with $r_{1}ldots r_{n-1}min Nsetminusphi(N)$, then $r_{1...

Let In = {1, 2, . . . , n} and x : In 7→ R be a map such that ∑ i∈In x(i) ≥ 0. (For any i, its image is denoted by x(i).) Let F = {J ⊂ In : |J | = k, and ∑ j∈J x(j) ≥ 0}. In [25] Manickam and Singhi have conjectured that |F| ≥ ( n−1 k−1 ) whenever n ≥ 4k and showed that the conclusion of the conjecture holds when k divides n. For any two integers r and ` let [r]` denote the smallest positive in...