Let H be a Krull monoid with class group G and suppose that each class contains a prime divisor. Then every element a ∈ H has a factorization into irreducible elements, and the set L(a) of all possible factorization lengths for a is the set of lengths of a. We consider the system L(H) = {L(a) | a ∈ H} of all sets of lengths, and we characterize (in terms of the class group G) when L(H) is addit...

Let D = (Dn)n≥1 be an elliptic divisibility sequence. We study the set S(D) of indices n satisfying n | Dn. In particular, given an index n ∈ S(D), we explain how to construct elements nd ∈ S(D), where d is either a prime divisor of Dn, or d is the product of the primes in an aliquot cycle for D. We also give bounds for the exceptional indices that are not constructed in this way.

A famous conjecture about group algebras of torsion-free groups states that there is no zero divisor in such algebras. recent approach to settle the show non-existence divisors with respect length possible ones, where by we mean size support an element algebra. The case $2$ cannot be happen. first unsettled existence $3$. Here study $3$ rational and over field $p$ elements for some prime $p$.

A random binary search tree grown from the uniformly random permutation of [n] is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction ck of vertices of a fixed rank k ≥ 0 is shown to decay exponentially with k. Notoriously hard to compute, the exact fractions ck had been determined for k ≤ 3 only. We computed c4 a...

The set $\mathbb{Z}_p[x]$ consists of all polynomials with coefficients in the field $\mathbb{Z}_p$, where $p$ is prime. If a polynomial $f(x)$ irreducible over $\mathbb{Z}_p$ then $\frac{\mathbb{Z}_p[x] }{ \langle f(x) \rangle}$ field. reducible polynomial, every non-zero element }{\langle either zero-divisor or unit. we exclude zero-divisors and zero, have finite Abelian group under multiplic...

/ = /i$i. Since (to)i = 5ii is in (s), and ni is prime to the order of Si, Si is a power of s. Thus h, as well as t, corresponds to t in the isomorphism of G with G/(s) ; but h and /' are of the same order tii. Every element of G/(s) whose order is a divisor of m/mi corresponds to an element of G whose order is a divisor of m. It follows that t', and hence every element of G/(s) whose order div...

Let G be a cyclic group generated by g whose group operations are written additively. Assume that the order of G is a prime p. Let d be a divisor of p+1. Let c ∈ Fp. Given cg, c g, . . ., cg, Cheon[4] gave an algorithm to compute c more efficiently than solving ordinal discrete logarithm problems. With improvement by Kozaki, Kutsuma and Matsuo[5], the algorithm runs with O(max(d, p p/d)) group ...

Let K be the field of rational functions in 2 variables over an algebraically closed field k of characteristic 0. Let D be a finite dimensional K-central division algebra whose ramification divisor on the projective plane over k is a singular cubic curve. It is shown that D is cyclic and that the exponent of D is equal to the degree of D. Let k be an algebraically closed field of characteristic...

Assume N is prime, so X0(N)Q has two cusps; these are labelled 0 and ∞. The two cusps are Q-rational points on X0(N)Q and the divisor (0) − (∞) on X0(N)Q generates a finite cyclic subgroup C in J0(N)(Q) called the cuspidal divisor group. Denote by C/Z the finite flat subgroup scheme of J generated by C ⊂ J0(N)(Q). Let C be the FN -valued points of CZ (“the specialization” of C in J × FN ). It i...

In 1988 P. Erdös asked if the prime divisors of xn−1 for all n = 1, 2, . . . determine the given integer x; the problem was affirmatively answered by Corrales-Rodorigáñez and R. Schoof [2] in 1997 together with its elliptic version. Analogously, K. Yamanoi [14] proved in 2004 that the support of the pull-backed divisor f∗D of an ample divisor on an abelian variety A by an algebraically non-dege...