We describe a new deterministic algorithm for the problem of constructing k-th power nonresidues in nite elds GF(p n), where p is prime and k is a prime divisor of p n ?1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for xed n and p ! 1, our algorithm runs in polynomial time. Unlike previous algorithms for this problem, this polynomial time bound holds even if k ...

There exist many characterizations for the sporadic simple groups. In this paper we give two new characterizations for the Mathieu sporadic groups. Let M be a Mathieu group and let p be the greatest prime divisor of |M|. In this paper, we prove that M is uniquely determined by |M| and |NM(P)|, where P ∈ Sylp(M). Also we prove that if G is a finite group, then G ∼= M if and only if for every pri...

A construction is given for associating to any symmetric (v, k, X) design a self-dual code of length v+ 1 over GF (p), where p is any divisor of the square-free part of k — k. These codes are then used to obtain a substantial strengthening of a result of Hughes on automorphisms of designs. Specifically, suppose a symmetric {v, k, X) design possesses an automorphism a of odd prime order q. If an...

We consider the primes which divide the denominator of the x-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive prime divisor, one that has not appeared in any earlier term. Proofs of this are known in only a few cases. Weaker results in the general direction are give...

1. Let (a, b) and [a, b] be the greatest common divisor and the least common multiple of a and b, respectively . p„ denotes the n'th prime ; p, q, q1 , q2 , . . . are prime numbers. A sum Z and a product f f denote a summation and a multiplication, respectively, over primes indicated . The symbol # { . . .} denotes the number of elements indicated in the bracket 11. Pμ is the product of the fir...

Let X be a curve over a number field K with genus g > 2, p a prime of OK over an unramified rational prime p > 2r, J the Jacobian of X, r = rank J(K), and X a regular proper model of X at p. Suppose r < g. We prove that #X(K) 6 #X (Fp) + 2r, extending the refined version of the Chabauty–Coleman bound to the case of bad reduction. The new technical insight is to isolate variants of the classical...

We use the hyperelliptic curve C : y + xy = x + tx + x + t over the field F2[t]/(t 113 + t + 1) which was generated by Wouter Castryck, Katholieke Universiteit Leuven, Belgium. The choice of the finite field takes into account three aspects: Firstly, it allows for an order of the divisor class group of appropriate size for the desired security level. Secondly, the extension degree of F2 was cho...

Let Γ be a congruence subgroup of SL(2,Z). The modular curve X(Γ) and its Jacobian variety J(Γ) are very important objects in number theory. For instance, the problem of determining all possible structures of (Q-)rational torsion subgroup of elliptic curves over Q is equivalent to that of determining whether the modular curves X1(N) have non-cuspidal rational points. Also, the celebrated theore...

All rings in this paper are assumed to be commutative with identity, and they will generally also be Noetherian. In several recent papers the asymptotic theory of ideals in Noetherian rings has been introduced and developed. In this new theory the roles played in the standard theory by associated primes, i?-sequences, classical grade, and Cohen-Macaulay rings are played by, respectively, asympt...

Let R be a commutative ring and Z(R) the set of all zero divisors R. Γ(R) is said to divisor graph if x,y ∈ V (Γ(R)) = (x,y) E(Γ(R)) only x.y 0. In this paper, we determine total vertex irregularity strength graphs associated with rings ℤp2 × Zq for p,q prime numbers.