Proof. First note that the zero element of the ring R/I is 0 + I = I and that a + I = I if and only if a ∈ I. Now suppose that I ⊆ R is a prime ideal and consider nonzero cosets a+ I and b+ I in R/I (i.e. consider a 6∈ I and b 6∈ I). Since I is prime this implies that ab 6∈ I, hence ab + I 6= I and we conclude that R/I is an integral domain. Conversely, let R/I be an integral domain and conside...

We continue our study on the elliptic curve discrete logarithm problem over finite extension fields. We show, among other results, the following two results: For sequences of prime powers (qi)i∈N and natural numbers (ni)i∈N with ni −→ ∞ and ni log(qi) −→ 0 for i −→ ∞, the discrete logarithm problem in the groups of rational points of elliptic curves over the fields Fqi i can be solved in subexp...

Theorem 12.2. Let p and q be prime divisors of N , and let `p and `q be the largest prime divisors of p− 1 and q− 1, respectively. If `p ≤ B and `p < `q then Algorithm 12.1 succeeds with probability at least 1− 1 `q . Proof. If a ≡ 0 mod p then the algorithm succeeds in step 2, so we may assume a ⊥ p. When the algorithm reaches ` = `p in step 3 we have b = a m, where m = ∏ `≤`p ` e is a multipl...

Introduction. We shall call a square matrix A of order n an Hadamard matrix or for brevity an iî-matrix, if each element of A has the value ± 1 and if the determinant of A has the maximum possible value w. It is known that such a matrix A is an iï-matrix [ l] if, and only if, AA'~nEn where A f is the transpose of A and En is the unit matrix of order n. It is also known that, if an iï-matrix of ...

Let R be a 2-torsion free σ-prime ring. It is shown here that if U 6⊂ Z(R) is a σ-Lie ideal of R and a, b in R such that aUb = σ(a)Ub = 0, then either a = 0 or b = 0. This result is then applied to study the relationship between the structure of R and certain automorphisms on R. To end this paper, we describe additive maps d : R −→ R such that d(u) = 2ud(u) where u ∈ U, a nonzero σ-square close...

where a, b and (yn)n>o take values in some specified ring and a and b are fixed elements which do not depend on the Integer index n. A solution (yn)„^0 ^ completely specified once the values of yQ and yx are given. It is customary to denote by Fn(a, b) the solution corresponding to the choice y0 = 0, yt = l, and by Ln(a, b) the solution corresponding to the choice y0 = 2, yx = a. Loosely speaki...

Using an almost complete relativistic method based on the Bethe-Salpeter equation, we study mixing angle $\theta$, mass splitting $\bigtriangleup M$, strong decay widths $\Gamma(D^{({\prime})}_1)$ and weak production rates $Br(B\to D^{({\prime})}_1\ell\nu_{\ell})$ of $D_1(2420)$ $D_1^{\prime}(2430)$. We find there is cancellation between $^1P_1$ $^3P_1$ partial waves in $D_1^{\prime}(2430)$ wit...

The theorem about primitive divisors in algebraic number fields is generalized in the following manner. Let A, B be algebraic integers, (A, B) = 1 , AB ^ 0, A/B not a root of unity, and Çk a primitive root of unity of order k . For all sufficiently large n , the number A" C,kB" has a prime ideal factor that does not divide Am £,'kBm for arbitrary m < n and j < k . The analogue of Zsigmondy's th...

Let Fq be the finite field with q = p m elements, where p is an odd prime and m is a positive integer. Let u be a positive integer and Tr be the trace function from Fq to Fp. We define a p-ary linear codes CD = {c(a, b) = (Tr(ax+ by))(x,y)∈D : a, b ∈ Fq}, where D = {(x, y) ∈ Fq\{(0, 0)} : Tr(x1 + y u+1) = 0} and N1 = 1 or 2. In this paper, we use Weil sums to investigate weight distributions of...