Let q denote a large prime number, a be any integer. Let N(q, a) denote the smallest positive integer n for which φ(n) ≡ a (mod q). The number N(q, a) exists. Indeed, if a+1 ≡ 0 (mod q) then one can take n = q. Otherwise, one can take n to be a prime from the arithmetical progression a + 1 (mod q). The problem of upper bound estimates for N(q, a) has been a subject of study of the work of Fried...

Let N be a prime Γ-near ring with multiplicative center Z. Let σ and τ be automorphisms of N and δ be a Γ− (σ, τ)-derivation of N such that N is 2-torsion free. In this paper the following results are proved: (1) If σγδ = δγσ and τγδ = δγτ and δ(N) ⊆ Z, or [δ(x), δ(y)]γ = 0, for all x, y ∈ N and γ ∈ Γ, then N is a commutative ring. (2) If δ1 is a Γ-derivation, δ2 is a Γ − (σ, τ) derivation of N...

in this article‎, ‎we have characterized ideals in $c(x)$ in which‎ ‎every ideal is also an ideal (a $z$-ideal) of $c(x)$‎. ‎motivated by‎ ‎this characterization‎, ‎we observe that $c_infty(x)$ is a regular‎ ‎ring if and only if every open locally compact $sigma$-compact‎ ‎subset of $x$ is finite‎. ‎concerning prime ideals‎, ‎it is shown that‎ ‎the sum of every two prime (semiprime) ideals of e...

Let p > 3 be a prime. We derive the following new congru-ences: p−1 n=0 (2n + 1)A n ≡ p (mod p 4) and p−1 n=0

We investigate one-point algebraic geometric codes CL(D, G) associated to maximal curves recently characterized by Tafazolian and Torres given by the affine equation yl = f(x), where f(x) is a separable polynomial of degree r relatively prime to l. We mainly focus on the curve y4 = x3 +x and Picard curves given by the equations y3 = x4-x and y3 = x4 -1. As a result, we obtain exact value of min...

Suppose that {Un}n≥0 is a Lucas sequence, and suppose that l1, . . . , lt are primes. We show that the equation Un1 · · ·Unm = ±l x1 1 · · · l xt t y , p prime, m < p, has only finitely many solutions. Moreover, we explain a practical method of solving these equations. For example, if {Fn}n≥0 is the Fibonacci sequence, then we solve the equation Fn1 · · ·Fnm = 2 x1 · 32 · 53 · · · 541100y under...

Let R be an hereditary Noetherian prime ring, let S be a "Dedekind closure" of R and let T be the category of finitely generated S-torsion R-modules. It is shown that for all i ~ 0, there is an exact sequence 0 -> Ki(T) -> Ki(R) -> Ki(S) ...... O. If i = 0, or R has finitely many idempotent ideals then this sequence splits. A notion of ''right ideal class group" is then introduced for hereditar...