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...

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

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...

Input: An integer n > 1. 0: if n is a power then output composite fi; 1: r := 2; 2: while (r < n) do 3: if gcd(r, n) 6= 1 then output composite fi; 4: if r is prime then 5: q := largest prime factor of r − 1; 6: if (q ≥ 4√r log n) and (n(r−1)/q 6≡ 1 mod r) then break fi; 7: fi; 8: r := r + 1; 9: od; 10: for a = 1 to 2 √ r log n do 11: if (x− a)n 6≡ (xn − a) mod (xr − 1, n) then output composite...

Let f(n) or the base-2 logarithm of f(n) be either d(n) (the divisor function), σ(n) (the divisor-sum function), φ(n) (the Euler totient function), ω(n) (the number of distinct prime factors of n) or Ω(n) (the total number of prime factors of n). We present good lower bounds for ∣ ∣M N − α ∣ ∣ in terms of N , where α = [0; f(1), f(2), . . .].

A ring $A$ is (principally) nilary, denoted (pr-)nilary, if whenever $XY=0,$ then there exists a positive integer $n$ such that either $X^n=0$ or $Y^n=0$ for all (principal) ideals $X$, $Y$ of $A$. We determine necessary and/or sufficient conditions the group $A[G]$ to be nilary in terms on $G$. For example, we show that: (1) If (pr-)nilary and $G$ prime order each finite nontrivial normal subg...

1. We say that g(n) is additive if g(mn)=g(m)+g(n) holds for every coprime pairs m, n of positive integers . If, moreover, g(p') =g(p)' for every prime power pa , then g(n) is called strongly additive . By p, pi , p2 , . . ., q, qr , q2 , . . . we denote prime numbers, c, Cl, c2i . . . are suitable positive constants . F(n) and x(n) denote the largest and the smallest prime factor of n . The sy...