On some rings of arithmetical functions.
On some special forms of simultaneous Pell equations
On some special quartic reciprocity laws
On some subgroup chains related to Kneser’s theorem
A recent result of Balandraud shows that for every subset of an abelian group there exists a non trivial subgroup such that holds only if . Notice that Kneser’s Theorem only gives .This strong form of Kneser’s theorem follows from some nice properties of a certain poset investigated by Balandraud. We consider an analogous poset for nonabelian groups and, by using classical tools from Additive Number Theory, extend some of the above results. In particular we obtain short proofs of Balandraud’s...
On some subgroups of the multiplicative group of finite rings
Let be a subset of , the field of elements and a polynomial of degree with no roots in . Consider the group generated by the image of in the group of units of the ring . In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of the projective...
On some sums involving the largest prime divisor of n
On some sums involving the largest prime divisor of n, II
On some tauberian theorems related to the prime number theorem
On Some Ternary Quartic Diophantine Equations.
On some theorems of Littlewood and Selberg, IV
On some topics connected with Waring's problem.
On some truncated units in algebraic number fields of degree n ... 3.
On some universal sums of generalized polygonal numbers
For m = 3,4,... those pₘ(x) = (m-2)x(x-1)/2 + x with x ∈ ℤ are called generalized m-gonal numbers. Sun (2015) studied for what values of positive integers a,b,c the sum ap₅ + bp₅ + cp₅ is universal over ℤ (i.e., any n ∈ ℕ = 0,1,2,... has the form ap₅(x) + bp₅(y) + cp₅(z) with x,y,z ∈ ℤ). We prove that p₅ + bp₅ + 3p₅ (b = 1,2,3,4,9) and p₅ + 2p₅ + 6p₅ are universal over ℤ, as conjectured by Sun. Sun also conjectured that any n ∈ ℕ can be written as and 3p₃(x) + p₅(y) + p₇(z) with x,y,z ∈ ℕ; in...
On Sophie Germain type criteria for Fermat's Last Theorem
On special values of generalized p-adic hypergeometric functions
On special values of theta functions of genus two
We study a certain finitely generated multiplicative subgroup of the Hilbert class field of a quartic CM field. It consists of special values of certain theta functions of genus 2 and is analogous to the group of Siegel units. Questions of integrality of these specials values are related to the arithmetic of the Siegel moduli space.
On Sprindzuk's classification of transcendental numbers.
On square classes in generalized Fibonacci sequences
Let P and Q be nonzero integers. The generalized Fibonacci and Lucas sequences are defined respectively as follows: U₀ = 0, U₁ = 1, V₀ = 2, V₁ = P and , for n ≥ 1. In this paper, when w ∈ 1,2,3,6, for all odd relatively prime values of P and Q such that P ≥ 1 and P² + 4Q > 0, we determine all n and m satisfying the equation Uₙ = wUₘx². In particular, when k|P and k > 1, we solve the equations Uₙ = kx² and Uₙ = 2kx². As a result, we determine all n such that Uₙ = 6x².
On square values of quadratics
On square values of the product of the Euler totient and sum of divisors functions
If n is a positive integer such that ϕ(n)σ(n) = m² for some positive integer m, then m ≤ n. We put m = n-a and we study the positive integers a arising in this way.