In this paper we are interested to show how to solve the diophantine equation x 2 + 3y 2 = z 2 by using the arithmetic technical.

In this paper, we show that if an automorphism α of an abelian torsion group, which is in fact a direct sum of its p-components, has the weak extension property then α = πid Ap + ρ, where p is a prime number, π is an invertible p-adic number and ρ ∈ Hom(A p , A 1 p ) with A 1 p is the first Ulm subgroup of the p-component A p of A.

An endomorphism f of an Abelian group A is said to be inessential (in the category of Abelian groups) if it can be extended to an endomorphism of any Abelian group which contains A as a subgroup. In this paper we show that f is as above if and only if (f − v id A )(A) is contained in the maximal divisible subgroup of A for some v ∈ Z.

In this paper. We are intersted to characterize the solutions of the diophantine equation x2 + y2 = 2z2 by using the arithmetic technicals.

No abstract