Kaarli, K., On certain classes of algebras generating arithmetical affine complete varieties, General Algebra and Applications , 152-161. Heldermann Verlag, Research and Exposition in Mathematics, Vol. 20 (Berlin, 1993). MR 93m :08006
We consider arithmetic progressions consisting of integers which are y-components of solutions of an equation of the form x2 − dy2 = m. We show that for almost all four-term arithmetic progressions such an equation exists. We construct a seven-term arithmetic
progression with the given property, and also several five-term arithmetic progressions which satisfy two different equations
of the given form. These results are obtained by studying the properties of a parametric family of elliptic curves.
For a large class of arithmetic functions f, it is possible to show that, given an arbitrary integer κ ≤ 2, the string of inequalities f(n + 1) < f(n + 2) < … < f(n + κ) holds for in-finitely many positive integers n. For other arithmetic functions f, such a property fails to hold even for κ = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of n satisfying σ2(n − 1) < σ2 < σ2(n + 1), where σ2(n) = ∑d|nd2. On the other hand, we prove that for the function f(n) := ∑p|np2, we do have f(n − 1) < f(n) < f(n + 1) in finitely often.