‎we obtain the asymptotic expansion of the sequence with general term $frac{a_n}{g_n}$‎, ‎where $a_n$ and $g_n$ are the arithmetic and geometric means of the numbers $d(1),d(2),dots,d(n)$‎, ‎with $d(n)$ denoting the number of positive divisors of $n$‎. ‎also‎, ‎we obtain some explicit bounds concerning $g_n$ and $frac{a_n}{g_n}$.

We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterization is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose si...

In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic progressions. It is well known and obvious that neither class must contain an infinite arithmetic progression. In fact, it is easy to see that for any sequence an there is another sequence bn9 with b...

In this note we prove that there is a linear ordering of the set of real numbers for which there is no monotonic 3-term arithmetic progression. This answers the question (asked by Erdős and Graham) of whether or not every linear ordering of the reals must have a monotonic k-term arithmetic progression for every k.

Iannucci considered the positive divisors of a natural number n that do not exceed and found all forms numbers whose such are in arithmetic progression. In this paper, we generalize Iannucci’s result by excluding trivial 1 (when is square). Surprisingly, length our progression cannot 5.

Let p(n) denote the ordinary partition function. Subbarao conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N ≡ r (mod t) for which p(N) is even, and infinitely many integers M ≡ r (mod t) for which p(M) is odd. We prove the conjecture for every arithmetic progression whose modulus is a power of 2.

Let p(n) denote the number of partitions of a non-negative integer n. A well-known conjecture asserts that every arithmetic progression contains infinitely many integers M for which p(M) is odd, as well as infinitely many integers N for which p(N) is even (see Subbarao [22]). From the works of various authors, this conjecture has been verified for every arithmetic progression with modulus t whe...

We study arithmetic progression in the x-coordinate of rational points on genus two curves. As we know, there are two models for the curve C of genus two: C : y = f5(x) or C : y = f6(x), where f5, f6 ∈ Q[x], deg f5 = 5, deg f6 = 6 and the polynomials f5, f6 do not have multiple roots. First we prove that there exists an infinite family of curves of the form y = f(x), where f ∈ Q[x] and deg f = ...

We prove that if A ⊆ { 1 , … N } does not contain any non-trivial three-term arithmetic progression, then | ≪ ( log ⁡ ) 3 + o .

Suppose a, b, c, and d are rational numbers such that a2, b2, c2, and d2 form an arithmetic progression: the differences b2−a2, c2−b2, and d2−c2 are equal. One possibility is that the arithmetic progression is constant: a2, a2, a2, a2. Are there arithmetic progressions of four rational squares which are not constant? This question was first raised by Fermat in 1640. There are no such progressio...