Suppose that f : Fpn → [0, 1] satisfies Σaf(m) = θF ∈ [F , F ], where F = |Fpn | = p. In this paper we will show the following: Let fj denote the size of the jth largest Fourier coefficient of f . If fj < θ j1/2+δF, for some integer j satisfying J0(δ, p) < j < F , then S = support(f) contains a non-trivial three-term arithmetic progression. Thus, the result is asserting that if the Fourier tran...

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.

‎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 $frac{A_n}{G_n}$.

MR1631259 (2000d:11019) 11B25; 11N13 Gowers, W. T. A new proof of Szemerédi’s theorem for arithmetic progressions of length four. Geometric and Functional Analysis 8 (1998), no. 3, 529–551. This remarkable paper gives a new proof that every subset of the integers with positive density must contain arithmetic progressions of length four. This was conjectured by P. Erdős and P. Turán [J. London M...

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.

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 = ...

Several authors have investigated the problem of finding elliptic curves over Q that contain rational points whose x-coordinates are in arithmetic progression. Traditionally, the elliptic curve has been taken in the form of an elliptic cubic or elliptic quartic. Moody studied this question for elliptic curves in Edwards form, and showed that there are infinitely many such curves upon which ther...