Abstract. Let l be a positive integer and s = (s1, . . . , sl) be a sequence of positive integers. In this paper we shall study the arithmetic properties of multiple harmonic sum H(s;n) which is the n-th partial sum of multiple zeta value series ζ(s). We conjecture that for every s and every prime p there are only finitely many p-integral partial sums H(s;n). This generalizes a conjecture of Es...

Given a finite nonempty sequence S of integers, write it as XY k, where Y k is a power of greatest exponent that is a suffix of S: this k is the curling number of S. The curling number conjecture is that if one starts with any initial sequence S, and extends it by repeatedly appending the curling number of the current sequence, the sequence will eventually reach 1. The conjecture remains open. ...

In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group Sym(n), the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points.

In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group Sym(n), the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points.

The Sylvester-Gallai theorem asserts that every finite set S of points in two-dimensional Euclidean space includes two points, a and b, such that either there is no other point in S is on the line ab, or the line ab contains all the points in S. V. Chvátal extended the notion of lines to arbitrary metric spaces and made a conjecture that generalizes the Sylvester-Gallai theorem. In the present ...

Very recently, Bialostocki et al. proposed the following conjecture. Let r, s be two nonnegative integers and let G = (V (G), E(G)) be a graph with |V (G)| ≥ 3r + 4s and minimum degree δ(G) ≥ 2r + 3s. Then G contains a collection of r cycles and s chorded cycles, all vertex-disjoint. We prove that this conjecture is true.

Let GF be a finite group of Lie type, where G is reductive defined over F¯q and F Frobenius root. Lusztig’s Jordan decomposition parametrises the irreducible characters in rational series E(GF,(s)G∗F∗) s∈G∗F∗ by E(CG∗(s)F∗,1). We conjecture that Shintani twisting preserves space class functions generated union E(GF,(s′)G∗F∗) (s′)G∗F∗ runs semi-simple classes G∗F∗ geometrically conjugate to s; f...

It is conjectured by Erdős, Graham and Spencer that if 1 ≤ a1 ≤ a2 ≤ · · · ≤ as are integers with ∑s i=1 1/ai < n − 1/30, then this sum can be decomposed into n parts so that all partial sums are ≤ 1. This is not true for ∑s i=1 1/ai = n − 1/30 as shown by a1 = · · · = an−2 = 1, an−1 = 2, an = an+1 = 3, an+2 = · · · = an+5 = 5. In 1997 Sandor proved that Erdős–Graham–Spencer conjecture is true ...

We further recall that under Selberg orthonormality conjecture, has unique factorization into primitive functions, the only primitive function with a pole at s = 1 is the Riemann zeta function ζ(s), and Fθ(s) is a primitive function if θ ∈R and if F ∈ are primitive and entire (see [4, Section 4]). We say a primitive function F ∈ is normal if θF = 0. Assuming Selberg orthonormality conjecture, w...

ferred to in the literature as the Shimura-Taniyama-Weil conjecture, the Taniyama-Shimura conjecture, the Taniyama-Weil conjecture, or the modularity conjecture, it postulates a deep connection between elliptic curves over the rational numbers and modular forms. It has now been almost completely proved thanks to the fundamental work of A. Wiles and R. Taylor [W], [TW], and its further refinemen...