The famous Erd˝ os-Heilbronn conjecture plays an important role in the development of additive combinatorics. In 2007 Z. W. Sun made the following further conjecture (which is the linear extension of the Erd˝ os-Heilbronn conjecture): For any finite subset A of a field F and nonzero elements a where p(F) is the additive order of the multiplicative identity of F , and δ ∈ {0, 1} takes the value ...

We use Greither’s recent results on Brumer’s Conjecture to prove Rubin’s integral version of Stark’s Conjecture, up to a power of 2, for an infinite class of CM extensions of totally real number fields, called “nice extensions”. As a consequence, we show that the Brumer–Stark Conjecture is true for “nice extensions”, up to a power of 2.

For Turán’s (3, 4)-conjecture, in the case of n = 3k +1 vertices, 1 2 6k−1 non-isomorphic hypergraphs are constructed that attain the conjecture. In the case of n = 3k + 2 vertices, 6k−1 non-isomorphic hypergraphs are constructed that attain the conjecture.

We refine the conjecture of Coppersmith and Steinberger [2] on nonnegative integer linear combinations and prove the new conjecture for three classes of coset combinations, including a class for which the original conjecture was still open.

We verify the skew weighing matrix conjecture for orders 2t .13, t 2:: 5, and give new results for 2t.15 proving the conjecture for t :2: 3.

We prove Stanley’s plethysm conjecture for the 2×n case, which composed with the work of Black and List provides another proof of Foulkes conjecture for the 2× n case. We also show that the way Stanley formulated his conjecture, it is false in general, and suggest an alternative formulation.

for a finite group $g$‎, ‎let $cent(g)$ denote the set of centralizers of single elements of $g$‎. ‎in this note we prove that if $|g|leq frac{3}{2}|cent(g)|$ and $g$ is 2-nilpotent‎, ‎then $gcong s_3‎, ‎d_{10}$ or $s_3times s_3$‎. ‎this result gives a partial and positive answer to a conjecture raised by a‎. ‎r‎. ‎ashrafi [on finite groups with a given number of centralizers‎, ‎algebra‎ ‎collo...

The Erdős–Faber–Lovász conjecture states that if a graph G is the union of n cliques of size n no two of which share more than one vertex, then χ(G) = n. We provide a formulation of this conjecture in terms of maximal partial clones of partial operations on a set.

Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...