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 $...

We study k dimensional Latin hypercubes of order n. We describe the automorphism groups of the hypercubes and define the parity of a hypercube and relate the parity with the determinant of a permutation hypercube. We determine the parity in the orbits of the automorphism group. Based on this definition of parity we make a conjecture similar to the Alon-Tarsi conjecture. We define an orthogonali...

Assuming finiteness of the Tate–Shafarevich group, we prove that Birch–Swinnerton–Dyer conjecture correctly predicts parity rank semistable principally polarised abelian surfaces. If surface in question is Jacobian a curve, require curve has good ordinary reduction at 2-adic places.

We study the parity of the class number of the pth cyclotomic field for p prime. By analytic methods we derive a parity criterion in terms of polynomials over the field of 2 elements. The conjecture that the class number is odd for p a prime of the form 2q +1, with q prime, is proved in special cases, and a heuristic argument is given in favor of the conjecture. An implementation of the criteri...

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 $...

The number of domino tilings of the 2n× 2n square with a centered hole of size 2m×2m, a figure known as the holey square and denoted H(m, n), was conjectured by Edward Early to be 2n−m(2km,n + 1) . Although the conjecture has remained unsolved until now, specific cases were known, for example see [4]. In this paper, we answer the general conjecture in the affirmative, primarily via a theorem ab...

Czap and Jendrol’ introduced the notions of strong parity vertex coloring and the corresponding strong parity chromatic number χs. They conjectured that there is a constant bound K on χs for the class of 2-connected plane graphs. We prove that the conjecture is true with K = 97, even with an added restriction to proper colorings. Next, we provide simple examples showing that the sharp bound is ...

In 1979 Goldfeld conjectured: 50% of the quadratic twists an elliptic curve defined over rationals have analytic rank zero. this expository article we present a few recent developments towards conjecture, especially its first instance - congruent number curves.

Given a graph G, we investigate the problem of determining the parity of the number of homomorphisms from G to some other fixed graph H. We conjecture that this problem exhibits a complexity dichotomy, such that all parity graph homomorphism problems are either polynomial-time solvable or ⊕P–complete, and provide a conjectured characterisation of the easy cases. We show that the conjecture is t...