Related papers: Almost prime Pythagorean triples in thin orbits

We consider almost-primes of the form $f(p)$ where $f$ is an irreducible polynomial over $\mathbb Z$ and $p$ runs over primes. We improve a result of Richert for polynomials of degree at least $3$. In particular we show that, when the…

For any numerical semigroup $S$, there are infinitely many numerical symmetric semigroups $T$ such that $S=\frac{T}{2}$ is their half. We are studying the Betti numbers of the numerical semigroup ring $K[T]$ when $S$ is a 3-generated…

We define operations that give the set of all Pythagorean triples a structure of commutative monoid. In particular, we define these operations by using injections between integer triples and $3 \times 3$ matrices. Firstly, we completely…

Let Gamma < SL(2,Z) be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive non-zero vectors v0, w0 in Z^2. We consider the set S of all integers occurring in <v0 gamma,w0>, for gamma in…

For families of 4-regular directed circulant graphs with $n$ vertices, we count the number of primitive periodic orbits of length up to at least $n$. The relevant counting techniques are then extended to count the number of primitive pseudo…

Let X be a projective scheme over a finite field. In this paper, we consider the asymptotic behavior of the number of effective cycles on X with bounded degree as it goes to the infinity. By this estimate, we can define a certain kind of…

We prove the Oppenheim conjecture for indefinite ternary diagonal forms of the type $x^{2}+y^{2} -\alpha z^{2}$ where $ \alpha $ is an irrational number. Our method is explicit in the sense that we are able to construct a solution to the…

We prove lower bounds of the form $\gg N/(\log N)^{3/2}$ for the number of primes up to $N$ primitively represented by a shifted positive definite integral binary quadratic form, and under the additional condition that primes are from an…

We calculate numerically the periodic orbits of pseudointegrable systems of low genus numbers $g$ that arise from rectangular systems with one or two salient corners. From the periodic orbits, we calculate the spectral rigidity…

The complete classification of the finite simple groups that are $(2,3)$-generated is a problem which is still open only for orthogonal groups. Here, we construct $(2, 3)$-generators for the finite odd-dimensional orthogonal groups…

We establish a joint distribution result concerning the fractional part of $\alpha p^\theta$ for $\theta \in (0,1), \ \alpha>0$, where $p$ is a prime satisfying a Chebotarev condition in a fixed finite Galois extension over $\mathbb{Q}$. As…

This study investigates a generalisation of the Pythagorean theorem to the lengths of conic arcs constructed symmetrically on the sides of a right triangle. It is demonstrated that the theorem remains valid whenever the conic eccentricity…

We investigate pairwise quasi-orthogonal subalgebras in $M_{p^{kn}}$ which are isomorphic to $M_{p^{k}}$ for $k \ge 1$, $n \ge 2$ and a prime number $p$ with $p \ge 3$. We prove there exist $p^{2kn}-1/p^{2k}-1$ such subalgebras and they…

In this paper, we give explicit equations for homogeneous spaces corresponding to a rational isogeny of degree $3$. An explicit set of elliptic curves with elements of order $3$ in their Tate-Shafarevich group is constructed. Combining this…

Let $G$ be a connected, simply connected, simple, complex, linear algebraic group. Let $P$ be an arbitrary parabolic subgroup of $G$. Let $X=G/P$ be the $G$-homogeneous projective space attached to this situation. Let $d\in H_2(X)$ be a…

This paper studies equable parallelograms whose vertices lie on the integer lattice. Using Rosenberger's Theorem on generalised Markov equations, we show that the g.c.d. of the side lengths of such parallelograms can only be 3, 4 or 5, and…

Let $E$ be an elliptic curve defined over $\mathbb Q$. Let $\Gamma$ be a subgroup of $E(\mathbb Q)$ and $P\in E(\mathbb Q)$. In [1], it was proved that if $E$ has no nontrivial rational torsion points, then $P\in\Gamma$ if and only if $P\in…

Let $m \in \mathbb{N}$ be large. We show that there exist infinitely many primes $q_{1}< \cdot\cdot\cdot < q_{m+1}$ such that \[ q_{m+1}-q_{1}=O(e^{7.63m}) \] and $q_{j}+2$ has at most \[ \frac{7.36m}{\log 2} + \frac{4\log m}{\log 2} + 21…

Solutions to special Lagrangian equations near infinity, with supercritical phases or with semiconvexity on solutions, are known to be asymptotic to quadratic polynomials for dimension $n\ge 3$, with an extra logarithmic term for $n=2$. Via…

We use the theory of self-similar groups to enumerate all combinatorial classes of non-exceptional quadratic Thurston maps with fewer than five postcritical points. The enumeration relies on our computation that the corresponding maps on…