Related papers: The Uniform Primality Conjecture for the Twisted F…
In this paper, we study configurations of three rational points on the Hermitian curve over $\mathbb{F}_{q^2}$ and classify them according to their Weierstrass semigroups. For $q>3$, we show that the number of distinct semigroups of this…
Take a rational elliptic curve defined by the equation $y^2=x^3+ax$ in minimal form and consider the sequence $B_n$ of the denominators of the abscissas of the iterate of a non-torsion point; we show that $B_{5m}$ has a primitive divisor…
We extend non-emtpyness and irreducibility of Hassett divisors to the moduli spaces of $M$-polarizable cubic fourfolds for higher rank lattices $M$, which in turn provides a systematic approach for describing the irreducible components of…
In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions.…
It is shown that every Leavitt path algebra L of an arbitrary directed graph E over a field K is an arithmetical ring, that is, the two-sided ideals of L form a distributive lattice. It is also shown that L is a multiplication ring, that…
For certain elliptic curves $E/\mathbb{Q}$ with $E(\mathbb{Q})[2]=\mathbb{Z}/2 \mathbb{Z}$, we prove a criterion for prime twists of $E$ to have analytic rank 0 or 1, based on a mod 4 congruence of 2-adic logarithms of Heegner points. As an…
In this paper we describe three different variations of prime ideals: strongly irreducible ideals, strongly prime ideals and insulated prime ideals in the context of Leavitt path algebras. We give necessary and sufficient conditions under…
A rationality condition is derived for the existence of odd perfect numbers involving the square root of a product, which consists of a sequence of repunits, multiplied by twice the base of one of the repunits. This constraint also provides…
Let f(t) be a rational function of degree at least 2 with rational coefficients. For a given rational number x_0, define x_{n+1}=f(x_n) for each nonnegative integer n. If this sequence is not eventually periodic, then the difference…
We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…
We show that both primes and smooth numbers are equidistributed in arithmetic progressions to moduli up to $x^{5/8 - o(1)}$, using triply-well-factorable weights for the primes (we also get improvements for the well-factorable linear sieve…
We show under the assumption that the Tate-Shafarevich group of any elliptic curve over the rational numbers is finite that the cubic surface $x_1^3 + p_1p_2x_2^3 + p_2p_3x_3^3 + p_3p_1x_4^3 = 0$ has a rational point, where $p_1, p_2$ and…
We study an LCM-based analogue of Rowland's GCD-based prime-generating recurrence, introduced by the author in 2008. The multiplicative increments of this sequence are conjectured always to be $1$ or prime, but a complete proof requires a…
Up to isomorphism there are six fixed-point free crystallographic groups in Euclidean Space generated by twists (screw motions). In each case, an orientable 3-manifold is obtained as the quotient of E3 by such a group. The cubic…
In this paper we consider elliptic divisibility sequences generated by a point on an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$ given by an integral short Weierstrass equation. For several different such elliptic…
Let p be a prime number which is split in an imaginary quadratic field k. Let \mathfrak{p} be a place of k above p. Let k_\infty be the unique Z_p-extension of k which unramified outside of \mathfrak{p}, and let K_\intfy be a finite…
Let $\{nP+Q\}_{n\geq0}$ be a sequence of points on an elliptic curve defined over a number field $K$. In this paper, we study the denominators of the $x$-coordinates of this sequence. We prove that, if $Q$ is a torsion point of prime order,…
With a simple transformation of the three exponents the generalized Fermat equation can be put into the same form as the Fermat equation. When it is rewritten into this new altered form any real solutions to the altered equation equal a…
In this paper we study the problem of detecting prime numbers between all consecutive cubes. Firstly, we use a large computation to show that there is always a prime between $n^3$ and $(n+1)^3$ for $n^3\leq 1.649\cdot 10^{40}$. In addition,…
In the projective space $\mathrm{PG}(3,q)$, we consider the orbits of lines under the stabilizer group of the twisted cubic. It is well known that the lines can be partitioned into classes every of which is a union of line orbits. All types…