arXiv subjects

Michael Bennett

Publications and source records attributed to Michael Bennett.

18 recordsLinked to original sources

arXiv2021

Odd values of the Ramanujan tau function

We prove a number of results regarding odd values of the Ramanujan $\tau$-function. For example, we prove the existence of an effectively computable positive constant $\kappa$ such that if $\tau(n)$ is odd and $n \ge 25$ then either \[ P(\tau(n)) \; > \; \kappa \cdot \frac{\log\log\log{n}}{\log\log\log\log{n}} \] or there exists a prime $p \mid n$ with $\tau(p)=0$. Here $P(m)$ denotes the largest prime factor of $m$. We also solve the equation $\tau(n)=\pm 3^{b_1} 5^{b_2} 7^{b_3} 11^{b_4}$ and the equations $\tau(n)=\pm q^b$ where $3\le q < 100$ is prime and the exponents are arbitrary nonnegative integers. We make use of a variety of methods, including the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue--Mahler equations due to Bugeaud and Gy\H{o}ry, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves.

math.NT↗

arXiv2018

Shifted powers in Lucas-Lehmer sequences

We develop a general framework for finding all perfect powers in sequences derived by shifting non-degenerate quadratic Lucas-Lehmer binary recurrence sequences by a fixed integer. By combining this setup with bounds for linear forms in logarithms and results based upon the modularity of elliptic curves defined over totally real fields, we are able to answer a question of Bugeaud, Luca, Mignotte and the third author by explicitly finding all perfect powers of the shape $F_k \pm 2 $ where $F_k$ is the $k$-th term in the Fibonacci sequence.

math.NT↗

arXiv2018

Bounds on sizes of general caps in $AG(n,q)$ via the Croot-Lev-Pach polynomial method

In 2016, Ellenberg and Gijswijt employed a method of Croot, Lev, and Pach to show that a maximal cap in $AG(n, q)$ has size $O(q^{cn})$ for some $c < 1$. In this paper, we show more generally that if $S$ is a subset of $AG(n, q)$ containing no $m$ points on any $(m - 2)$- flat, then $|S| < q^{c_mn}$ for some $c_m < 1$, as long as $q$ is odd or $m$ is even.

math.CO↗

arXiv2016

Perfect powers that are sums of consecutive cubes

Euler noted the relation $6^3=3^3+4^3+5^3$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular Stroeker determined all squares that can be written as a sum of at most $50$ consecutive cubes. We generalize Stroeker's work by determining all perfect powers that are sums of at most $50$ consecutive cubes. Our methods include descent, linear forms in two logarithms, and Frey-Hellegouarch curves.

math.NT↗

arXiv2016

Sums of two ${s}$-units via frey-hellegouarch curves

In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational S-units, where S is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for the most part, does not require lower bounds for linear forms in logarithms. Its main virtue is that it enables to carry out such a program explicitly, at least for certain small sets of primes S; we do so for S = {2, 3} and S = {3, 5, 7}.

math.NT↗

arXiv2015

Occurrence of Right Angles in Vector Spaces Over Finite Fields

Here we examine some Erdos-Falconer-type problems in vector spaces over finite fields involving right angles. Our main goals are to show that a) a subset A of F_q^d of size >> q^[(d+2)/3] contains three points which generate a right angle, and b) a subset A of F_q^d of size >> q^[(d+2)/2] contains two points which generate a right angle with the vertex at the origin. We will also prove that b) is sharp up to constants and provide some partial results for similar problems related to spread and collinear triples.

math.CO↗

arXiv2015

Rational Points on Erdos-Selfridge Superelliptic Curves

Given $k \geq 2$, we show that there are at most finitely many rational numbers $x$ and $y \neq 0$ and integers $\ell \geq 2$ (with $(k,\ell) \neq (2,2)$) for which $$ x (x+1) \cdots (x+k-1) = y^\ell. $$ In particular, if we assume that $\ell$ is prime, then all such triples $(x,y,\ell)$ satisfy either $y=0$ or $\log \ell < 3^k$.

math.NT↗

arXiv2012

Three-point configurations determined by subsets of $\mathbb{F}_q^2$ via the Elekes-Sharir paradigm

We prove that if $E \subset {\mathbb F}_q^2$, $q \equiv 3 \mod 4$, has size greater than $Cq^{7/4}$, then $E$ determines a positive proportion of all congruence classes of triangles in ${\mathbb F}_q^2$. The approach in this paper is based on the approach to the Erd\H os distance problem in the plane due to Elekes and Sharir, followed by an incidence bound for points and lines in ${\mathbb F}_q^3$. We also establish a weak lower bound for a related problem in the sense that any subset $E$ of ${\mathbb F}_q^2$ of size less than $cq^{4/3}$ definitely does not contain a positive proportion of {\bf translation} classes of triangles in the plane. This result is a special case of a result established for $n$-simplices in ${\mathbb F}_q^d$. Finally, a necessary and sufficient condition on the lengths of a triangle for it to exist in $\mathbb{F}^2$ for any field $\mathbb F$ of characteristic not equal to 2 is established as a special case of a result for $d$-simplices in ${\mathbb F}^d$.

math.CO↗

arXiv2010

Convective-reactive proton-C12 combustion in Sakurai's object (V4334 Sagittarii) and implications for the evolution and yields from the first generations of stars

Depending on mass and metallicity as well as evolutionary phase, stars occasionally experience convective-reactive nucleosynthesis episodes. We specifically investigate the situation when nucleosynthetically unprocessed, H-rich material is convectively mixed with a He-burning zone, for example in convectively unstable shell on top of electron-degenerate cores in AGB stars, young white dwarfs or X-ray bursting neutron stars. Such episodes are frequently encountered in stellar evolution models of stars of extremely low or zero metal content [...] We focus on the convective-reactive episode in the very-late thermal pulse star Sakurai's object (V4334 Sagittarii). Asplund etal. (1999) determined the abundances of 28 elements, many of which are highly non-solar, ranging from H, He and Li all the way to Ba and La, plus the C isotopic ratio. Our simulations show that the mixing evolution according to standard, one-dimensional stellar evolution models implies neutron densities in the He that are too low to obtain a significant neutron capture nucleosynthesis on the heavy elements. We have carried out 3D hydrodynamic He-shell flash convection [...] we assume that the ingestion process of H into the He-shell convection zone leads only after some delay time to a sufficient entropy barrier that splits the convection zone [...] we obtain significantly higher neutron densities (~few 10^15 1/cm^3) and reproduce the key observed abundance trends found in Sakurai's object. These include an overproduction of Rb, Sr and Y by about 2 orders of magnitude higher than the overproduction of Ba and La. Such a peculiar nucleosynthesis signature is impossible to obtain with the mixing predictions in our one-dimensional stellar evolution models. [...] We determine how our results depend on uncertainties of nuclear reaction rates, for example for the C13(\alpha, n)O16 reaction.

astro-ph.SR↗

arXiv2008

Difficulties in Probing Nuclear Physics: A Study of $^{44}$Ti and $^{56}$Ni

The nucleosynthetic yield from a supernova explosion depends upon a variety of effects: progenitor evolution, explosion process, details of the nuclear network, and nuclear rates. Especially in studies of integrated stellar yields, simplifications reduce these uncertainties. But nature is much more complex, and to actually study nuclear rates, we will have to understand the full, complex set of processes involved in nucleosynthesis. Here we discuss a few of these complexities and detail how the NuGrid collaboration will address them.

astro-ph↗

arXiv2008

NuGrid: Toward High Precision Double-Degenerate Merger Simulations with SPH in 3D

We present preliminary results from recent high-resolution double-degenerate merger simulations with the Smooth Particle Hydrodynamics (SPH) technique. We put particular emphasis on verification and validation in our effort and show the importance of details in the initial condition setup for the final outcome of the simulation. We also stress the dynamical importance of including shocks in the simulations. These results represent a first step toward a suite of simulations that will shed light on the question whether double-degenerate mergers are a viable path toward type 1a supernovae. In future simulations, we will make use of the capabilities of the NuGrid collaboration in post-processing SPH particle trajectories with a complete nuclear network to follow the detailed nuclear reactions during the dynamic merger phase.

astro-ph↗

arXiv2008

Nucleosynthesis Calculations from Core-Collapse Supernovae

We review some of the uncertainties in calculating nucleosynthetic yields, focusing on the explosion mechanism. Current yield calculations tend to either use a piston, energy injection, or enhancement of neutrino opacities to drive an explosion. We show that the energy injection, or more accurately, an entropy injection mechanism is best-suited to mimic our current understanding of the convection-enhanced supernova engine. The enhanced neutrino-opacity technique is in qualitative disagreement with simulations of core-collapse supernovae and will likely produce errors in the yields. But piston-driven explosions are the most discrepant. Piston-driven explosion severely underestimate the amount of fallback, leading to order-of-magnitude errors in the yields of heavy elements. To obtain yields accurate to the factor of a few level, we must use entropy or energy injection and this has become the NuGrid collaboration approach.

astro-ph↗

arXiv2008

Nucleosynthetic Yields from "Collapsars"

The "collapsar" engine for gamma-ray bursts invokes as its energy source the failure of a normal supernova and the formation of a black hole. Here we present the results of the first three-dimensional simulation of the collapse of a massive star down to a black hole, including the subsequent accretion and explosion. The explosion differs significantly from the axisymmetric scenario obtained in two-dimensional simulations; this has important consequences for the nucleosynthetic yields. We compare the nucleosynthetic yields to those of hypernovae. Calculating yields from three-dimensional explosions requires new strategies in post-process nucleosynthesis; we discuss NuGrid's plan for three-dimensional yields.

astro-ph↗

arXiv2008

44Ti and 56Ni in core-collapse supernovae

We investigate the physical conditions where 44Ti and 56Ni are created in core-collapse supernovae. In this preliminary work we use a series of post-processing network calculations with parametrized expansion profiles that are representative of the wide range of temperatures, densities and electron-to-baryon ratios found in 3D supernova simulations. Critical flows that affect the final yields of 44Ti and 56Ni are assessed.

astro-ph↗

arXiv2008

Nucleosynthesis simulations for a wide range of nuclear production sites from NuGrid

Simulations of nucleosynthesis in astrophysical environments are at the intersection of nuclear physics reaction rate research and astrophysical applications, for example in the area of galactic chemical evolution or near-field cosmology. Unfortunately, at present the available yields for such applications are based on heterogeneous assumptions between the various contributing nuclear production sites, both in terms of modeling the thermodynamic environment itself as well as the choice of specifc nuclear reaction rates and compilations. On the other side, new nuclear reaction rate determinations are often taking a long time to be included in astrophysical applications. The NuGrid project addresses these issues by providing a set of codes and a framework in which these codes interact. In this contribution we describe the motivation, goals and first results of the NuGrid project. At the core is a new and evolving post-processing nuclesoynthesis code (PPN) that can follow quiescent and explosive nucleosynthesis following multi-zone 1D-stellar evolution as well as multi-zone hydrodynamic input, including explosions. First results are available in the areas of AGB and massive stars.

astro-ph↗

arXiv2008

NuGrid: s process in massive stars

The s-process production in massive stars at very low metallicities is expected to be negligible due to the low abundance of the neutron source 22Ne, to primary neutron poisons and decreasing iron seed abundances. However, recent models of massive stars including the effects of rotation show that a strong production of 22Ne is possible in the helium core, as a consequence of the primary nitrogen production (observed in halo metal poor stars). Using the PPN post-processing code, we studied the impact of this primary 22Ne on the s process. We find a large production of s elements between strontium and barium, starting with the amount of primary 22Ne predicted by stellar models. There are several key reaction rate uncertainties influencing the s-process efficiency. Among them, 17O(alpha,gamma) may play a crucial role strongly influencing the s process efficiency, or it may play a negligible role, according to the rate used in the calculations. We also report on the development of a new parallel (MPI) post-processing code (MPPNP) designed to follow the complete nucleosynthesis in stars on highly resolved grids. We present here the first post-processing run from the ZAMS up to the end of helium burning for a 15 solar mass model.

astro-ph↗

arXiv2008

Spatial Distribution of Nucleosynthesis Products in Cassiopeia A: Comparison Between Observations and 3D Explosion Models

We examine observed heavy element abundances in the Cassiopeia A supernova remnant as a constraint on the nature of the Cas A supernova. We compare bulk abundances from 1D and 3D explosion models and spatial distribution of elements in 3D models with those derived from X-ray observations. We also examine the cospatial production of 26Al with other species. We find that the most reliable indicator of the presence of 26Al in unmixed ejecta is a very low S/Si ratio (~0.05). Production of N in O/S/Si-rich regions is also indicative. The biologically important element P is produced at its highest abundance in the same regions. Proxies should be detectable in supernova ejecta with high spatial resolution multiwavelength observations.

astro-ph↗

arXiv2008

Complete nucleosynthesis calculations for low-mass stars from NuGrid

Many nucleosynthesis and mixing processes of low-mass stars as they evolve from the Main Sequence to the thermal-pulse Asymptotic Giant Branch phase (TP-AGB) are well understood (although of course important physics components, e.g. rotation, magnetic fields, gravity wave mixing, remain poorly known). Nevertheless, in the last years presolar grain measurements with high resolution have presented new puzzling problems and strong constraints on nucleosynthesis processes in stars. The goal of the NuGrid collaboration is to present uniform yields for a large range of masses and metallicities, including low$-$mass stars and massive stars and their explosions. Here we present the first calculations of stellar evolution and high-resolution, post-processing simulations of an AGB star with an initial mass of 2 M_sun and solar-like metallicity (Z=0.01), based on the post-processing code PPN. In particular, we analyze the formation and evolution of the radiative 13C-pocket between the 17th TP and the 18th TP. The s-process nucleosynthesis profile of a sample of heavy isotopes is also discussed, before the next convective TP occurrence.

astro-ph↗