Related papers: Tables of record gaps between prime constellations
Let $M(x)$ be the length of the largest subinterval of $[1,x]$ which does not contain any sums of two squareful numbers. We prove a lower bound \[ M(x)\gg \frac{\ln x}{(\ln\ln x)^2} \] for all $x\geq 3$. The proof relies on properties of…
Using a sieve-theoretic argument, we show that almost all gaps $(p_n, p_{n+1})$ between consecutive primes $p_n, p_{n+1}$ contain a natural number $m$ whose least prime factor $p(m)$ is at least the length $p_{n+1} - p_n$ of the gap,…
We investigate logarithmic and square-root types of bounds for the general difference of two primes, $P_{k+q}-P_k$, $k, q\in\mathbb{N}$.
For every positive integer $k$, a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-tuple dominating set of $G$ if every vertex of $V-S$ is adjacent to least $k$ vertices and every vertex of $S$ is adjacent to least $k-1$ vertices in $S$.…
The intermediate period gap, discovered by Kepler, is an observed dearth of stellar rotation periods in the temperature-period diagram at $\sim$ 20 days for G dwarfs and up to $\sim$ 30 days for early-M dwarfs. However, because Kepler…
We report in this paper a new exponential relation distance of planets in the newly discovered exoplanetary system of the Trappist-1 star, and we comment on near orbital mean motion resonances among the seven planets. We predict that…
Transitional disks are protoplanetary disks with opacity gaps/cavities in their dust distribution, a feature that may be linked to planet formation. We perform Bayesian modeling of the three transitional disks SZ Cha, CS Cha and T25…
The cross-ratio degree problem asks for the number of configurations of $n$ points in $\mathbb{P}^1$ that satisfy $n-3$ specified cross-ratio conditions. It is known that the maximal cross-ratio degree for 8 points is at least 4. In this…
An interval $k$-graph is the intersection graph of a family $\mathcal{I}$ of intervals of the real line partitioned into at most $k$ classes with vertices adjacent if and only if their corresponding intervals intersect and belong to…
We investigate via numerical modeling the effects of two planets locked in resonance, and migrating outward, on the dust distribution of the natal circumstellar disk. We aim to test whether the dust distribution exhibits peculiar features…
We investigate the evolution of two-planet systems embedded in a protoplanetary disc, which are composed of a Jupiter-mass planet plus another body located further out in the disc. We consider outermost planets with masses ranging from 10…
Constraining the strength of gas turbulence in protoplanetary discs is an open problem that has relevant implications for the physics of gas accretion and planet formation. In this work, we gauge the amount of turbulence in 6 of the discs…
Color-color diagrams for the clump and interclump emission in 10 clump-cluster galaxies of the Ultra Deep Field are made from B,V,i, and z images and compared with models to determine redshifts, star formation histories, and galaxy masses.…
We investigate pattern speeds in spiral galaxies where the structure is induced by an interaction with a companion galaxy. We perform calculations modeling the response of the stellar and/or gaseous components of a disc. Generally we do not…
We study gravitational instabilities in disks, with special attention to the most massive clumps that form because they are expected to be the progenitors of globular-type clusters. The maximum unstable mass is set by rotation and depends…
The degree of insulation of a prime $p$ is defined as the largest interval around it in which no other prime exists. Based on this, the $n$-th prime $p_{n}$ is said to be insulated if and only if its degree of insulation is higher than its…
Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n, 2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1} - p_n = O(log^{1+\epsilon}p_n)$…
It is not understood whether long-lived star clusters possess a continuous range of sizes and masses (and hence densities), or if rather, they should be considered as distinct types with different origins. Utilizing the Hubble Space…
We have used the full radial extent of images from the Hubble Space Telescope's Advanced Camera for Surveys and Wide Field Planetary Camera 2 to extract surface brightness profiles from a sample of six, local lenticular galaxy candidates.…
Based on the primes less than $4 \times 10^{18}$, Oliveira e Silva et al. (2014) conjectured an asymptotic formula for the sum of the $k$th power of the gaps between consecutive primes less than a large number $x$. We show that the…