相关论文: On primitive Pythagorean triples of special forms
It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years. In this survey article we discuss some of the…
We study the possible Picard numbers of abelian varieties of given dimension $g$. If $R_g$ denotes the set of realizable Picard numbers, then $R_g$ is bounded by $g^2$. We show that, for $g$ at least $3$, the set $R_g$ always has gaps and…
For each integer $n\ge 2$, we identify new infinite families of monogenic trinomials $f(x)=x^n+Ax^m+B$ with non-squarefree discriminant, many of which have small Galois group. These families are thus different from many previous…
For every odd natural number g=2d+1 we prove the existence of a countably infinite family of special Lagrangian cones in C^3 over a closed Riemann surface of genus g, using a geometric PDE gluing method.
In their paper "Pythagorean Boxes", Raymond A.Beauregard and E.R.Suryanarayan define the concept or notion of Pythagorean Rectangle as one with sidelengths and integer diagonal lengths(see [1]);they also introduce the concept of a…
We study additive properties of the set $S$ of bijections (or permutations) $\{1,\dots,n\}\to G$, thought of as a subset of $G^n$, where $G$ is an arbitrary abelian group of order $n$. Our main result is an asymptotic for the number of…
The sets of primitive, quasiprimitive, and innately transitive permutation groups may each be regarded as the building blocks of finite transitive permutation groups, and are analogues of composition factors for abstract finite groups. This…
We extend the theory of Interacting Hopf algebras with an order primitive, and give a sound and complete axiomatisation of the prop of polyhedral cones. Next, we axiomatise an affine extension and prove soundness and completeness for the…
Ever since it was published by Neugebauer and Sachs in 1945, the Old Babylonian tablet known as Plimpton 322 has been the subject of numerous studies leading to different and often conflicting interpretations of it. Overall, the tablet is…
After a review of the results in arXiv:1203.3184 [math-ph] about Pythagorean inequalities for products of spectral triples, I will present some new results and discuss classes of spectral triples and states for which equality holds.
We establish mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive…
We classify 0-dimensional spectral triples over complex and real algebras and provide some general statements about their differential structure. We investigate also whether such spectral triples admit a symmetry arising from the Hopf…
In this paper, we analyze the density of the Fibonacci word and its derived forms by examining the morphisms associated with each. It offers a comparative analysis of the density of Fibonacci numbers alongside other words derived from…
A solution $(x,y,z) \in \mathbb{Z}^3-\{(0,0,0)\}$ to a generalized Fermat equation \[ Ax^a + By^b + Cz^c = 0, \] is called \emph{primitive} if $\gcd(x,y,z) = 1$. By work of Beukers, we know that in the \emph{spherical} regime (that is, when…
If $D$ is the definite quaternion algebra over $\qu$ of discriminant $p$, we compute, for any prime $p>3$, the number of infinite dimensional cusp forms on $D^*$ which are trivial at infinity, tamely ramified at $p$, and have given…
Let $a$, $b$, $c$ be distinct primes with $a<b$. Let $S(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x + b^y = c^z$. In a previous paper \cite{LeSt} it was shown that if $(a,b,c)$ is a triple of…
The simple symplectic triple systems over the real numbers are classified up to isomorphism, and linear models of all of them are provided. Besides the split cases, one for each complex simple Lie algebra, there are two kinds of non-split…
A classification of twin primes implies special twin primes. When applied to triplets, it yields exceptional prime number triplets. These generalize yielding exceptional prime number multiplets.
We study the difference between the number of primitive roots modulo $p$ and modulo $p+k$ for prime pairs $p,p+k$. Assuming the Bateman-Horn conjecture, we prove the existence of strong sign biases for such pairs. More importantly, we prove…
We call the \emph{$p$-fundamental string} of a complex simple Lie algebra to the sequence of irreducible representations having highest weights of the form $k\omega_1+\omega_p$ for $k\geq0$, where $\omega_j$ denotes the $j$-th fundamental…