Related papers: Strong Eulerian triples

We prove a Lagrangian analogue of the Conley conjecture: given a 1-periodic Tonelli Lagrangian with global flow on a closed configuration space, the associated Euler-Lagrange system has infinitely many periodic solutions. More precisely, we…

In this paper we prove that all irrational numbers from totally real cubic number fields are well approximable by rationals (i.e. the partial quotients in the continued fraction expansion of such a number are unbounded). This settles the…

In this work, we show that for a certain class of threefolds in positive characteristics, rational-chain-connectivity is equivalent to supersingularity. The same result is known for K3 surfaces with elliptic fibrations. And there are…

A rational triangle is a triangle with rational sides and rational area. A Heron triangle is a triangle with integral sides and integral area. In this article we will show that there exist infinitely many rational parametrizations, in terms…

We show that it is possible to generate an infinite set of solvable rational extensions from every exceptional first category translationally shape invariant potential. This is made by using Darboux-B\"acklund transformations based on…

We study rationality properties of real singular cubic threefolds.

We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this…

Let $ (G_n)_{n=0}^{\infty} $ be a polynomial power sum, i.e. a simple linear recurrence sequence of complex polynomials with power sum representation $ G_n = f_1\alpha_1^n + \cdots + f_k\alpha_k^n $ and polynomial characteristic roots $…

A perfect cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. The existence of such cuboids is neither proved, nor disproved. A rational perfect cuboid is a natural…

We present a short ultrafilter proof of the existence of monochromatic exponential triples $\{a, b, b^a\}$ in any finite coloring of the natural numbers. The proof is given from scratch and uses only Ramsey's theorem, the notion of…

The abc conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying a+b=c cannot simultaneously have significant repetition among their prime factors; in particular, the product of the…

One unsolved mathematical problem remains the perfect cuboid problem. A perfect cuboid is a rectangular parallelepiped whose edges, face diagonals and space diagonal are all expressed as integers. No such cuboid has yet been discovered and…

We provide an upper bound on the uniform exponent of approximation to a triple (xi, xi^2, xi^3) by rational numbers with the same denominator, valid for any transcendental real number xi. This upper bound refines a previous result of…

Let $\varepsilon>0$ be a small constant. In the present paper we prove that whenever $\eta$ is real and constants $\lambda _i$ satisfy some necessary conditions, then there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying…

We classify all positive integers n and r such that (stably) non-rational complex r-fold quadric bundles over rational n-folds exist. We show in particular that for any n and r, a wide class of smooth r-fold quadric bundles over projective…

Let $[\,\cdot\,]$ denote the floor function. Assume that $\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is a real number.…

A perfect cuboid is a rectangular parallelepiped with integer edges, integer face diagonals, and integer space diagonal. Such cuboids have not yet been found, but nor has their existence been disproved. Perfect cuboids are described by a…

In this paper we study the Palais-Smale sequences of the conformal Dirac-Einstein problem. After we characterize the bubbling phenomena, we prove an Aubin type result leading to the existence of a positive solution. Then we show the…

We show that all triples $(x_1,x_2,x_3)$ of singular moduli satisfying $x_1 x_2 x_3 \in \mathbb{Q}^{\times}$ are "trivial". That is, either $x_1, x_2, x_3 \in \mathbb{Q}$; some $x_i \in \mathbb{Q}$ and the remaining $x_j, x_k$ are distinct,…

We show that there are only finitely many triples of integers $ 0 < a < b < c $ such that the product of any two of them is the value of a given polynomial with integer coefficients evaluated at an $ S $-unit that is also a positive…