Related papers: Nonexistence of $D(4)$-quintuples

The third unramified cohomology group is shown to vanish on certain varieties equipped with a pencil of quadrics or of smooth complete intersections of two quadrics. Over the complex field, this shows that the integral Hodge conjecture in…

Let $a$ and $b=ka$ be positive integers with $k\in \{2, 3, 6\},$ such that $ab+4$ is a perfect square. In this paper, we study the extensibility of the $D(4)$-pairs $\{a, ka\}.$ More precisely, we prove that by considering three families of…

We study totally positive definite quadratic forms over the ring of integers $\mathcal{O}_K$ of a totally real biquadratic field $K=\mathbb{Q}(\sqrt{m}, \sqrt{s})$. We restrict our attention to classical forms (i.e., those with all…

Dujella and Peth\H{o}, generalizing a result of Baker and Davenport, proved that the set $\{1, 3\}$ cannot be extended to a Diophantine quintuple. As a consequence of our main result, it is shown that the Diophantine pair $\{1, b\}$ cannot…

We show that the classical algebra of quaternions is a commutative $\Z_2\times\Z_2\times\Z_2$-graded algebra. A similar interpretation of the algebra of octonions is impossible.

In this paper, we show that an equilateral triangle cannot be dissected into finitely many smaller equilateral triangles, no two of which share two vertices. We do this without the use of Electrical Networks.

We prove the existence of classical solutions to the Dirichlet problem for a class of fully nonlinear elliptic equations of curvature type on Riemannian manifolds. We also derive new second derivative boundary estimates which allows us to…

Let n > 1 be an integer, and let F denote a field of p elements for a prime p = 1 (mod n). By 2015, the question of existence or nonexistence of n-th power residue difference sets in F had been settled for all n < 24. We settle the case n =…

We give a proof of Pythagoras' theorem which does not use neither squares nor similarity of triangles.

We consider Clifford algebras over the field of real or complex numbers as a quotient algebra without fixed basis. We present classification of Clifford algebra elements based on the notion of quaternion type. This classification allows us…

In this paper, we give an example of a chiral 4-polytope in projective 3-space. This example naturally yields a finite chiral 4-polytope in Euclidean 4-space, giving a counterexample to Theorem 11.2 of [2].

In a previous work, we introduced the notion of uniform generators of the Brauer group and proved that general diagonal cubic surfaces do not have such generators. In this paper, we prove that a similar non-existence result holds for affine…

Recently the problem of constructing a perfect Euler cuboid was related with three conjectures asserting the irreducibility of some certain three polynomials depending on integer parameters. In this paper a partial result toward proving the…

We propose a generalization of a conjecture of D. Quillen, on the vanishing of Andr\'e-Quillen homology, to simplicial commutative rings. This conjecture characterizes a notion of local complete intersection, extended to the simplicial…

We establish that finite-time singularities do not occur in four-dimensional Yang-Mills flow, confirming the conjecture of Schlatter, Struwe, and Tahvildar-Zadeh. The proof relies on a weighted energy identity and sharp decay estimates in…

A perfect cuboid is a rectangular parallelepiped whose edges, whose face diagonals, and whose space diagonal are of integer lengths. The problem of finding such cuboids or proving their non-existence is not solved thus far. The second…

We find a new gauge in which U(1) noncommutative instantons are explicitly non-singular on the whole noncommutative R^4, thus resolving the previous confusions of the author. We start with the pedagogical introduction to the noncommutative…

In this paper we solve the Diophantine equation $\binom{m}{l}-\binom{n}{k}=d$ (where m,n are positive integers unknowns) when (k,l)=(3,6) for various values of d and when (k,l)=(8,2) and d=1. As a byproduct of our results we will obtain…

A non-existence theorem of classical electrodynamics in odd-dimensional spacetimes is shown to be invalid. The source of the error is pointed out, and is then demonstrated during the derivation of the fields generated by a uniformly moving…

After having shown that the formula which describes the Doppler effect in the general case holds only in the case of the "very high" frequency assumption, we derive free of assumptions Doppler formulas for two scenarios presented in the…