Related papers: $(1^3+1)(2^3+1)\cdots(n^3+1)$ is not a cube

Triangulations of the cube into a minimal number of simplices without additional vertices have been studied by several authors over the past decades. For $3\leq n\leq 7$ this so-called simplexity of the unit cube $I^n$ is now known to be…

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 note we describe a procedure of calculating the number all regular tetrahedra that have coordinates in the set {0,1,...,n}. We develop a few results that may help in finding good estimates for this sequence which is twice A103158 in…

We study the problem of acute triangulations of convex polyhedra and the space R^n. Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the n-cube do not exist…

Let $(U_n)_{n\geq 0}$ be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer $\ell$, let $\ell \cdot 2^{\ell} + 1$ be a Cullen number. Recently in \cite{bmt}, generalized Cullen numbers in…

In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of the curves X^2+3Y^2 = 4DZ^3 and…

It is known that the $n$-qubit system has no unextendible product bases (UPBs) of cardinality $2^n-1$, $2^n-2$ and $2^n-3$. On the other hand the $n$-qubit UPBs of cardinality $2^n-4$ exist for all $n\ge3$. We prove that they do not exist…

In 1997 we proved that if $n$ is of the form $$ 4k, \quad 8k-1\quad {\rm or} \quad 2^{2m+1}(2k-1)+3, $$ where $k,m\in \mathbb N,$ then there are no positive rational numbers $x,y,z$ satisfying $$ xyz = 1, \quad x+y+z = n. $$ Recently, N. X.…

We consider Diophantine quintuples $\{a, b, c, d, e\}$, sets of distinct positive integers the product of any two elements of which is one less than a perfect square. Triples of the first kind are the subsets $\{a, b, d\}$ with $d> b^{5}$.…

We show that the diophantine equation $n^\ell+(n+1)^\ell + ...+ (n+k)^\ell=(n+k+1)^\ell+ ...+ (n+2k)^\ell$ has no solutions in positive integers $k,n \ge 1$ for all $\ell \ge 3$.

A poset is (3+1)-free if it does not contain the disjoint union of chains of length 3 and 1 as an induced subposet. These posets are the subject of the (3+1)-free conjecture of Stanley and Stembridge. Recently, Lewis and Zhang have…

What is the maximum number of points that can be selected from an $n \times n$ square lattice such that no $k+1$ of them are in a line? This has been asked more than $100$ years ago for $k=2$ and it remained wide open ever since. In this…

Let n be a positive odd integer and let p>n+1 be a prime. We mainly derive the following congruence: $$\sum_{0<i_1<...<i_n<p}(i_1/3)(-1)^{i_1}/(i_1...i_n)=0 (mod p).$$

A long-standing conjecture states that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, 454 is a sum of at most seven positive cubes. This was first observed by Jacobi in 1851 on…

Let k_1,...,k_d be positive integers, and D be a subset of [k_1]x...x[k_d], whose complement can be decomposed into disjoint sets of the form {x_1}x...x{x_{s-1}}x[k_s]x{x_{s+1}}x...x{x_d}. We conjecture that the number of elements of D can…

Let $F_{k,d}(n)$ be the maximal size of a set ${A}\subseteq [n]$ such that the equation \[a_1a_2\dots a_k=x^d, \; a_1<a_2<\ldots<a_k\] has no solution with $a_1,a_2,\ldots,a_k\in {A}$ and integer $x$. Erd\H{o}s, S\'ark\"ozy and T. S\'os…

Given integers $k,l\geq 2$, where either $l$ is odd or $k$ is even, let $n(k,l)$ denote the largest integer $n$ such that each element of $A_n$ is a product of $k$ many $l$-cycles. In 2008, M. Herzog, G. Kaplan and A. Lev proved that if…

Let $k \geq 1$ be an integer. A set $A \subset \mathbb{Z}$ is a $k$-fold Sidon set if $A$ has only trivial solutions to each equation of the form $c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 = 0$ where $0 \leq |c_i | \leq k$, and $c_1 + c_2 + c_3…

Let $f(n)$ denote the maximum sum of the side lengths of $n$ non-overlapping squares packed inside a unit square. We prove that $f(n^2+1) = n$ for all positive integers $n$ if and only if the sum $\sum_{k\geq 1}(f(k^2+1)-k)$ converges. We…

Let $p\equiv 2,5\mod 9$ be an odd prime. In this paper, we prove that at least one of $3p$ and $3p^2$ is a cube sum by constructing certain nontrivial Heegner points. We also establish the explicit Gross-Zagier formulae for these Heegner…