Related papers: Square-free values of $\mathbf{n^2+n+1}$

Let $n=a^2b$, where $b$ is square-free. In this paper we present an algorithm based on class groups of binary quadratic forms that finds the square-free decomposition of $n$, i.e. $a$ and $b$, in heuristic expected time: $$…

We give formulas for the number of representations of non negative integers by various quadratic forms. We also give evaluations in the case of sum of two cubes (cubic case) and the quintic case, as well. We introduce a class of generalized…

We determine all modular curves $X_0(N)/\langle w_d\rangle$ that admit infinitely many cubic points over the rational field $\mathbb{Q}$, when $N$ is square-free.

We study the asymptotics of the average number of squares (or quadratic residues) in Z_n and Z_n^*. Similar analyses are performed for cubes, square roots of 0 and 1, and cube roots of 0 and 1.

Let $F$ be a non-degenerate quadratic form on an $n$-dimensional vector space $V$ over the rational numbers. One is interested in counting the number of zeros of the quadratic form whose coordinates are restricted in a smoothed box of size…

Let $q$ be a perfect power of a prime number $p$ and $E({\mathbb F}_q)$ be an elliptic curve over ${\mathbb F}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer $n$ we denote by $ \# E({\mathbb F}_{q^n})$ the number of…

Let $k\ge 2$ be a fixed integer. We consider sums of type $\sum_{n_1^2+\cdots+ n_k^2\le x} F(n_1,\ldots,n_k)$, taken over the $k$-dimensional spherical region $\{(n_1,\ldots,n_k)\in {\Bbb Z}^k: n_1^2+\cdots+ n_k^2\le x\}$, where $F:{\Bbb…

In this article, we prove that the density of integers $a, b$ such that $a^4+b^3$ is squarefree, when ordered by $\max\{|a|^{1/3},|b|^{1/4}\}$, equals the conjectured product of the local densities. We show that the same is true for…

Let $f(x_1, x_2, x_3)$ be an indefinite anisotropic integral quadratic form with determinant $d(f)$, and $t$ a non-zero integer such that $d(f)t$ is square-free. It is proved in this paper that, as long as there is one integral solution to…

Given any positive integer M, we show that there are infinitely many real quadratic fields that do not admit universal quadratic forms in M variables.

We study a multiplicative function analogue of Linnik's problem on the least prime in an arithmetic progression. Let $h\colon \mathbb{N}\to\mathbb{R}\setminus\{0\}$ be a multiplicative function, and let $a \pmod q$ be a reduced residue…

We give a partial answer to a problem of Harju by constructing an infinite ternary squarefree word $w$ with the property that for every $k \geq 3312$ there is an interior length-$k$ factor of $w$ that can be deleted while still preserving…

Triangular numbers that are multiple of other triangular numbers are investigated. It is known that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers which are triangular numbers. If the…

Let $k \geq 1$ be a cube-free integer with $k \equiv 1 \pmod {9}$ and $\gcd(k, 7\cdot 571)=1$. In this paper, we prove the existence of infinitely many triples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$…

Let k, r > 1 be two integers. We prove an asymptotic formula for the number of k-free values of the r variables polynomial t_1...t_r - 1 over the integral points of [1, x]^r.

In this note, we show that there are many infinity positive integer values of $n$ in which, the following inequality holds $$ \left\lfloor{1/2}(\frac{(n+1)^2}{\log(n+1)}-\frac{n^2}{\log n})-\frac{\log^2 n}{\log\log…

In a recent paper, Harju posed three open problems concerning square-free arithmetic progressions in infinite words. In this note we solve two of them.

In this paper, we study additively indecomposable quadratic forms over real biquadratic and simplest cubic fields. In particular, we show that over these fields, we can always find such a classical form in 2 variables, which differs from…

A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\alpha n]+1$, where $1<\alpha<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical…

We show that there exists $\eta > 0$ such that the interval $[X, X + X^{\frac 15 - \eta}]$ contains a squarefree number for all large $X$. This improves on an earlier result of Filaseta and Trifonov who showed that there is a squarefree…