Related papers: The Diophantine Frobenius Problem revisited

Let $1<c<37/18,\,c\neq2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in(N,2N],$ the Diophantine inequality $|p_1^c+p_2^c+p_3^c-R|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3.$…

In this paper, we completely solve the Diophantine equations $F_{n_1} + F_{n_2} = 2^{a_1} + 2^{a_2} + 2^{a_3}$ and $ F_{m_1} + F_{m_2} + F_{m_3} =2^{t_1} + 2^{t_2} $, where $F_k$ denotes the $k$-th Fibonacci number. In particular, we prove…

In [FI09], Friedlander and Iwaniec studied the so-called Hyperbolic Prime Number Theorem, which asks for an infinitude of elements gamma in SL(2,Z) such that the norm squared |gamma|^2 = a^2 + b^2 + c^2 + d^2 = p, a prime. Under the…

For non-zero integers $n$ and $k\geq2$, a generalized Diophantine $m$-tuple with property $D_k(n)$ is a set of $m$ positive integers $S = \{a_1,a_2,\ldots, a_m\}$ such that $a_ia_j + n$ is a $k$-th power for $1\leq i< j\leq m$. Define…

We show that if $k\ge 2$ is an integer and $(F_n^{(k)})_{n\ge 0}$ is the sequence of $k$-generalized Fibonacci numbers, then there are only finitely many triples of positive integers $1<a<b<c$ such that $ab+1,~ac+1,~bc+1$ are all members of…

In this paper we show that for every positive integer $n$ there exists a prime number in the interval $[n,9(n+3)/8]$. Based on this result, we prove that if $a$ is an integer greater than 1, then for every integer $n>14.4a$ there are at…

In this paper, we study the diophantine equation ${{\sigma }_{2}}(n)-{{n}^{2}}=An+B$. We prove that except for finitely many computable solutions, all the solutions to this equation with $(A,B)=({{L}_{2m}},F_{2m}^{2}-1)$ are…

For two odd primes $p$ and $q$ such that $p<q$, let $A(p,q):=(a_k)_{k=1}^{\infty}$ be the arithmetic progression whose $k$th term is given by $a_k=(k-1)(q-p)+p$ (i.e., with $a_1=p$ and $a_2=q$). Here we conjecture that for every positive…

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a…

The number of solutions of the diophantine equation $\sum_{i=1}^k \frac{1}{x_i}=1,$ in particular when the $x_i$ are distinct odd positive integers is investigated. The number of solutions $S(k)$ in this case is, for odd $k$: \[\exp \left(…

Let $N$ be an odd perfect number. Then, Euler proved that there exist some integers $n, \alpha$ and a prime $q$ such that $N = n^{2}q^{\alpha}$, $q \nmid n$, and $q \equiv \alpha \equiv 1 \bmod 4$. In this note, we prove that the ratio…

We are motivated by a result of Alzer and Luca who presented all the integer solutions to the relations $(k!)^n-k^n=(n!)^k-n^k$ and $(k!)^n+k^n=(n!)^k+n^k$. We modify the equations by considering the double factorial instead and present all…

Let $(F_n)$ be the sequence of Fibonacci numbers and, for each positive integer $k$, let $\mathcal{P}_k$ be the set of primes $p$ such that $\gcd(p - 1, F_{p - 1}) = k$. We prove that the relative density $\text{r}(\mathcal{P}_k)$ of…

Let m>=1 be an arbitrary fixed integer and let N_m(x) count the number of odd integers u<=x such that the order of 2 modulo u is not divisible by m. In case m is prime estimates for N_m(x) were given by H. Mueller that were subsequently…

Let $u_n$ be a fixed non-degenerate binary recurrence sequence with positive discriminant, $w$ a fixed non-zero integer and $p_1,p_2,\dots,p_s$ fixed, distinct prime numbers. In this paper we consider the Diophantine equation $u_n+u_m=w…

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. For a $k$-tuple of positive integers $\underline{\lambda} = (\lambda_{1}, \dots{} ,\lambda_{k})$ with $1 \le \lambda_{1} < \lambda_{2} < \dots{} < \lambda_{k}$, we…

We study the number $\nu(n)$ of representations of a positive integer $n$ by the form $x^3+y^3+z^3-3xyz$ in the conditions $0\leq x\leq y\leq z; z\geq x+1.$ We proved the following results: (i) for every positive $n,$ except for…

Let $d\ge 2$ and $n\ge d$ with $(d,n)\notin \{(2,2),(3,3)\}$. We consider homogeneous Diophantine equations of degree $d$ in $n+1$ variables and whether they have solutions in the primes. In particular, we show that a certain local-global…

Using elementary number theory we study Diophantine equations over the rational integers of the following form, $y^2=(x+a)(x+a+k)(x+b)(x+b+k)$, $y^2=c^2x^4+ax^2+b$ and $y^2=(x^2-1)(x^2-\alpha^2)(x^2-(\alpha+1)^2).$ We express their integer…

For $h \ge 2$ and an infinite set of positive integers $A$, let $R_{A,h}(n)$ denote the number of solutions of the equation $a_{1} + a_{2} + \dots{} + a_{h} = n, a_{1} \in A, \dots{} ,a_{h} \in A, a_{1} < a_{2} < \dots{} < a_{h}.$ In this…