We study the Diophantine equations ${(k!)}^n-k^n={(n!)}^k-n^k$ and ${(k!)}^n+k^n={(n!)}^k+n^k,$ where $k$ and $n$ are positive integers. We show that the first one holds if and only if $k=n$ or $(k,n)=(1,2),(2,1)$ and that the second one holds if and only if $k=n$.

In this paper, we find all the solutions of the Diophantine equation $B_1^p+2B_2^p+\cdots +k{B}_k^p=B_n^q$ in positive integer variables $(k,n)$, where $B_i$ is the $i^{th}$ balancing number if the exponents $p$, $q$ are included in the set $\{1,2\}$.

Let $p$ be an odd prime. By using the elementary methods we prove that: (1) if $2\nmid x$, $p\equiv \pm 3(mod8),$ the Diophantine equation $(2^x-1)(p^y-1)=2z^2$ has no positive integer solution except when $p=3$ or $p$ is of the form $p=2a_0^2+1$, where $a_0>1$ is an odd positive integer. (2) if $2\nmid x$, $2\mid y$, $y\ne 2,4,$ then the Diophantine equation $(2^x-1)(p^y-1)=2z^2$ has no positive integer solution.

Let $F_n$ denote the $n^{th}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_1^p+2F_2^p+\cdots +k{F}_k^p=F_n^q$ in the positive integers $k$ and $n$, where $p$ and $q$ are given positive integers. A complete solution is given if the exponents are included in the set $\{1,2\}$. Based on the specific cases we could solve, and a computer search with $p,q,k\le 100$ we conjecture that beside the trivial solutions only $F_8=F_1+2F_2+3F_3+4F_4$, $F_4^2=F_1+2F_2+3F_3$, and $F_4^3=F_1^3+2F_2^3+3F_3^3$ satisfy the title equation.

S. S. Pillai proved that for a fixed positive integer $a$, the exponential Diophantine equation $x^y-y^x=a$, $min(x,y)>1$, has only finitely many solutions in integers $x$ and $y$. We prove that when $a$ is of the form $2z^2$, the above equation has no solution in integers $x$ and $y$ with $gcd(x,y)=1$.

The triples $(x,y,z)=(1,z^z-1,z)$, $(x,y,z)=(z^z-1,1,z)$, where $z\in \mathbb{N}$, satisfy the equation $x^y+y^x=z^z$. In this paper it is shown that the same equation has no integer solution with $min\{x,y,z\}>1$, thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.

Let g ≥ 2 be an integer and ${}_g\subset \mathbb{N}$ be the set of repdigits in base g. Let ${}_g$ be the set of Diophantine triples with values in ${}_g$; that is, ${}_g$ is the set of all triples (a,b,c) ∈ ℕ³ with c < b < a such that ab + 1, ac + 1 and bc + 1 lie in the set ${}_g$. We prove effective finiteness results for the set ${}_g$.

Let ${\left(L_n\right)}_{n\ge 0}$ be the Lucas sequence. We show that the Diophantine equation $L_n-L_m=M_k$ has only the nonnegative integer solutions $(n,m,k)=(2,0,1)$, $(3,1,2)$, $(3,2,1)$, $(4,3,2)$, $(5,3,3)$, $(6,2,4)$, $(6,5,3)$ where $M_k=2^k-1$ is the $k$th Mersenne number and $n>m$.

Let $F_n$ be the $n$-th Fibonacci number. Put $\varphi =\frac{1+\sqrt{5}}{2}$. We prove that the following inequalities hold for any real $\alpha$: 1) ${inf}_{n\in \mathbb{N}}\left|\right|F_n\alpha \left|\right|\le \frac{\varphi -1}{\varphi +2}$, 2) ${lim\; inf}_{n\to \infty }\left|\right|F_n\alpha \left|\right|\le \frac{1}{5}$, 3) ${lim\; inf}_{n\to \infty }\left|\right|{\varphi }^n\alpha \left|\right|\le \frac{1}{5}$. These results are the best possible.

We consider systems of equations of the form $x_i+x_j=x_k$ and $x_i·x_j=x_k$, which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.

We consider $N$, the number of solutions $(x,y,u,v)$ to the equation ${(-1)}^{u}r{a}^x+{(-1)}^{v}s{b}^y=c$ in nonnegative integers $x,y$ and integers $u,v\in \{0,1\}$, for given integers $a\&gt;1$, $b\&gt;1$, $c\&gt;0$, $r\&gt;0$ and $s\&gt;0$. When $gcd(ra,sb)=1$, we show that $N\le 3$ except for a finite number of cases all of which satisfy $max(a,b,r,s,x,y)\&lt;2·{10}^{15}$ for each solution; when $gcd(a,b)\&gt;1$, we show that $N\le 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving $N=3$ solutions.

Suppose that ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4$ are nonzero real numbers, not all negative, $\delta >0$, $𝒱$ is a well-spaced set, and the ratio ${\lambda }_1/{\lambda }_2$ is algebraic and irrational. Denote by $E(𝒱,N,\delta )$ the number of $v\in 𝒱$ with $v\le N$ such that the inequality $$|{\lambda }_1{p}_1^2+{\lambda }_2{p}_2^3+{\lambda }_3{p}_3^4+{\lambda }_4{p}_4^5-v|<v^{-\delta }$$ has no solution in primes $p_1$, $p_2$, $p_3$, $p_4$. We show that $$E(𝒱,N,\delta )\ll N^{1+2\delta -1/72+\epsilon }$$ for any $\epsilon >0$.

In this paper, we find all integer solutions $(x,y,n,a,b,c)$ of the equation in the title for non-negative integers $a,b$ and $c$ under the condition that the integers $x$ and $y$ are relatively prime and $n\ge 3$. The proof depends on the famous primitive divisor theorem due to Bilu, Hanrot and Voutier and the computational techniques on some elliptic curves.

Let $P_m$ and $E_m$ be the $m$-th Padovan and Perrin numbers respectively. Let $r,s$ be non-zero integers with $r\ge 1$ and $s\in \{-1,1\}$, let ${\left\{U_n\right\}}_{n\ge 0}$ be the generalized Lucas sequence given by $U_{n+2}=r{U}_{n+1}+s{U}_n$, with $U_0=0$ and $U_1=1.$ In this paper, we give effective bounds for the solutions of the following Diophantine equations $$P_m=U_n{U}_k\text{and}{E}_m=U_n{U}_k,$$ where $m$, $n$ and $k$ are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing sequences.

Let $W(m,n;\underline{\psi })$ denote the set of ${\psi }_1,...,{\psi }_n$–approximable points in ${ℝ}^{mn}$. The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions $\underline{\psi }$. Removing monotonicity from the Khintchine–Groshev theorem is attributed to different authors for different cases of $m$ and $n$. It can not be removed for $m=n=1$ as Duffin–Schaeffer provided the counter example. We deal with the only remaining case $m=2$ and thereby remove all unnecessary conditions from the Khintchine–Groshev theorem. ...

Consider the system $x^2-a{y}^2=b$, $P(x,y)=z^2$, where $P$ is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation $7X^2+Y^7=Z^2$ if $(X,Y)=(L_n,F_n)$ (or $(X,Y)=(F_n,L_n)$) where $\left\{F_n\right\}$ and $\left\{L_n\right\}$ represent the sequences of Fibonacci numbers and Lucas numbers...