Let r,m be positive integers with r > 1, m even, and A,B be integers satisfying $A+B\surd (-1)={(m+\surd (-1))}^r$. We prove that the Diophantine equation ${\left|A\right|}^x+{\left|B\right|}^y={(m²+1)}^z$ has no positive integer solutions in (x,y,z) other than (x,y,z) = (2,2,r), whenever $r>{10}^{74}$ or $m>{10}^{34}$. Our result is an explicit refinement of a theorem due to F. Luca.

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 $k=n$.

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 $gcd(x,y)=1$.

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.

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 ${\left(L_n\right)}_{n\ge 0}$ 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 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 ${}_g$.

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.

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. ...

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 $\epsilon >0$.

Let $[·]$ be the floor function. In this paper, we prove by asymptotic formula that when $1<c<\frac{3441}{2539}$, then every sufficiently large positive integer $N$ can be represented in the form $$N=\left[p_1^c\right]+\left[p_2^c\right]+\left[p_3^c\right]+\left[p_4^c\right]+\left[p_5^c\right],$$ where $p_1$, $p_2$, $p_3$, $p_4$, $p_5$ are primes such that $p_1=x^2+y^2+1$.