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For any given positive integer k, and any set A of nonnegative integers, let $r_{1,k}(A,n)$ denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ and $r_{1,l}(A,n)=r_{1,l}(\mathbb{N}\setminus A,n)$ hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.

Recently, Miyazaki and Togbé proved that for any fixed odd integer b ≥ 5 with b ≠ 89, the Diophantine equation $b^x+2^y={(b+2)}^z$ has only the solution (x,y,z) = (1,1,1). We give an extension of this result.

Let $n>m\ge 2$ be positive integers and $n=(m+1)\ell +r$, where $0\le r\le m.$ Let $C$ be a subset of $\{0,1,\cdots ,n\}$. We prove that if $$\left|C\right|>\left\{\begin{array}{cc}\lfloor n/2\rfloor +1\hfill & \text{if}m\text{is}\text{odd},\hfill \\ m\ell /2+\delta \hfill & \text{if}m\text{is}\text{even},\hfill \end{array}\right.$$ where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ and $\delta$ denotes the cardinality of even numbers in the interval $[0,min\{r,m-2\left\}\right]$, then $C-C$ contains a power of $m$. We also show that these lower bounds are best possible.