<s>[INST] Infinite number of ways to write $1=\frac{1}{n}+\frac{1}{a_1}+\cdots+\frac{1}{a_k}$ How can I show that there is an infinite number of ways in which $1$ can be written in the form $$1=\frac{1}{n}+\frac{1}{a_1}+\cdots+\frac{1}{a_k},$$ where $n>1$ is an integer (this number is fixed), each $a_i$ is an integer and $n<a_1<\cdots<a_k.$
I don't know what to do.. 
Thank you for your help. 
 [/INST] Hint: For any integer $a_k > 1$, we have $\dfrac{1}{a_k} = \dfrac{1}{a_k+1} + \dfrac{1}{a_k^2+a_k}$, where $a_k+1 < a_k^2+a_k$. 
So, if $1 = \dfrac{1}{n}+\dfrac{1}{a_1}+\cdots+\dfrac{1}{a_{k-1}}+\dfrac{1}{a_k}$ for some integers $n < a_1 < \cdots < a_{k-1} < a_k$, then we also have $1 = \dfrac{1}{n}+\dfrac{1}{a_1}+\cdots+\dfrac{1}{a_{k-1}}+\dfrac{1}{a_k+1} + \dfrac{1}{a_k^2+a_k}$, where $n < a_1 < \cdots < a_{k-1} < a_k+1 < a_k^2+a_k$.
You should be able to finish the proof from this.
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