TITLE: If $\frac{a_{1}a_{2}a_{3}\cdots a_{n}-1}{(a_{1}-1)(a_{2}-1)(a_{3}-1)\cdots(a_{n}-1)}\in \mathbb{N}^{+}$, how to find $a_{i}$
QUESTION [13 upvotes]: Question:

Let $a_{1},a_{2},\cdots,a_{n}$ is positive integer, and such $1<a_{1}<a_{2}<\cdots<a_{n}$ and 
  $$\dfrac{a_{1}a_{2}a_{3}\cdots a_{n}-1}{(a_{1}-1)(a_{2}-1)(a_{3}-1)\cdots(a_{n}-1)}\in \mathbb{N}^{+}$$
  find the $a_{1},a_{2},\cdots,a_{n}$?

I know these particular cases:
When $n=3$

If $1<a_{1}<a_{2}<a_{3}$,and 
  $$\dfrac{a_{1}a_{2}a_{3}-1}{(a_{1}-1)(a_{2}-1)(a_{3}-1)}\in \mathbb{N}^{+}$$
  then we have 
  $(a_{1},a_{2},a_{3})=(2,4,8)$ or $(3,5,15)$, and this full solution can see
  link solution

and when $n=4$,

This is 1996 Monthly Problem:
  Find all sets of distinct integers $1 < a < b < c < d$, such
  $$\dfrac{abcd-1}{(a-1)(b-1)(c-1)(d-1)}\in \mathbb{N}^{+}$$
  then we have  $(a,b,c,d)=(2,4,10,80)$ or $(3,5,17,255)$, and the full solution can be found here: http://www.mat.uniroma2.it/~tauraso/AMM/AMM10523.pdf

What about the general case? 
Thank you

REPLY [0 votes]: Suppose $x_k=a_k-1$. Then $x_1,x_2,\cdots,x_n$ are positive integers such that $0<x_1<x_2<\cdots<x_n$. Further we have $x_1 \geq 1$ and $x_m \geq x_{m-1}+1$. Our function
$$R(\mathbf{x}) = \frac {\prod_{k=1}^{n}(x_k+1)-1}{\prod_{k=1}^{n}x_k}, \quad R \in \mathbb{N}^+$$
To cut the long story short here are the solutions:
Case $n=1$: $R( \mathbf{x})=1$. Solutions: $x_1=1,2,\dots$
Case $n=2$: No Solutions
Case $n=3$: 2 Solutions $(x,y,z,R)=(2,4,14,2)$ or $(1,3,7,3)$
Case $n=4$: 2 Solutions $(w,x,y,z,R)=(2,4,16,254,2)$ and $(1,3,9,79,3)$
Case $n=5$: 2 Solutions: $(v,w,x,y,z,R)=(2,4,16,256,65534,2)$ and $(1,3,9,81,6559,3)$
Can you spot the pattern? R is always 2 or 3. For the even solution you start with $2$, then square it, then square that and so on except at the end you square and subtract $2$. Similarly for the odd case starting at 1 and using powers of $3$. Easily generalizable. But these numbers are enormous: the even solution for n=8 is 
$2$,
$4$,
$16$,
$256$,
$65536$,
$4294967296$,
$18446744073709551616$,
$340282366920938463463374607431768211456$,
$115792089237316195423570985008687907853269984665640564039457584007913129639934$.
I have to get some sleep now and will post the proof tomorrow.
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