Konigsberg wrote: Find the number of all infinite sequences $a_1$, $a_2$, ... of positive integers such that $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2005$ for all positive integers $n$. Let $b_n=a_n-a_{n+2}\in\mathbb Z$ $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2005$ $a_{n+1}+a_{n+2}=2a_{n+3}a_{n+4}+2005$ Subtracting, we get $b_n=2a_{n+3}b_{n+2}$ and so $b_1=2^kb_{2k+1}\prod_{i=1}^ka_{2k+2}$ and $b_2=2^kb_{2k+2}\prod_{i=1}^ka_{2k+3}$ And so $b_i=0$ $\forall i$ and $a_{n+2}=a_n$ and the sequence is $a,b,a,b,a,b,....$ where $a+b=2ab+2005$ $\iff$ $(2a-1)(2b-1)+4009=0$, impossible with $a,b>0$ Hence no such sequences.