This is equivalent to saying $ p_{i}\equiv\pm 1\mod{p_{i+1}}$. So just take the smallest $ p_{i}$: this is clearly impossible unless that $ p_{i}$ is $ 2$. Then, WLOG, $ p_{1}= 2$. Then $ p_{2}= 3$, $ p_{3}= 2$, $ p_{4}= 3$, ... This is impossible as $ 2007$ is odd.