MathNet 2011 Question 50

Determine, with proof, whether there is any odd integer $n \ge 3$ and $n$ distinct prime numbers $p_1, p_2, \dots, p_n$, such that all $p_i + p_{i-1}$ ($i=1, 2, \dots, n$, and $p_{n-1} = p_1$) are perfect squares?