How find prime numbers $p_{i}$ such $p_{1}+p_{2},p_{2}+p_{3},p_{3}+p_{4},\cdots,p_{n}+p_{1}$ is square number

Question

Question:

Let $n\ge 5$ be an odd number, show that: there exist (or does not exist) primes $p_{i}\:;\:i=1,2,\cdots,n$ such that $$p_{1}+p_{2},p_{2}+p_{3},p_{3}+p_{4},\cdots,p_{n}+p_{1}$$ all are square numbers.

I think these primes do not exist, but I can't prove it.

I think this problem is interesting.

I have only taken the case $n=2$ .

It is said that this problem is a hard problem from a competition.

Answer 1

If they are all odd, then their remainders mod 4 must alternate 1 and 3, and it is impossible for an odd cycle to alternate.
So $p=2$ must appear. It can appear $n$ times; or $n-1$ times and 7 appear once.
If they must all be different, then the neighbours of 2 must both be $3\mod 4$, and again alternation is impossible.