Consecutive Quadratic Residues

For all prime numbers \(p\) where \(p > 5\), define \(C_p\) to be the set of all positive integers \(k\) such that \(k \le p-2\) with \(k\) and \(k+1\) as quadratic residues modulo \(p\). For example, \(C_{11} = \{3, 4\}\), because \(3,4,5\) are quadratic residues modulo 11 (\(5^2 \equiv 3, 2^2 \equiv 4, 4^2 \equiv 5 \pmod{11}\)).

It can be proven that \(C_p\) is non-empty for all \(p\). Let \(m_p\) be the smallest element of \(C_p\). Find the maximum value of \(m_p\) among all \(p\).

If this maximum doesn't exist, enter 0 as your answer.

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