Consecutive Quadratic Residues

For all prime numbers p,p, where p>5p > 5, define CpC_p to be the set of all positive integers kk such that kp2k \le p-2 with kk and k+1k+1 as quadratic residues modulo pp. For example, C11={3,4}C_{11} = \{3, 4\}, because 3,4,53,4,5 are quadratic residues modulo 11 (523,224,425(mod11)).\big(5^2 \equiv 3, 2^2 \equiv 4, 4^2 \equiv 5 \pmod{11}\big).

It can be proven that CpC_p is non-empty for all pp. Let mpm_p be the smallest element of CpC_p. Find the maximum value of mpm_p among all pp.

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

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