A probabilistic coincidence: the set of primitive roots modulo a prime number p, when p > 61, contains two roots r1,r2 whose sum is the next prime to p

I wrote about this observation at Mathematics Stack Exchange, and basically it is the weirdest thing I have seen so far, but finally it is based on probabilities as one fellow at MSE explained!

Take the set of Primitive Roots Modulo p (link to definition here) of a prime number p, Pr(p). For those primes p > 61 there is always a pair of primitive roots r1 and r2 in Pr(p) whose sum is the next prime to the current prime p, I will call it N(p).

I have tested this with Python, in the interval [62,7000] being always true. E.g.:

For p∈[1,61] there are only five counterexamples: {2,3,5,7,61} and for the rest of primes the observation is true as well.