Higher order Wierferich prime pairs
Suppose odd primes p and q are Wierferich prime pairs of order k if and only if:
p^(q1) = 1 modulo q^k
q^k = 1 modulo p
where p ≠ 1 modulo q, q ≠ 1 modulo p, or equivalent restriction k > 1.
The first such pair (p,q) of order k = 2 is (3,11) because
3^10 = 1 modulo 11^2 and 11^2 = 1 modulo 3
The first pair of order k = 3 is (19,7) because
19^6 = 1 modulo 7^3 and 7^3 = 1 modulo 19
Which is the first such prime pair (p,q) or order k = 4, in other words primes p, q such that
p^(q1) = 1 modulo q^4 and q^4 = 1 modulo p
It is easy to find any such primes p, q obviously is it not, but what are the smallest such primes?
What about orders k = 5, 6, 7 and so on?
It doesn't seem that easy to find such pairs as (3,11) k = 2 and (19,7) k = 3 are rare cases.
