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Old 2018-04-14, 05:21   #1
carpetpool
 
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"Sam"
Nov 2016

22·83 Posts
Post Higher order Wierferich prime pairs

Suppose odd primes p and q are Wierferich prime pairs of order k if and only if:

p^(q-1) = 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^(q-1) = 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.
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