Example: Let p = 43. It can be verified that k = 14 since 214 mod 43 = 1 and 14 is the smallest such integer so 14 is the order of 2 mod 43. Since 27 mod 43 = 42 then 43 is a complimentary prime.
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| Verified result |
The first few complimentary primes are 5,11,13,17,19,29,37,41,43,53,59,61,67,83,97,101.
This is a sequence documented in oeis.org (more or less) as the sequence of primes p where the order of 2 mod p is even. What isn't documented there and what I find special about complimentary primes is the following conjecture that I made:
Conjecture: If p is a complimentary prime and k is the order of 2 mod p then
(2i (mod p)) + (2(k/2)+i(mod p)) = p for all i greater or equal to 0
Example: Let p = 43. Since k is the order of 2 and k = 14, then k/2 = 7
20 mod 43 = 1 and 27 mod 43 = 42,
21 mod 43 = 2 and 28 mod 43 = 41,
22 mod 43 = 4 and 29 mod 43 = 39
23 mod 43 = 8 and 210 mod 43 = 35,
24 mod 43 = 16 and 211 mod 43 = 27
25 mod 43 = 32 and 212 mod 43 = 11,
26 mod 43 = 21 and 213 mod 43 = 22
27 mod 43 = 42 and 214 mod 43 = 1