Definition: Complimentary primes are prime integers p such that 2
^{k/2} mod p = p1 where k is the order of 2 mod p.
Example: Let p = 43. It can be verified that k = 14 since 2
^{14} mod 43 = 1 and 14 is the smallest such integer so 14 is the order of 2 mod 43. Since 2
^{7} mod 43 = 42 then 43 is a complimentary prime.

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
(2^{i} (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
2
^{0} mod 43 = 1 and 2
^{7} mod 43 = 42,
2
^{1} mod 43 = 2 and 2
^{8} mod 43 = 41,
2
^{2} mod 43 = 4 and 2
^{9} mod 43 = 39
2
^{3} mod 43 = 8 and 2
^{10} mod 43 = 35,
2
^{4} mod 43 = 16 and 2
^{11} mod 43 = 27
2
^{5} mod 43 = 32 and 2
^{12} mod 43 = 11,
2
^{6} mod 43 = 21 and 2
^{13} mod 43 = 22
2
^{7} mod 43 = 42 and 2
^{14} mod 43 = 1
You can use the iFrame below to find powers of 2 modulo an odd integer.