**Definition:**Complimentary primes are prime integers p such that 2

^{k/2}mod p = p-1 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