If k == 2 mod 6, then:
either
* k+1 is power of 3, k is of the form 2^r*p^s with p prime, k1 is prime power, the rvalue of k must be 1 or the k will have algebra factors and cannot be prime, the only known such kvalues are 26 and 1326168790943636873463383702999509006710931275809481594345135568419247032683280476801020577006926016883473704238442000000602205815896338796816029291628752316502980283213233056177518129990821225531587921003213821170980172679786117182128182482511664415807616402, and it is conjectured that all other such kvalues
or
* k is power of 2, k1 and (k+1)/3 are both prime powers (k+1 cannot be divisible by 9 or the (k+1)/(3^r) will have algebra factors and cannot be prime), the only known such kvalues are 8, 32, 128, 8192, 131072, 524288, 2147483648, 2305843009213693952, 170141183460469231731687303715884105728, and it is conjectured that all other such kvalues (related to New Mersenne Conjecture)
If k == 4 mod 6, then:
either
* k1 is power of 3, k is of the form 2^r*p^s with p prime, k+1 is prime power, the rvalue of k must be 1 or 2 or the k will have algebra factors and cannot be prime, the only known such kvalues with r=1 are 10 and 82, and it is conjectured that all other such kvalues, the only known such kvalues with r=2 are 28, and it is conjectured that all other such kvalues
or
* k is power of 2, k+1 and (k1)/3 are both prime powers, the only such k is 16, since (k1)/3 has algebra factors
