Groups of prime number-1

There are several groups of prime number. I will show you some.
1. Twin prime numbers: when p and p+2 are both prime numbers, human call them 'twin prime numbers.'
ex)  (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43)
2. Cousin prime numbers: when p and p+4 are both prime numbers, human call them 'cousin prime numbers.'
ex) (3, 7), (7, 11), (13, 17), (19, 23), (37, 41)
3. Mersenne prime numbers: when 2^a is a prime number, human call the numder 'Mersenne prime numbers.'
ex) 3, 7, 31, 127, 511, 2047

Prime Number Story #3

So, the soldiers of RPNK fought, and they didn't win, but they didn't losed! If ten soldiers were dead, they were replaced by another ten soldiers. That went again and again. Why? Because the numbers of soldiers were infinite! So, NNK's soldiers went back to their land. RPNK was surrounded by 'Yes!' and 'Wow!' and 'We won!' They were happy. However, their lands were destructed, so they had to rebuild houses. But, there were bunches of people, and when somebody tried to move, a lot of people were in front, besides, behind them, so they actually couldn't move! So, the Queen thought.
'How can I make them move better?'
At that moment, 3 said,
"Ma'am, I have a good idea."
"For what?" Queen asked.
"I can divide them in some groups, then they will move better, ma'am."
"How...?" Queen asked again.
"Prime numbers, as we are, can be divided in few groups. Humans, who are living in another world, divided prime numbers. The groups' names are..."