Great Grandfather has american citizenship, can I obtain american citizenship?

My great grandfather lived in Georgia for a long time. He had one son, X, in the US. He had my grandfather when he came back from the US who did not apply for the citizenship and did not live in the US. My father also did not do anything about that. NOW I am trying.
I would like to note that X , my grandfathers brother who was born in the US had many children who all are American based on X's birth certificate, which my grandfather does not have.

What is the best reasoning to follow to make my case. Are there any laws related to this issue???

Answer:
My best answer is that if you father is US citizen the is yes you can apply well he is the one that have to apply for you...work this way the closes in the family is the only one that can apply for an status for a relative like me to my brothers and father...My kids are citizens so I don't need....About you grandfather if he odesn't have a birth certicate is not much to do...Unless is a prove of his borth here like the hospital paper=release of his mother that will prove his birth here and then can take it to the birth certificate office vital office and the he will get a birth certificate...That is a prove of his birth here other wisw is not much to do for his status...Or if he had kids here then they can make a petition for him and he will get his legal status fix here....
I am afraid the relationship isn't close enough to get you a preference. You will need to try a different category.

USCIS.gov
even if you had one us parent it would be a paperwork trail but not close.

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