Masters w.o. thesis on the area I'm interested vs. masters w/ thesis on a different area I am currently a graduate student and it's my first semester. My advisor is not assigned yet and it must be done soon. There is a prof. that I would like to work with on analytic number theory, however, he says that he is so busy and already have 3 students. So, he does not accepting me with a thesis program but instead, he says that he can only accept me if I choose not to write a thesis. If I decide to work with him, I will be alone on my research. I really want to make a research on analytic number theory and do not want to make a research on something different like probability or numerical analysis even if they are also a good research option. 

In conclusion, should I accept him and try to write a thesis alone or look for a different advisor on a different area? Also, is it important to have a masters degree with thesis to apply for a phD program?
 Thank you for your valuable suggestions.
Should I accept him and try to write a thesis alone or look for a different adviser on a different area?

This is up to you. You are pursuing a graduate degree and you have to take charge of your educational path. You have to weigh both options. A) Writing a thesis alone with no supervision and potentially having to struggle, B) Working under a different adviser who will supervise you, but it may be a boring/uninteresting topic. 

Is it important to have a masters degree with thesis to apply for a PhD program?

This is dependent on the program. Some programs are more research heavy than others. I would scope out your future plans and tentatively look at what PhD programs you would like to apply for and see what their requirements are. I do believe that it is impossible to do "too much" work when applying for a PhD program, so a thesis may be in your favor.