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.