If the applicant claims patent priority, there is no need to make a written statement when applying, right?

Wrong. According to Article 30 of the Patent Law of People's Republic of China (PRC), if a patent applicant claims priority, he shall make a written statement at the time of application and submit a copy of the first patent application document within three months; Failing to submit a written statement or a copy of the patent application documents within the time limit shall be regarded as not claiming priority, so it is wrong.