If anyone knew what the cause of the pain was without needing to carry out an X-ray, your friend wouldn't need the X-ray.
Some of the most common causes of buttock pain are listed here:
https://www.healthline.com/health/pain-in-buttocks
Sciatica is probably the most common of all. You've suggested another one (arthritis) is the title of your post. My friend, who is also 78, is currently recovering from a very bad period of intense pain in that area though, aided by a lot of physiotherapy. The cause of his problem was related to problems with the discs in his spine, which required an MRI scan to produce a definitive diagnosis for him.