What is the ruling on light brown discharge that appears as spots after the cessation of menstrual bleeding that lasted 23 days? Is it considered menstruation? And how can a woman know that she has become pure in this situation? And what is the ruling on prayer, fasting, and sexual intercourse with this discharge?

The maximum duration of menstruation is fifteen days. If the bleeding exceeds this period, it is considered irregular bleeding (istihada). What you have seen of blood is not all menstrual blood; rather, only the blood that coincides with the duration of your usual period is considered menstruation. Anything beyond that is istihada. You must make up the prayers you missed, thinking it was menstruation. As for brown drops outside the time of your usual period and not connected with blood, they are not menstruation, do not necessitate a ghusl (ritual bath), and do not prevent prayer, fasting, or intercourse. However, they are impure, and purification (istinja') is required after them.