It is proven that the Prophet used to pray four Rak`ahs after `Isha’. As for the Hadiths which speak of the virtue of doing so, they are all extremely inauthentic; the best and strongest of them is the Hadith of Ibn `Umar, despite the fact that it is inauthentic.

1-The Hadith which reads “Fasting on one of these days (the first ten days of Dhul-Hijjah) is equivalent to fasting for a year” has a da‘if isnad (weak chain of transmission) because of an-Nahhas ibn Qaham and and Mas‘ud ibn Wasil 2- The hadith which reads “There are five things that the Prophet (peace and blessings of Allah be upon him) never failed to do…” is based on the report narrated by Hunaydah ibn Khalid al-Khuza‘i, from whom it was narrated via many isnads, in several versions. The critics of hadith differed as to the classification of this hadith because each of these versions has its own isnad and text. For more, see the detailed answer.

The Hadith which reads: “Whoever recites Ayat Al-Kursi after Wudu, Allah will give him the reward of forty scholars, and raise him forty degrees in status, and give him in marriage forty of Al-hur Al-`Iyn” is false and fabricated.