When is the fasting of ayyam al-beed (white days) and what is its importance?

Ayyam al-beed (the White Days) are the 13th, 14th and 15th nights of hijri months when the moon is brightest (Bukhari, Sawm, 60). This name was given because the moon is seen as full on these nights and brightens the night more than usual. The Prophet Muhammad (saw) advised to fast on these days in every month (Abu Dawud, Sawm, 69; Tirmidhi, Sawm, 54) and stated that fasting on these days is considered like fasting for the whole year (Ibn Majah, Siyam, 29).

Source: Presidency Of Religious Affairs The Turkey, High Board of Religious Affairs FATWAS

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