Mut'ah is a special marriage, temporary in essence, where the husband and wife sets in advance the duration of said marriage (see
4:24, especialy the tafsir of ibn Kathir). It was forbidden by Mohamed (see
1405f). What is the difference with prostitution? Can Mohamed forbade what was hallal in the Coran? Is it possible for the prophet to abrogate the Quran?
And [also prohibited to you are all] married women except those your right hands possess. [This is] the decree of Allah upon you. And lawful to you are [all others] beyond these, [provided] that you seek them [in marriage] with [gifts from] your property, desiring chastity, not unlawful sexual intercourse. So for whatever you enjoy [of marriage] from them, give them their due compensation as an obligation. And there is no blame upon you for what you mutually agree to beyond the obligation. Indeed, Allah is ever Knowing and Wise. - Quran An-Nisa(The Women) 4:24 (islam)
Chapter: Mut'ah Marriage: It was permitted then abrograted, then permitted then abrogated, and it will remain Forbidden until the day of resurrection
Iyas b. Salama reported on the authority of his father that Allah's Messenger (ﷺ) gave sanction for contracting temporary marriage for three nights in the year of Autas 1847 and then forbade it. - Hadith Muslim 1405f (islam)
Chapter: Allah's Messenger (saws) prohibited Nikah-al-Mut'a lately.
Salama bin Al-Akwa` said:
Allah's Messenger (ﷺ)'s said, "If a man and a woman agree (to marry temporarily), their marriage should last for three nights, and if they like to continue, they can do so; and if they want to separate, they can do so." I do not know whether that was only for us or for all the people in general. Abu `Abdullah (Al-Bukhari) said: `Ali made it clear that the Prophet said, "The Mut'a marriage has been cancelled (made unlawful). - Hadith Al Bukhari 5119 (islam)
Chapter: Allah's Messenger (saws) prohibited Nikah-al-Mut'a lately.
Narrated Abu Jamra:
I heard Ibn `Abbas (giving a verdict) when he was asked about the Mut'a with the women, and he permitted it (Nikah-al-Mut'a). On that a freed slave of his said to him, "That is only when it is very badly needed and women are scarce." On that, Ibn `Abbas said, "Yes." - Hadith Al Bukhari 5116 (islam)
Chapter: The Thrice Divorced Woman Cannot Return To Her Husband Until She Re-Marries
Narrated Aisha, Ummul Mu'minin:
The Messenger of Allah (ﷺ) was asked about a man who divorced his wife three times, and she married another who entered upon her, but divorced her before having intercourse with her, whether she was lawful for the former husband. She said: The Prophet (ﷺ) replied: She is not lawful for the first (husband) until she tastes the honey of the other husband and he tastes her honey. - Hadith Abu Dawud 2309 (islam)
Hadith Ibn Majah 1933 (en en2) Could she go back to her first man? No, not until he had tasted her sweetness.
from the Prophet, concerning a man who had a wife then divorced her, then another man married her but divorced her before consummating the marriage. Could she go back to the first man? He said: “No, not until he tastes her sweetness.” - Hadith Ibn Majah 1933 (islam)
Chapter: To divorce one's wife thrice (at a time)
Narrated `Aisha:
The wife of Rifa`a Al-Qurazi came to Allah's Messenger (ﷺ) and said, "O Allah's Messenger (ﷺ)! Rifa`a divorced me irrevocably. After him I married `Abdur-Rahman bin Az-Zubair Al-Qurazi who proved to be impotent." Allah's Messenger (ﷺ) said to her, "Perhaps you want to return to Rifa`a? Nay (you cannot return to Rifa`a) until you and `Abdur-Rahman consummate your marriage." - Hadith Al Bukhari 5260 (islam)