Q- Is the following Hadith authentic?“The one who fasts on the twenty seventh (27th) day of Rajab will receive the reward of fasting for sixty years“
Have not come across this narration. However Hafiz Ibn Hajar (rahimahullah) has cited a similar narration as the statement of Sayyiduna Abu Hurayrah (radiyallahu ‘anhu) -not as as the statement of Nabi (sallallahu ‘alayhi wa sallam) and has declared it as unreliable. (Tabyinul ‘Ajab, pg. 45, Hadith: 28)
It should be noted that the Hadith Masters have agreed that there exists no Hadith that mentions the virtues of fasting on any specific date in Rajab. In Fact Hafiz Ibn Rajab (rahimahullah) has stated the following: “There exists no Hadith, reported from Nabi (sallallahu ‘alayhi wa sallam) or the Sahabah that mentions the virtues of fasting on any specific date in Rajab”.(Lataiful Ma’arif, pg. 228)
And Allah Ta’ala Knows best
https://hadithanswers.com/a-virtue-of-fasting-on-the-twenty-seventh-27th-of-rajab/