When declining ordinal numbers (date). Would nominative or genitive be more natural?
Either пятый день... or "Пятое"?
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"Пятый день декабря" is archaic, with Bible connotations, never used now, except in highly stylised fiction. "Пятое декабря" is standard modern Russian and used by everybody.
You can use "пятый день (time period in genitive)" to mean the fifth day of a period. For instance:
пятый день отпуска = fifth day of a/the vacation
пятый день забастовки = fifth day of a/the strike
пятый день войны = fifth day of a/the war
пятый день сбора урожая = fifth day of a/the harvest
But you would not usually use this for months, to mean a date. In that case you use
пятое [число] декабря
The neuter case is used because the noun число (number) is implyed but usually omitted.
But note that you cannot use this form with some unusual, allien, ancient calendar where days are numered different way so you fall back to the first form "пятый день of something" in this case.
This means that you cannot use the term число (number) as a synonym for day for any Moon-based calendar, including the one used in Bible. Compare English as well: would you say "Nissan, 15th"? I doubt. It would be better to say "15th day of the month Nissan" because the listener may not even know what Nissan is.