I had a hard time formulating the question, but I wonder if a month with 31 days has a specific name, Likewise months with 30, 29 and 28 days.
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I've never heard any terms that fit your criteria, so I'd say these are not common words that most people would understand. But the Wikipedia article about months states that the lunar months (as determined by the moon) are 30 days or 29 days, which are respectively called a full month and a hollow month. However, since lunar calendars are difficult to reconcile with the seasons and the sun, the modern (Gregorian) calendar is a solar calendar and its months do not really have anything to do with the lunar cycle. In that case we have 31- and 30-day months, and the allocation of days is arbitrary and historic.
No,the only thing related which has name is "leap year" which means year where February has 29 days.
I would gladly be corrected, anyhow you might be interested to read this, as it describes how the days got "distributed" between the different months: