Answer: 23-SKIDOO
Solution: All of the prime-numbered squares spell out, in order, THEBIGLEBOWYABBADABBA, whose missing "endings" signal SKI and DOO, respectively. This gives us SKIDOO, the second half of an old-fashioned, early 20th century slang expression 23-SKIDOO, the meta answer. Note that 23, of course, is a prime number, the current year, and 23-SKIDOO was an antiquated phrase that meant to "skedaddle," "scram," or "get out of here." This is reinforced at 23-Across with BEATIT. If you didn't know the expression, quickly googling "numerical expression skidoo" or something similar gets you what you need.