23 times 253 = 5619. 23 is a prime, 253 is not.
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No it can not
The 625th prime number is equal to 4637.
A prime number
A balanced prime number is a prime number that is equal distance from the last prime before it and the next prime after it. Example: 53 is a balanced prime. The last prime number before it was 47 and the next prime after it is 59. (53 - 47) and (59 - 53) both equal 6, so 53 is a balanced prime.
Since the product number has 4 factors, it cannot be a prime.