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The coworker gave the definition of semiprime as a number divisible by itself, 1, and one other number (for instance, 9....1X9 and 3X3).

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Assuming that the true definition of semiprime numbers is the first one, the only numbers available are odd, square numbers whose square root = a prime number. Seems like a small amount, huh? But another way to put it is: take any prime, multiply it by itself, and you get a semisquare number. Since there are an infinite number of primes, there must, logically, be an infinite number of semiprimes.

They're somewhat more common than this. The definition doesn't restrict itself only to the squares of prime numbers. Any number that could be expressed as P to the Nth power, where P is a prime number and N is any integer greater than one, is a semiprime number (assuming that this definition is correct). So for every prime number there would be an infinite amount of corresponding semiprime numbers.