# Catalan semiprimes

franktaw at netscape.net franktaw at netscape.net
Wed Apr 12 23:53:01 CEST 2006

```A comment to A000108 (the Catalan numbers) reads:

Koshy and Salmassi give an elementary proof that the only prime Catalan numbers are a(2) = 2 and a(3) = 5. Is the only semiprime Catalan number a(4) = 14? - Jonathan Vos Post (jvospost2(AT)yahoo.com), Mar 06 2006

The answer is yes.  Using the formula C_n = C(2n,n)/(n+1), it is immediately clear that C_n can have no prime factor greater than 2n.  For n >= 7, C_n > (2n)^2, so it cannot be a semiprime.d

Given that the Catalan numbers grow exponentially, the above consideration implies that the number of prime divisors of C_n, counted with multiplicity, must grow without limit.

The number of distinct prime divisors must also grow without limit, but this is more difficult.  Any prime between n+1 and 2n (exclusive) must divide C_n.  That the number of such primes grows without limit follows from the prime number theorem.

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