Catalan semiprimes

franktaw at franktaw at
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), 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.
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
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