20050724, 23:34  #1 
Aug 2004
Melbourne, Australia
2^{3}·19 Posts 
Primorial question
For the question I need a slightly modified definition of primorial. Let n# = the product of all primes less than or equal to n, for all natural numbers n. Note that this disobeys the convention that n must be prime for n# to be valid.
Does the sequence 3#, 3##, 3###, ... get arbitarily large? Or could there exist a (nontrivial) natural number k such that k#=k##? 
20050727, 11:59  #2 
Feb 2005
2^{2}·3^{2}·7 Posts 
Note that if n >= 16 then n# > n:
according to Chebyshev theorem there is a prime p between [n/2] and n, and there is a prime q between [n/4] and [n/2]. Therefore, #n >= pq > [n/4]^2 >= sqrt(n)^2 = n. Since 3# = 2*3 = 6 3## = 2*3*5 = 30 which is >= 16 from this point the sequence must be strictly increasing: 3### > 3## 3#### > 3### and so on. 
20050728, 13:13  #3 
Aug 2004
Melbourne, Australia
152_{10} Posts 
Thanks.

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