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 2005-07-24, 23:34 #1 Dougy     Aug 2004 Melbourne, Australia 23·19 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 (non-trivial) natural number k such that k#=k##?
 2005-07-27, 11:59 #2 maxal     Feb 2005 22×32×7 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.
 2005-07-28, 13:13 #3 Dougy     Aug 2004 Melbourne, Australia 23·19 Posts Thanks.

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