I have been thinking about it lately. Let's think of prime number sequence: $$q_1,q_2,...q_n$$ where $q_1=2, q_2=3$ and onwards. Can we find an n such as the inequality $$q_n \gt q_1q_2q_3...q_{n-1}.$$ I thought since the prime number gap is increasing continually this might hold however I could not prove it or disprove. And the only method which comes to me about prime number gap is the one concerning factorials which says if you want to create an n-number prime number gap use $$(n!)+2, (n!)+3, ..., (n!)+n.$$ However, I could not get to anywhere from there neither. Any help would be appreciated.
• It seem that $q_n \gt q_1q_2q_3...q_{n-1}$ implies $q_n=q_1q_2q_3...q_{n-1}+1$. :-) – Alex Ravsky Apr 18 '13 at 14:31
no. Bertrand's Postulate. There is always a prime between $n$ and $2n.$