April 16th, 2018, 10:14 AM  #1 
Senior Member Joined: Dec 2015 From: Earth Posts: 256 Thanks: 28  Prime or not ?
Is $\displaystyle 14n3$ prime for each positive integer $\displaystyle n$ if so then how to prove it ? 
April 16th, 2018, 10:35 AM  #2 
Senior Member Joined: May 2016 From: USA Posts: 1,186 Thanks: 489 
$\dfrac{14n  3}{3} = 14 * \dfrac{n}{3}  1.$ Do you suppose there are some positive integers evenly divisible by 3? 
April 16th, 2018, 11:58 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,865 Thanks: 1833 
It's easier to disprove it: 14 × 2  3 = 5², which is composite. One can similarly deal with $14^n  3$. 
April 16th, 2018, 02:19 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,626 Thanks: 622 
There is no simple polynomial formula for primes.

April 16th, 2018, 05:35 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 19,865 Thanks: 1833 
Where "simple" means what?

April 16th, 2018, 09:34 PM  #6 
Senior Member Joined: Sep 2016 From: USA Posts: 502 Thanks: 280 Math Focus: Dynamical systems, analytic function theory, numerics  
April 16th, 2018, 11:29 PM  #7 
Senior Member Joined: Aug 2012 Posts: 2,076 Thanks: 593  Any. Mathman meant to say that there is no polynomial that produces only primes. Simple or not. I think it was a rhetorical imprecision, not a mathematical one. There is no such thing as a simple polynomial, it's not a definition I've ever heard. So mathman was using simple as an intensifier ... a simple polynomial, as in a mere polynomial. No mere polynomial could do what you want. No simple polynomial etc. That's how I interpreted mathman's remark. Simple as in a rhetorical flourish; not at all as specifying some subset of all the polynomials. Last edited by Maschke; April 16th, 2018 at 11:32 PM. 
April 17th, 2018, 12:46 PM  #8 
Senior Member Joined: Dec 2015 From: Earth Posts: 256 Thanks: 28 
How can we show that the prime formula exists or not ? In math we always must prove the existence first 
April 17th, 2018, 01:46 PM  #9 
Global Moderator Joined: May 2007 Posts: 6,626 Thanks: 622  

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