20190601, 22:43  #1 
Feb 2017
Nowhere
2^{5}·139 Posts 
Not eleven smooth (four consecutive integers >11))
The entry for the number 11 in the Penguin Dictionary of Curious and Interesting Numbers says, "Given any four consecutive integers greater than 11, there is at least one of them that is divisible by a prime greater than 11."
Your mission, should you decide to accept it, is to prove this. 
20190602, 00:56  #2 
"Matthew Anderson"
Dec 2010
Oregon, USA
2·3·7·17 Posts 
It is not a proof but it would be interesting to see if this property holds up to, say 10^6.
Regards, Matt 
20190602, 01:41  #3  
Sep 2017
3^{2}·11 Posts 
Quote:
I checked all the 4consecutive integers in the range [12,10^10] and couldn't find a violation of this property. 

20190602, 01:41  #4 
"Matthew Anderson"
Dec 2010
Oregon, USA
2×3×7×17 Posts 
This looks like a solid conjecture.

20190602, 03:36  #5  
May 2019
113 Posts 
Quote:
To prove the conjecture, we can solve 31 Pell’s equations using the procedure described in https://en.m.wikipedia.org/wiki/St%C...er%27s_theorem. Last fiddled with by 2M215856352p1 on 20190602 at 04:13 

20190602, 05:16  #6  
Sep 2017
3^{2}·11 Posts 
Quote:


20190602, 09:24  #7  
May 2019
113 Posts 
Quote:
I have written a Python 3 script which verified the conjecture up to 10^{30}. The script could take a few seconds to run, hence you would need to be patient. Please notify me if the program has a bug. The Python script does not generate a proof because there still remains the possibility of a solution beyond 10^{30}. To really prove the conjecture, the only method I have now is to solve the 31 Pell's equations involved, which is going to be very tedious. To run the script, please change the file extension from .txt to .py. I don't know why .py file extension is not supported. 

20190602, 09:32  #8 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
5×1,223 Posts 

20190602, 09:35  #9 
May 2019
113 Posts 

20190602, 09:37  #10 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
5×1,223 Posts 

20190602, 09:46  #11 
May 2019
161_{8} Posts 
A good enough hint for me. Thanks. Should I spoil the solution myself if I find it?
Last fiddled with by 2M215856352p1 on 20190602 at 09:57 
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