[seqfan] Re: Smallest prime with a substring of exactly n zeros

[Bob Selcoe]

Hi Felix and Seqfans,

Interesting question - I imagine the answer is no.

There can only be at most 36 possible combinations of digits with completely 
contiguous zeros and number of digits = n+2, ({1..9} for the first digit, 
{1,3,7,9} for the last).  If those combinations are exhausted, then there 
are only 36 possibilities for first digit 1 and the second-to-last digit 
{1..9} with last digit {1,3,7,9} being prime for n+3 digits.  If those 
combinations are exhausted, it would make possible many other combinations 
without completely contiguous zeros.

So I would conjecture there are many (perhaps infinite) counterexamples as n 
gets quite large.

Is someone with programming skills interested in finding the smallest such 
counterexample (presuming it exists)?

Cheers,
Bob Selcoe

--------------------------------------------------
From: "Felix Fröhlich" <felix.froe at gmail.com>
Sent: Friday, February 19, 2016 9:44 AM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] Smallest prime with a substring of exactly n zeros

> Dear Sequence fans
>
> a(n) is the smallest prime p containing a substring of exactly n zeros. 
> The
> sequence (with offset 1) starts
>
> 101, 1009, 10007, 100003, 1000003, 20000003, 100000007, 1000000007
>
> Does the sequence always coincide with A037053? A counterexample would 
> have
> to be a prime having exactly n zeros in its decimal expansion but where
> those zeros do not form a contiguous string. That prime would also need to
> be the smallest prime with exactly n zeros. Does such a counterexample
> exist?
>
> Best regards
> Felix Fröhlich
>
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