[seqfan] Re: Jumping off the primes thread

israel at math.ubc.ca israel at math.ubc.ca
Sun Mar 22 18:38:32 CET 2020

The n'th prime p(n) has about log_{10}(p(n)) decimal digits.
The probability of its matching a random number of this size is 
about 1/10^log_{10}(p(n)) = 1/p(n). Since sum_n 1/p(n) diverges,
we should expect it to match infinitely many times.  However, 
since that sum diverges extremely slowly, these matches should
be very sparse.


On Mar 22 2020, Daniel via SeqFan wrote:

> The title of the current conversation about "primes describing digit 
> positions" gave me an idea: When does the nth prime number appear 
> starting with the nth digit of pi? A good starting point is to look for 
> cases where A064467(n)=n, but there are none in that sequence's b-file. 
> However, a given prime might appear both before and at the nth digit of 
> pi, so I compared the primes and the digits of pi directly.
> My Excel spreadsheet couldn't find any examples up to n=10,000. But I did 
> find a few near-misses: * Obviously the 2nd prime number (3) appears in 
> the 1st digit of pi. * The 41st prime number is 179 - and digits 41-43 of 
> pi are 169. Just one off on the middle digit! * The 63rd prime number 
> (307) misses by only two places: it appears in digits 65-67 of pi. * The 
> 118th prime number (647) misses by only one place: it appears in digits 
> 119-121 of pi.
> Obviously the probability of finding one shrinks as the numbers climb. 
> Are there any examples at all? If so, are they sequence-worthy? If not, 
> is there an underlying reason why not? Daniel Sterman
>Seqfan Mailing list - http://list.seqfan.eu/

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