Re: [PrimeNumbers] Fascinating little result
- This problem entailed finding primes such that one digit dominates the combined digits of a) the prime itself, b) the prime's index, and c) the number of digits in all primes through it (or, alternatively, the length of the Smarandache-Wellin number associated with (ending in) it) beyond 50%. As I finally recognized, this is a large but probably finite list. Reporting the first instance, and perhaps last, in which 8 is the digit dramatically over-represented. 141888488887 is the 5759888858th prime, and adding its twelve digits to the total number of all primes up to it gives a total of 64488195888 digits. No example yet for 0, and my guess is that statistics would argue no such exists. Probably the largest outlier in statistical terms (limited to those exceeding 50% for some digit, so that a very large 45% example beyond the list would be excluded) is the nineteen 1s in 35117121691 with index 1511411519 and length of corresponding Smarandache-Wellin
number of 16113111114, but don't hold me to that. That would mean the last to reach or exceed 60% is given by the triple (32161111,1983151,15111113), all others being small. The absolute percentage maximum goes to 1979, where ten digits are involved seven of which are 9. Astronomical flukes are assumed non-existent, of course, in making this statement.
On Sat Aug 6th, 2011 12:27 AM EDT James Merickel wrote:
>Wrong! DIGITS TEND TOWARD EACH OTHER FOR ALL THREE. Whether or not the trend is totally down toward 1/3+2/30=0.4 is unclear to me at this point. Done!
>On Sat Aug 6th, 2011 12:12 AM EDT James Merickel wrote:
>>Okay, definitely last unless it is discussed. It's going to be relatively trivial to get limsup on the maximum ratio at 0.6, heuristically at least and probably with full proof. Most likely one can prove that there are infinitely many for which the ratio exceeds this. Dirichlet's theorem plus not too much ordinary statistics considering that factor between the prime and total digits would seem likely to accomplish this and no more.
>>On Fri Aug 5th, 2011 11:34 PM EDT James Merickel wrote:
>>>Well, don't want to spam this forum, so last. I don't stand by that expected gap without computation, and this is the first prime for which 6 is the digit: 1324666661 (with index 66403276 and total digits 606669676). Apologies.
>>>On Fri Aug 5th, 2011 11:23 PM EDT James Merickel wrote:
>>>>The final values in the total digits of 26 range after the list given are 1,3,3,4,4, and I expect 27 will also have. In retrospect, it seems certain the list is infinite and even 0 would eventually appear, with the number of digits in the first N primes going to 1/ln(10) times that prime, with essentially always the same number of digits within 1. I would expect a long gap to arise while the index remains relatively significant, however.
>>>>On Fri Aug 5th, 2011 9:12 PM EDT James Merickel wrote:
>>>>>The digits for successive cases of a digit being more than half those in the combination of prime, index, and total digits of all primes through, are:
>>>>>1,2,3,4,1,9,9,3,9,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,3,3,3,3,3,3,4,1,1,4. May or may not be more, and full examples are more interesting, but I will leave that to somebody set up to copypaste his or her own version.