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Re: [PrimeNumbers] Half period primes

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  • Ben Bradley
    ... I wondered that myself, and I found the answer at, of all places, the primepages.org website: http://primes.utm.edu/glossary/page.php?sort=PeriodOfAPrime
    Message 1 of 3 , Jul 15, 2004
      At 09:41 AM 7/15/04 -0700, Cletus Emmanuel wrote:
      >What is the period of a prime?....And what is meant by half period? I think I know, but I am not sure...

      I wondered that myself, and I found the answer at, of all places, the primepages.org website:

      http://primes.utm.edu/glossary/page.php?sort=PeriodOfAPrime

      In short, it's the "period" of the decimal expansion of the reciprocal of a prime, but even here, "period" has a special meaning: If the reciprocal of P repeats at P-1, it's a "full period. Primes such as 7, 17 and 19 are examples. Some other primes have reciprocals that repeat at (P-1)/2 digits, such as 13's reciprocal repeating every six digits (1/13 = 0.076923 076923 076923 ...) and these are called "half period."
      The "period" of a prime, as studied here, is an artifact of the base 10 number system. Has any study been done on prime periods in different number systems? Many numbers have a finite, exact-value finite-digit reciprocal in one base, and an infinite series of repeating digits in another base, so it seems this research would have different and perhaps interesting results in other number bases. I see no mention of other bases in relation to period on the Prime Pages.

      >----Cletus
      >
      >julienbenney <jpbenney@...> wrote:
      >Such websites as "web.usna.navy.mil/~wdj/book/node43.html" show that
      >the proportion of full period primes is asymptotic to 37%. I wonder if
      >it is possible to extend this theorem to other possible periods,
      >beginning with half period primes - those with the period equal to (p-
      >1)/2.
      >
      >I have tried to count (that is all I can do, actually) the number of
      >half period primes (first few: 13, 31, 43, 67, 71, 83, 89, 107...) and
      >found that there were 44 half period primes out of 168 less than 1000,
      >and 33 out of 135 between 1000 and 2000. This suggests that the
      >proportion of half period primes falls off with size to a certain
      >asymptotic proportion - half period primes amount to 26.19 percent of
      >primes less than 1000 and 24.44 percent between 1000 and 2000 -
      >suggesting a levelling to say, 20-22 percent of all primes.
      >
      >What study has been done about the distribution of actual versus
      >possible periods for primes of arbitrary size??
      >
      >
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      ---
      http://mindspring.com/~benbradley
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