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The decimal expansion of 1/13 has a period of six digits. Find all primes p for which 1/p has a period of exactly four digits.

 Apr 13, 2019
 #1
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I could find only one such prime: 101

1/101 =0.00990099009900............etc.

 Apr 13, 2019
 #2
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The reciprocal of every prime p (other than two and five) has a period, that is the decimal expansion of 1/p repeats in blocks of some set length. This period is the period of p. ... For example: 3, 11, 37 and 101 are the only primes with periods one, two, three and four respectively−so they are unique-primes.

 

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.693.5325&rep=rep1&type=pdf

 Apr 14, 2019
 #3
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Thank you!

 Apr 14, 2019

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