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Repunits and bases (Posted on 2013-07-24) Difficulty: 3 of 5
112=3 and 1113=13 are prime. However, 11114=85=5*17 and 111115=781=11*71 are not prime. What is the least number n>3 such that the repunit consisting of n 1's in base n is prime?

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Answer: | Comment 1 of 6

The general formula is x=(n^n-1)/(n-1), which gives a base 10 value that can quickly be checked for primality. Obviously, only prime n need be checked. Using this method,

{{n == 2, x == 3}

 {n == 3, x == 13}

 {n == 5, x == 781}

 {n == 7, x == 137257}

 {n == 11, x == 28531167061}

 {n == 13, x == 25239592216021}

 {n == 17, x == 51702516367896047761}

 {n == 19, x == 109912203092239643840221}...}

109912203092239643840221 is prime, and in base 19 is:

1111111111111111111.

Edited on July 24, 2013, 11:45 am
  Posted by broll on 2013-07-24 11:29:46

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