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Prime repunits (Posted on 2016-11-24) Difficulty: 2 of 5
A number like 11, 111, or 1111 i.e. a number containing only the digit 1
is called repunit.

List all base-8 prime repunits.

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Non-trivial. | Comment 4 of 5 |

Let 8(n) denote the nth repunit, mod 8.

The even cases are trivial; each is divisible by 3, which has period 2 mod the sequence.

Other primes having very short periods mod the sequence are 31 (period 5) 127 (period 7) etc. 3,31,127 etc are Mersenne primes. It would be tempting to suggest a rule that 8(n) is divisible by Mersenne(n) with period p. Certainly, 8191 divides 8(13) and 131071 divides 8(17) while 524287 divides 8(19); but this is not true of 7 (period 7), nor does it account for 73, which is not a Mersenne Prime, yet has a period length of 3.

The rule 8(n) is of the form (an+1)(bn+1)(cn+1)..., for as many prime factors as needed applies to all the prime entries I have checked, 7 again excepted - 299593 has factors 7, (18*7+1), (48*7+1).

Edited on November 25, 2016, 11:16 pm
  Posted by broll on 2016-11-25 22:20:47

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