Testwiki:Reference desk/Archives/Mathematics/2013 November 20

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November 20

Cyclic numbers

The article states that there are none in square bases. Why? Double sharp (talk) 05:37, 20 November 2013 (UTC)

If you look at how to form them then if the numerator $b^{p-1}-1$ is divisible by p, and b is the square of s then p will divide s so you don't get the longest length. Just factor the numerator and use Euler's theorem. Dmcq (talk) 10:22, 20 November 2013 (UTC)

Not quite... p is a prime that does not divide b, so p cannot divide s. However, if you expand the numerator: $b^{p-1}-1=s^{2(p-1)}-1=(s^{p-1}-1)(s^{p-1}+1)$, so any cyclic number of length p-1 in base b would have to be a multiple of a cyclic number of length p-1 in base s. The multiplying factor, expressed in base p, is 1[0]1, where [0] is a number of 0s (actually ((p-1)/2)-1 0s), which means that the base b cyclic number is the base s cyclic number (expressed in base b) repeated twice. Since the number is cyclic in base s, all pemuations of that number in base s are multiples. This will result in two sets of digits in base b (one for the even permutations, and one for the odd permuations), since two digits in s make up each digit in b. Note that the even and odd permuations are not necessarily the same as the even and odd multiples. MChesterMC (talk) 16:37, 20 November 2013 (UTC)

Sorry yes, I should have just given the theorem as a clue and left the rest as an exercise for the reader. Much less chance of me getting it wrong that way ;-) Dmcq (talk) 16:55, 20 November 2013 (UTC)