Yeah, 1 more than any prime power always has solutions. This has been known for a while. My notes are here. The Singer construction generates many such rulers. http://cube20.org/golomb/ -tom On Tue, Nov 10, 2020 at 10:47 AM ed pegg <ed@mathpuzzle.com> wrote:

A perfect cyclic difference set with n values has length L = (n-1)^2 + n and works mod L. All differences up to L-1 exist uniquely. 3 values is structurally unique, 1 solution. 4 values has 2 solutions. 5 values has 1 solution. 6 values has 5 solutions. 7 values has 0 solutions. 8 values has 1 solution. Do any subsequent values have more than 1 solution?

{{{7,3},{0,1,3}}, {{13,4},{0,1,3,9}},{{13,4},{0,1,4,6}}, {{21,5},{0,1,4,14,16}},

{{31,6},{0,1,3,8,12,18}},{{31,6},{0,1,3,10,14,26}},{{31,6},{0,1,4,6,13,21}},{{31,6},{0,1,4,10,12,17}},{{31,6},{0,1,8,11,13,17}}, {{57,8},{0,1,3,13,32,36,43,52}},

{{73,9},{0,1,12,20,26,30,33,35,57}},{{91,10},{0,1,5,7,27,35,44,67,77,80}},{{133,12},{0,1,3,17,21,58,65,73,100,105,111,124}},{{183,14},{0,1,3,24,41,52,57,66,70,96,102,149,164,176}},{{273,17},{0,1,22,33,83,122,135,141,145,159,175,200,226,229,231,238,246}},{{307,18},{0,1,3,30,37,50,55,76,98,117,129,133,157,189,199,222,293,299}},{{381,20},{0,1,13,16,21,45,75,82,86,92,128,142,180,206,229,231,264,286,354,363}}} _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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