permutations) listed in the Special Cases section above) In each of these cases, the digits across half the period add up to the base minus one. Thus for binary, the sum of the bits across half the period is 1; for ternary, it is 2, and so on. In binary, the sequence of cyclic numbers begins: 11 (3) → 01 101 (5) → 0011 1011 (11) → 0001011101 1101 (13) → 000100111011 10011 (19) → 000011010111100101 11101 (29) → 0000100011010011110111001011 In ternary: 2 (2) → 1 12 (5) → 0121 21 (7) → 010212 122 (17) → 0011202122110201 201 (19) → 001102100221120122 In quaternary, there are none. In quinary: 2 (2) → 2 3 (3) → 13 12 (7) → 032412 32 (17) → 0121340243231042 43 (23) → 0102041332143424031123 122 (37) → 003142122040113342441302322404331102 In senary: 15 (11) → 0313452421 21 (13) → 024340531215 25 (17) → 0204122453514331 105 (41) → 0051335412440330234455042201431152253211 135 (59) → 0033544402235104134324250301455220111533204514212313052541 141 (61) → 003312504044154453014342320220552243051511401102541213235335 In base 7: 2 (2) → 3 5 (5) → 1254 14 (11) → 0431162355 16 (13) → 035245631421 23 (17) → 0261143464055232 32 (23) → 0206251134364604155323 In octal: 3 (3) → 25 5 (5) → 1463 13 (11) → 0564272135 35 (29) → 0215173454106475626043236713 65 (53) → 0115220717545336140465103476625570602324416373126743 73 (59) → 0105330745756511606404255436276724470320212661713735223415 In nonary, the unique cyclic number is 2 (2) → 4 In base 11: 2 (2) → 5 3 (3) → 37 12 (13) → 093425A17685 16 (17) → 07132651A3978459 21 (23) → 05296243390A581486771A 27 (29) → 04199534608387A69115764A2723 In duodecimal: 5 (5) → 2497 7 (7) → 186A35 15 (17) → 08579214B36429A7 27 (31) → 0478AA093598166B74311B28623A55 35 (41) → 036190A653277397A9B4B85A2B15689448241207 37 (43) → 0342295A3AA730A068456B879926181148B1B53765 In ternary (b = 3), the case p = 2 yields 1 as a cyclic number. While single digits may be considered trivial cases, it may be useful for completeness of the theory to consider them only when they are generated in this way. It can be shown that no cyclic numbers (other than trivial single digits, i.e. p = 2) exist in any numeric base which is a perfect square, that is, base 4, 9, 16, 25, etc. See also Repeating decimal Fermat's little theorem Cyclic permutation of integer Parasitic number References Further reading Gardner, Martin. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments From Scientific American. New York: The Mathematical Association of America, 1979. pp. 111–122. Kalman, Dan; 'Fractions with