142857 is the best-known
cyclic number in base 10, being the six
repeating digits of (0.). If 142857 is
multiplied by 2, 3, 4, 5 or 6, the answer will be a
cyclic permutation of itself, and will correspond to the repeating digits of , , , or respectively: : 1 × 142,857 = 142,857 : 2 × 142,857 = 285,714 : 3 × 142,857 = 428,571 : 4 × 142,857 = 571,428 : 5 × 142,857 = 714,285 : 6 × 142,857 = 857,142 : 7 × 142,857 = 999,999 If multiplying by an integer greater than 7, there is a simple process to get to a cyclic permutation of 142857. By adding the rightmost six digits (ones through hundred thousands) to the remaining digits and repeating this process until only six digits are left, it will result in a cyclic permutation of 142857: : 142857 × 8 = 1142856 : 1 + 142856 = 142857 : 142857 × 815 = 116428455 : 116 + 428455 = 428571 : 1428572 = 142857 × 142857 = 20408122449 : 20408 + 122449 = 142857 Multiplying by a multiple of 7 will result in 999999 through this process: : 142857 × 74 = 342999657 : 342 + 999657 = 999999 If you square the last three digits and subtract the square of the first three digits, you also get back a cyclic permutation of the number. : 8572 = 734449 : 1422 = 20164 : 734449 − 20164 = 714285 It is the repeating part in the
decimal expansion of the
rational number = 0.. Thus, multiples of are simply repeated copies of the corresponding multiples of 142857: : \begin{align} \tfrac17 & = 0.\overline{142857}\ldots \\[3pt] \tfrac27 & = 0.\overline{285714}\ldots \\[3pt] \tfrac37 & = 0.\overline{428571}\ldots \\[3pt] \tfrac47 & = 0.\overline{571428}\ldots \\[3pt] \tfrac57 & = 0.\overline{714285}\ldots \\[3pt] \tfrac67 & = 0.\overline{857142}\ldots \\[3pt] \tfrac77 & = 0.\overline{999999}\ldots = 1 \\[3pt] \tfrac87 & = 1.\overline{142857}\ldots \\[3pt] \tfrac97 & = 1.\overline{285714}\ldots \\ & \,\,\,\vdots \end{align} == Connection to the enneagram ==