The number 142,857 is a

Kaprekar number In mathematics, a natural number in a given number base is a p-Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has p digits, that add up to the original number. The numbers are n ...

. 142857, the six repeating digits of (0.), is the best-known

cyclic number A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are :142857 × 1 = 142857 :14 ...

in base 10. If it is multiplied by 2, 3, 4, 5, or 6, the answer will be a

cyclic permutation In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...

of itself, and will correspond to the repeating digits of , , , , or respectively.

Calculation

: 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 A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\ldots b_0.a_1a_2\ldots Here is the decimal separator, is ...

of the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

= 0.. Thus, multiples of are simply repeated copies of the corresponding multiples of 142857: :

\tfrac97 & = 1.\overline\ldots \\ & \,\,\,\vdots \end

Connection to the enneagram

The 142857 number sequence is used in the enneagram figure, a symbol of the Gurdjieff Work used to explain and visualize the dynamics of the interaction between the two great laws of the Universe (according to G. I. Gurdjieff), the Law of Three and the Law of Seven. The movement of the numbers of 142857 divided by , . etc., and the subsequent movement of the enneagram, are portrayed in Gurdjieff's sacred dances known as the movements.

Other properties

The 142857 number sequence is also found in several decimals in which the denominator has a factor of 7. In the examples below, the numerators are all 1, however there are instances where it does not have to be, such as (0.). For example, consider the fractions and equivalent decimal values listed below: : = 0.... : = 0.0... : = 0.03... : = 0.0... : = 0.017... : = 0.0... The above decimals follow the 142857 rotational sequence. There are fractions in which the denominator has a factor of 7, such as and , that do not follow this sequence and have other values in their decimal digits.

References

* * * {{DEFAULTSORT:142857 Integers