142857 (number)

Calculations

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 (= 142857 + 857142)

If you multiply 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 you have only the six digits 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 1/7 = 0.142857. Thus, multiples of 1/7 are simply repeated copies of the corresponding multiples of 142857:

1 ÷ 7 = 0.142857 2 ÷ 7 = 0.285714 3 ÷ 7 = 0.428571 4 ÷ 7 = 0.571428 5 ÷ 7 = 0.714285 6 ÷ 7 = 0.857142 7 ÷ 7 = 0.999999 = 1 8 ÷ 7 = 1.142857 9 ÷ 7 = 1.285714 …

1/7 as an infinite sum

There is an interesting pattern of doubling, shifting and addition that gives 1/7.

1 / 7   = 0.142857142857142857 … = 0.14 + 0.0028 + 0.000056 + 0.00000112 + 0.0000000224 + 0.000000000448 + 0.00000000000896 + ⋯ = 14 100 + 28 100 2 + 56 100 3 + 112 100 4 + 224 100 5 + ⋯ + 7 × 2 N 100 N + ⋯ = ( 7 50 + 7 50 2 + 7 50 3 + 7 50 4 + 7 50 5 + ⋯ + 7 50 N + ⋯ ) = ∑ k = 1 ∞ 7 50 k

Each term is double the prior term shifted two places to the right.

Another infinite sum is

1 / 7   = 0.1 + 0.03 + 0.009 + 0.0027 + 0.00081 + 0.000243 + 0.0000729 + ⋯ = 3 0 10 1 + 3 1 10 2 + 3 2 10 3 + 3 3 10 4 + 3 4 10 5 + ⋯ + 3 N − 1 10 N + ⋯

Other bases

In some other bases, six-digit numbers with similar properties exist, given by (base⁶ − 1)/7. E.g. in base 12 it is 186A35 and base 24 3A6LDH.

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 1/7, 2/7. etc., and the subsequent movement of the enneagram, are portrayed in Gurdjieff's sacred dances known as the movements.