142857
cyclic number
142,857 is the natural number following 142,856 and preceding 142,858. It is both a Kaprekar number and a cyclic number.
Cyclic number
142857 is the best-known cyclic number in base 10, being the six repeating digits of 1/7 (0.142857).
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 2/7, 3/7, 4/7, 5/7 or 6/7 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 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.
This brief starts where responsible research should: with the source description of “142857” as cyclic number. Everything that follows is an evidence route, not borrowed authority.
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The subject matters to the general reference register because the source frames it as cyclic number. Its deeper value depends on whether names, dates, institutions and citations support that framing.
Vocabulary and entity names are the principal evidence signals here, because they determine the precision of every later search. The source revision retrieved here is dated Aug 7, 2026. The linked authority identifier is Q846329. None of the 0 selected statements returned an explicit reference. The first chronological checks are 1422.
The absence of detail may reflect summary conventions rather than a lack of surviving documentation. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
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