Cyclic Number¶

Encode a repeating unit fraction as a digit block whose consecutive nonzero multiples appear as cyclic rotations of that block in a fixed base.

Version: v3 · 2026-09-19 · History
Domain-specific #: 1611
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Number Theory → Mathematics
Aliases: Full reptend cyclic number

Core Idea¶

A cyclic number in base b is a finite digit block whose consecutive multiples rotate its digits cyclically rather than changing their multiset and order structure. The classic decimal example 142857 satisfies n×142857 for n=1,…,6 as the six rotations 142857, 285714, 428571, 571428, 714285, and 857142.

The phenomenon comes from recurring unit fractions. If a prime p has b as a primitive root modulo p, then 1/p has period p−1 in base b; its repetend supplies a cyclic number when leading-zero and ordering conventions are handled. Merely having rotations among some multiples is insufficient: the classical definition requires the designated consecutive multipliers and normally excludes trivial repeated blocks.

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The Merry-Go-Round Number

Some numbers do a magic trick. Take 142857 and multiply it by 2: you get 285714, the very same digits just going around in a circle starting at a different spot. Multiply by 3, 4, 5, or 6 and the same thing happens. A number that does this trick is a cyclic number.

Digits That Spin Around

A cyclic number is a number whose multiples just rotate its digits. The famous one is 142857: times 2 is 285714, times 3 is 428571, times 4 is 571428, times 5 is 714285, and times 6 is 857142. Each answer uses the same digits in the same circular order, just starting at a different spot. It is not enough for only some multiples to do this; it has to work for each multiplier in the row, from 1 up to the length of the number. These numbers come from the repeating digits of fractions like 1/7.

Rotating-Multiple Numbers

A cyclic number is a block of digits, in some base, such that multiplying it by 1, 2, 3, … up to its number of digits just rotates the digits cyclically. In base 10, 142857 is the classic case: multiplying by 1 through 6 gives its six rotations. Cyclic numbers come from repeating decimals of fractions 1/p for certain primes p. When the repeating part of 1/p is as long as possible, p−1 digits, that repeating block (keeping any leading zeros) is a cyclic number; for 1/7 the repeating block is 142857. Having a few multiples that happen to be rotations isn't enough: the definition requires the whole run of consecutive multipliers, and a block that just repeats one digit doesn't count.

A cyclic number in base b is a finite digit block such that multiplying it by 1, 2, …, up to its length produces exactly the cyclic permutations of its digits, preserving both the digit multiset and their circular order. The decimal paradigm is 142857, whose multiples by 1 through 6 are its six rotations. The phenomenon arises from recurring unit fractions: if p is prime and b is a primitive root modulo p, then the base-b expansion of 1/p has period exactly p − 1, and its repetend is a cyclic number. Because the repetend can begin with zeros in some bases, leading-zero conventions (and conventions about digit ordering) must be fixed for the block to qualify. The classical definition is strict: it requires the designated consecutive multipliers, not just some multiples that happen to be rotations, and it normally excludes trivial repeated-digit blocks.

Scope of Application¶

Cyclic numbers are literal in base-dependent number theory and recurring-expansion analysis.

Recurring decimals. Connecting repetends of unit fractions with digit rotations.
Modular arithmetic. Using multiplicative order and primitive roots.
Recreational number theory. Constructing and classifying striking digit identities.
Base-generalized numeration. Studying the property beyond decimal.
Repunit factorization. Relating blocks to (b^L−1)/p.
Algorithmic search. Testing period, rotation, and consecutive-multiple conditions.

Clarity¶

State the base, digit width, leading-zero policy, excluded trivial cases, and exact multiplier range. Distinguish rotation of a fixed-width string from numerical equality after dropping zeros. When deriving from 1/p, prove the period length and the order of b modulo p rather than inferring cyclicity from a few multiples.

Manages Complexity¶

The digit orbit turns a modular-order relation into a visually compact invariant. It connects multiplication, carries, fractions, and rotations in one example. Visual pattern matching can overaccept near-misses; modular arithmetic supplies the proof and cleanly separates full cycles from partial or out-of-order rotations.

The abstraction compresses a large multiplication table into an orbit. Instead of storing each product independently, one stores a word, a rotation action, and the correspondence between multiplier and rotation.

Abstract Reasoning¶

Fix a base and width convention.
Generate the cyclic rotations of the candidate block.
Compute its consecutive multiples under fixed-width representation.
Test equality between the ordered multiplier list and rotation orbit.
Relate the block to a unit-fraction repetend when available.
Compute the base's multiplicative order modulo the denominator.
Exclude trivial repetitions and convention artifacts.
Generalize or enumerate only within the stated base.

Knowledge Transfer¶

The strict parent is Cycle because digit rotation forms a closed orbit returning to the original arrangement. The number-theoretic identity adds multiplication, base representation, and modular order. Rotating schedules or strings elsewhere instantiate Cycle, not the cyclic-number abstraction.

Cycle is the strict parent because the observable products occupy one finite orbit under rotation and eventually return to the starting block. What transfers is the structure state word -> repeated shift action -> closed orbit, not the decimal appearance.

Relationships to Other Abstractions¶

Current abstraction Cyclic Number Domain-specific

Parents (1) — outgoing typed relations

Cyclic Number is a kind of Cycle Prime

Cycle is the strict parent because rotations form a closed orbit under consecutive multiplication, eventually returning to the original digit arrangement.

Hierarchy path (1) — routes to 1 parentless root

Cyclic Number → is a kind of Cycle → presupposes Network

Neighborhood in Abstraction Space¶

Cyclic Number sits in a sparse region of the domain-specific corpus (66^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Numeral Systems & Digit Algorithms (11 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25