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.
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.
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) — more general patterns this builds on
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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 → Cycle → Network → Reservoir-Flux Network → Conservation Laws → Invariance
Neighborhood in Abstraction Space¶
Cyclic Number sits in a sparse region of the domain-specific corpus (77th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Numeration & Arithmetic Representations (15 abstractions)
Nearest neighbors
- Multiplicative Digital Root — 0.85
- Normal Number — 0.84
- Lunar arithmetic — 0.83
- Complex-base system — 0.82
- Champernowne constant — 0.82
Computed from structural-signature embeddings · 2026-09-08