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: v2 · 2026-09-06 · History
Domain-specific #: 1611
Origin domain: mathematics
Subdomain: number theory
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.^[1]

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.

Structural Signature¶

The fixed numeral base. Digit representation and rotation are base-dependent.
The finite digit block. One nontrivial numeral of declared length is tested.
The cyclic-rotation orbit. Moving a prefix to the suffix generates the allowed representations.
The consecutive multipliers. Successive integers map the number through the rotation orbit.
The carry structure. Multiplication carries coordinate the global digit rotation.
The repeating-fraction source. The block often forms the full repetend of 1/p.
The multiplicative-order condition. The base cycles through all nonzero residues modulo p.
The leading-zero convention. Fixed-width representations determine whether some examples qualify.

What It Is Not¶

Not a cyclic group. The number exhibits a digit-rotation property, not an algebraic group structure.
Not any repeating decimal. The repetend must support the consecutive-multiple rotation property.
Not any number whose rotations are multiples. The multiplier order matters.
Not base-independent. Changing the base changes digits and modular order.
Not a repdigit or repeated cyclic block. Classical conventions exclude trivial repetition.
Not well-defined without leading-zero rules. Fixed-width rotations can begin with zero.

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.

Fix the numeral base and the exact digit block before testing the property. Leading zeros belong to the fixed-width block when a rotation moves them to the front; dropping them changes both length and rotation. The relevant multiples are the nonzero multiples specified by the definition, ordinarily through the digit length or denominator relation, and each result must be read with the same width. A repeating decimal can be periodic without being cyclic, and a rotation observed for one multiplier does not establish closure for the required multiplier range. Modular arithmetic provides the audit trail: multiplication of the repetend corresponds to a shift only when the powers of the base traverse the appropriate residue class pattern. State whether the number means the digit integer, the repetend of a unit fraction, or the underlying denominator, because historical sources sometimes move among these related objects.^[1]

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. That compression also supplies diagnostics. Repeated blocks reveal a shorter true period; missing rotations reveal that the relevant residue orbit is incomplete; carries or width changes reveal that ordinary multiplication is not acting as a pure cyclic shift. The base is constitutive rather than cosmetic, since changing it changes the power residues and the digit alphabet. Computation can enumerate products efficiently, but a finite list should be paired with a reason that the tested range is exhaustive. Number-theoretic order conditions explain why the pattern persists and distinguish an accidental collection of rotated products from the stable cyclic-number identity.

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. Similar cyclic words occur in coding and combinatorics, but a word with rotational symmetry is not automatically a cyclic number: the multiplication action and fixed-base digit interpretation remain indispensable. Periodicity is a neighbor because unit fractions repeat, yet a period can visit only a subset of rotations or map multipliers to nonrotational rearrangements. Group-theoretic language can describe the rotation action, but the node does not claim that the digit integers form a group under ordinary multiplication. The domain accent is precisely the compatibility between arithmetic multiplication and the rotation orbit.

Examples¶

Canonical¶

The decimal repetend of 1/7 is 142857. Multiplying by 1 through 6 generates every cyclic shift exactly once in a particular order, and multiplying by 7 yields 999999. The period six matches the multiplicative order of 10 modulo 7.^[1]

Mapped back: unit fraction 1/7 → six-digit repetend → consecutive multiplication → closed rotation orbit → return through modular period.

Applied / In Practice¶

A search program loops over primes p, computes the order of base b modulo p, and keeps full-period cases. It forms the fixed-width repetend of 1/p and verifies consecutive multiples against rotations. Leading zeros remain part of the string so candidates are not inconsistently accepted or rejected by integer rendering.

For the familiar base-ten block 142857, multiply with a fixed six-digit width. The products by successive allowed nonzero multipliers are checked against rotations of the same six symbols, and the return relation is recorded. A good audit does more than display the successful products: it identifies which rotation corresponds to each multiplier, verifies that no digit was dropped, and connects the six-place repetend to the order of ten modulo its denominator.^[1] Compare a different repeating fraction whose block has the same length but whose double is not a rotation. Both are periodic decimal expansions, yet only the first satisfies the multiplication–rotation coupling. Changing to another base requires a new block and new modular-order calculation; transliterating the decimal digits would preserve typography but not the abstraction.

Mapped back: base/prime → modular-order filter → fixed-width repetend → rotation test → cyclic-number roster.

Structural Tensions¶

Numeric value vs. digit representation. The property depends on how the value is written. Diagnostic: Is the base and width fixed?
Visual rotation vs. modular proof. Pattern recognition is immediate but incomplete. Diagnostic: Does multiplicative order certify the full cycle?
Leading zeros vs. integer notation. Dropping zeros changes the rotation orbit. Diagnostic: Which convention is applied consistently?
Partial multiples vs. consecutive completeness. Some numbers have rotated multiples out of order. Diagnostic: Are all required consecutive multipliers present?
Autonomous object vs. generic cycle. Cycle travels; numeral multiplication supplies the identity. Diagnostic: Does the closed orbit arise from consecutive multiples of one digit block?

Structural–Framed Character¶

Cyclic numbers are structural-leaning. Modular relationships are formal; base, digit string, and leading-zero conventions frame their presentation. They are evaluatively neutral and observer-independent after representation is fixed. They remain domain-specific because the property requires positional numeration and integer multiplication.

A recognition audit should be invariant under where the circular block is written. Choosing another rotation as the displayed start changes the integer notation but not the orbit, whereas changing digit order or shortening a repeated subblock changes the candidate. The audit should also distinguish the multiplier action from addition, concatenation, or reversal. These operations can generate attractive patterns without satisfying the same structure. When a program searches for examples, it should preserve base, width, denominator, and multiplier range in the output so a curator can reproduce the number-theoretic claim rather than trusting a visual animation of rotating digits. A further check compares the claimed multiplier range with the orbit length: duplicate rotations, an omitted multiplier, or an overflow beyond the fixed width can make a plausible digit display fail the definition. This audit is especially useful when leading zeros are part of the rotation but disappear under ordinary integer formatting.

Structural Core vs. Domain Accent¶

The skeleton is finite state → repeated operation → orbit through rotations → return. The accent is base-b digits, consecutive multiples, carries, repeating unit fractions, and primitive-root conditions. Removing those yields a generic cycle.

Cycle is the strict parent because rotations form a closed orbit under consecutive multiplication, eventually returning to the original digit arrangement. Permutation is related, but closure of the ordered orbit is the defining higher-level pattern.

The prospective workspace queue contains one strict upward edge to prime:cycle. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Cyclic Number Domain-specific

Parents (1) — more general patterns this builds on

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 (77^th 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

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Cyclically ordered group. An algebraic group equipped with a cyclic order relation.
Full reptend prime. A prime whose reciprocal has maximum period; it generates a cyclic number under suitable conventions.
Repunit. A numeral consisting only of repeated ones.
Cyclic permutation. The generic rotation operation rather than the special number.
Automorphic number. A number whose square ends in its own digits.

References¶

[1] David Wells, The Penguin Dictionary of Curious and Interesting Numbers, revised ed. (Penguin, 1997). registry ↩a ↩b ↩c ↩d