Keith number¶
A natural number whose base-b digits seed a k-step Fibonacci-like recurrence that later reproduces the number.
Core Idea¶
The property depends on numeral base, repeated numbers and trivial one-digit cases require convention and the recurrence must preserve digit order from most significant to least. The k digits initialize a sequence, every subsequent term sums the preceding k terms and membership is tested until terms reach or exceed the original number. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Keith number belongs to recreational number theory and is useful where the analyst can specify the typed recreational number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the natural number and base, ordered digit sequence and digit count, initial recurrence terms, k-term update rule, stopping rule, reproduced term index and convention for one-digit or repeated cases are explicit. The scope is broad within that domain but bounded by the need for the natural number and base, ordered digit sequence and digit count, initial recurrence terms, k-term update rule, stopping rule, reproduced term index and convention for one-digit or repeated cases are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the natural number and base, ordered digit sequence and digit count, initial recurrence terms, k-term update rule, stopping rule, reproduced term index and convention for one-digit or repeated cases are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Keith number. Keith number compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed recreational number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the natural number and base, ordered digit sequence and digit count, initial recurrence terms, k-term update rule, stopping rule, reproduced term index and convention for one-digit or repeated cases are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of recreational number theory because they reuse the typed recreational number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, The k digits initialize a sequence, every subsequent term sums the preceding k terms and membership is tested until terms reach or exceed the original number., and type the carrier, state every parameter and convention in the definition, test that the natural number and base, ordered digit sequence and digit count, initial recurrence terms, k-term update rule, stopping rule, reproduced term index and convention for one-digit or repeated cases are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Keith number Domain-specific
Parents (1) — more general patterns this builds on
-
Keith number is a kind of Recurrence Prime
The proposed strict upward parent is
prime:recurrence.
Hierarchy path (1) — routes to 1 parentless root
- Keith number → Recurrence
Neighborhood in Abstraction Space¶
Keith number sits in a crowded region of the domain-specific corpus (38th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Digit Properties & Recreational Numbers (6 abstractions)
Nearest neighbors
- Dudeney number — 0.93
- Amenable number — 0.92
- Taxicab number — 0.90
- 15 puzzle — 0.89
- Nonhypotenuse number — 0.89
Computed from structural-signature embeddings · 2026-09-08