Keith number¶

A natural number whose base-b digits seed a k-step Fibonacci-like recurrence that later reproduces the number.

Version: v2 · 2026-09-19 · History
Domain-specific #: 5184
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Recreational Number Theory → Mathematics
Aliases: Repfigit 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.

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Grow-Back Numbers

Take the number 14. Start with its digits, 1 and 4. Add them to get 5, then add the last two numbers, 4 and 5, to get 9, then 5 and 9 to get 14. We landed right back on 14! Numbers that grow themselves back from their own digits like this are called Keith numbers.

Numbers That Rebuild Themselves

A Keith number is a number that shows up in a special list made from its own digits. Write its digits in order, then keep adding: each new number is the sum of the last few numbers, where 'few' means how many digits the original number has. For 14 (two digits): 1, 4, 5, 9, 14, so 14 is a Keith number. If the list jumps past the number without hitting it, it's not a Keith number. One-digit numbers are a special case that people have to agree on how to treat.

Self-Reproducing Digit Recurrence

A Keith number is a natural number n with k digits that reappears in the sequence seeded by its own digits, where each new term is the sum of the previous k terms. The digits must be used in order from most significant to least, and you stop once a term reaches or exceeds n. For example, 14 gives 1, 4, 5, 9, 14. The property depends on the base you write numbers in, so a number can be a Keith number in base 10 but not in another base. One-digit numbers trivially appear and need a convention about whether they count. It's like a Fibonacci-style sequence, but the seed and the number of terms summed come from the number itself.

A Keith number is a natural number N with k digits in a chosen base such that N appears in the k-term linear recurrence seeded by its own digits. The ordered digits, most significant first, give the initial k terms; each later term is the sum of the preceding k terms. Since the sequence increases, one generates terms until reaching or exceeding N; membership holds exactly when N is hit. The definition is base-dependent, because the seed digits are base-dependent. Conventions must be stated for one-digit numbers, which are trivial cases, and for how repeated cases are counted. Reordering the digits changes the recurrence and therefore the answer. A precise claim specifies the base, digit sequence, k, update rule, stopping rule, and the index at which N is reproduced.

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.
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) — outgoing typed relations

Keith number is a kind of Recurrence Prime

It remains its own entry because its identity is fixed by 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.

Hierarchy path (1) — routes to 1 parentless root

Keith number → is a kind of Recurrence

Neighborhood in Abstraction Space¶

Keith number sits in a crowded region of the domain-specific corpus (24^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Numeral Systems & Integer Sequences (22 abstractions)

Nearest neighbors

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