Persistence of a number¶

The number of repeated applications of a specified digit operation needed for an integer to reach a fixed point, most commonly a single digit under repeated digit sums or products.

Version: v1 · 2026-09-08 · History
Domain-specific #: 6052
Origin domain: recreational number theory
Subdomain: specialized structures

Core Idea¶

Numerical persistence measures transient length under a digit-induced dynamical system. Each step rewrites the current integer as the sum or product of its digits in a fixed radix until the chosen operation no longer changes the state. 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.

The load-bearing residual is not the broad topic of recreational number theory. It is The number of repeated applications of a specified digit operation needed for an integer to reach a fixed point, most commonly a single digit under repeated digit sums or products.

Scope of Application¶

Persistence of a number belongs to recreational number theory and is useful where the analyst can specify a nonnegative integer, numeral radix, digitwise aggregation operation, iteration sequence, stopping set, persistence count and terminal digital root or product, then evaluate the radix, digit operation and stopping rule remain fixed and the count equals the number of transformations before the first terminal state. The scope is broad within that domain but bounded by the need for the radix, digit operation and stopping rule remain fixed and the count equals the number of transformations before the first terminal state. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the radix, digit operation and stopping rule remain fixed and the count equals the number of transformations before the first terminal state the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Persistence of a number can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

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 Persistence of a number. Persistence of a 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: a nonnegative integer, numeral radix, digitwise aggregation operation, iteration sequence, stopping set, persistence count and terminal digital root or product. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the radix, digit operation and stopping rule remain fixed and the count equals the number of transformations before the first terminal state independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of recreational number theory because they reuse a nonnegative integer, numeral radix, digitwise aggregation operation, iteration sequence, stopping set, persistence count and terminal digital root or product, Each step rewrites the current integer as the sum or product of its digits in a fixed radix until the chosen operation no longer changes the state., and type the carrier, state every parameter and convention in the definition, test that the radix, digit operation and stopping rule remain fixed and the count equals the number of transformations before the first terminal state, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Persistence of a number Domain-specific

Parents (1) — more general patterns this builds on

Persistence of a number is a kind of Iteration Prime

The proposed strict upward parent is prime:iteration.

Hierarchy path (1) — routes to 1 parentless root

Persistence of a number → Iteration

Neighborhood in Abstraction Space¶

Persistence of a number sits in a moderately populated region (47^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Digit Properties & Recreational Numbers (6 abstractions)

Nearest neighbors

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