Thabit number¶

An integer of the form 3·2^n−1 for a nonnegative integer n, historically linked through special primality conditions to constructions of amicable numbers.

Version: v1 · 2026-09-08 · History
Domain-specific #: 7108
Origin domain: number theory
Subdomain: integer sequences

Core Idea¶

The nth Thabit number is the integer obtained by multiplying 2^n by three and subtracting one. Exponential doubling produces a simple recurrence and binary pattern, while certain simultaneous primality conditions make selected Thabit expressions factors in amicable-number formulas. 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 number theory. It is named exponential integer sequence associated with an early amicable-number theorem. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that the number has exactly the form 3 times a power of two minus one for an integer n≥0 fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test.

Scope of Application¶

Thabit number belongs to number theory and is useful where the analyst can specify a nonnegative integer index n, power 2^n, integer T_n=3·2^n−1, binary representation, divisibility and primality, related Thabit–Ibn Qurra formulas, then evaluate the number has exactly the form 3 times a power of two minus one for an integer n≥0. The scope is broad within that domain but bounded by the need for the number has exactly the form 3 times a power of two minus one for an integer n≥0. 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 number has exactly the form 3 times a power of two minus one for an integer n≥0 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 Thabit 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 Thabit number. Thabit 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 index n, power 2^n, integer T_n=3·2^n−1, binary representation, divisibility and primality, related Thabit–Ibn Qurra formulas. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the number has exactly the form 3 times a power of two minus one for an integer n≥0 independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse a nonnegative integer index n, power 2^n, integer T_n=3·2^n−1, binary representation, divisibility and primality, related Thabit–Ibn Qurra formulas, Exponential doubling produces a simple recurrence and binary pattern, while certain simultaneous primality conditions make selected Thabit expressions factors in amicable-number formulas., and type the carrier, state every parameter and convention in the definition, test that the number has exactly the form 3 times a power of two minus one for an integer n≥0, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Thabit number Domain-specific

Parents (1) — more general patterns this builds on

Thabit number is a kind of Recursion Prime

The proposed strict upward parent is prime:recursion.

Hierarchy path (1) — routes to 1 parentless root

Thabit number → Recursion

Neighborhood in Abstraction Space¶

Thabit number sits in a moderately populated region (58^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Algebraic Number Theory & Reciprocity (28 abstractions)

Nearest neighbors

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