Fermat number

Generate the integer sequence F_n = 2⁽²n) + 1, whose product recurrence makes distinct terms pairwise coprime and whose rare prime members connect to constructible polygons.

Version

v1 · 2026-08-30 · History

Domain-specific #

1829

Origin domain

number theory

Subdomain

special integer sequences

Core Idea

The nth Fermat number is \(F_n=2^{2^n}+1\) for a nonnegative integer n. Double-exponential growth and the identity \(F_n=2+\prod_{i<n}F_i\) yield recurrences, pairwise coprimality, and strong restrictions on prime divisors. 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 the exact double-exponential sequence and its product identities, distinct from arbitrary numbers of form 2^m+1 or from the subset of Fermat primes.

Scope of Application

Fermat number belongs to number theory and is useful where the analyst can specify nonnegative integer indices and the positive integers they generate, then evaluate membership means exact equality to 2 raised to a power of two plus one for a nonnegative integer index. The scope is broad within that domain but bounded by the need for membership means exact equality to 2 raised to a power of two plus one for a nonnegative integer index. Current primality and factorization claims are time-sensitive and should be verified before canonical use; the stable entry emphasizes proved structural facts.

Clarity

The abstraction clarifies a crowded vocabulary by making membership means exact equality to 2 raised to a power of two plus one for a nonnegative integer index 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 Fermat's name appears in many unrelated theorems and number classes, so the defining formula is essential.

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 Fermat number. Fermat 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

1. Identify the carrier. State what the elements, states, objects, or observations are: nonnegative integer indices and the positive integers they generate. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express membership means exact equality to 2 raised to a power of two plus one for a nonnegative integer index independently of one notation or implementation. This step prevents the canonical example from becoming the definition.

Knowledge Transfer

Knowledge transfers strongly among subfields of number theory because they reuse nonnegative integer indices and the positive integers they generate, Double-exponential growth and the identity \(F_n=2+\prod_{i<n}F_i\) yield recurrences, pairwise coprimality, and strong restrictions on prime divisors., and recover or verify the index, compute with exact arithmetic, distinguish probable-prime evidence from proof, and validate claimed factors by divisibility and primality certificates. A theorem, diagnostic, or modeling warning can travel when those roles remain literal.

Relationships to Other Abstractions

Current abstraction Fermat number Domain-specific

Parents (1) — more general patterns this builds on

• Fermat number is a kind of Constraint Prime

  The proposed strict upward parent is prime:constraint.

Hierarchy path (1) — routes to 1 parentless root

• Fermat number → Constraint

Neighborhood in Abstraction Space

Fermat number sits in a moderately populated region (42^nd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Integer Functions & Special Numbers (13 abstractions)

Nearest neighbors

• Unusual number — 0.90
• Hyperperfect number — 0.90
• Dedekind psi function — 0.89
• Integer factorization — 0.89
• Wilson quotient — 0.89

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