Wall–Sun–Sun prime¶

A conjectural prime p for which p² divides the Fibonacci number indexed by p's Pisano period, equivalently a Fibonacci–Wieferich prime under standard formulations.

Version: v1 · 2026-09-08 · History
Domain-specific #: 7451
Origin domain: number theory
Subdomain: fibonacci congruences

Core Idea¶

A Wall–Sun–Sun prime is a prime exhibiting exceptional lifting of a Fibonacci divisibility congruence from p to p squared. Fibonacci residues repeat modulo p, ensuring p divides the term at a related period; the exceptional prime would make the same term vanish to at least second p-adic order. 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¶

Wall–Sun–Sun prime belongs to number theory and is useful where the analyst can specify a prime p, Fibonacci sequence modulo p and p², Pisano period π(p), divisibility of F_{π(p)}, p-adic valuation, equivalent rank-of-apparition formulation and computational search bound, then evaluate p is prime and the exact period or equivalent Fibonacci quotient congruence satisfies the squared-divisibility condition under the stated exceptional cases. The scope is broad within that domain but bounded by the need for p is prime and the exact period or equivalent Fibonacci quotient congruence satisfies the squared-divisibility condition under the stated exceptional cases. The class is conjectured to be nonempty, but no member is known; the entry records the definition rather than asserting existence.

Clarity¶

The abstraction clarifies a crowded vocabulary by making p is prime and the exact period or equivalent Fibonacci quotient congruence satisfies the squared-divisibility condition under the stated exceptional cases 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 Wall–Sun–Sun prime 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 Wall–Sun–Sun prime. Wall–Sun–Sun prime 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 prime p, Fibonacci sequence modulo p and p², Pisano period π(p), divisibility of F_{π(p)}, p-adic valuation, equivalent rank-of-apparition formulation and computational search bound. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express p is prime and the exact period or equivalent Fibonacci quotient congruence satisfies the squared-divisibility condition under the stated exceptional cases independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse a prime p, Fibonacci sequence modulo p and p², Pisano period π(p), divisibility of F_{π(p)}, p-adic valuation, equivalent rank-of-apparition formulation and computational search bound, Fibonacci residues repeat modulo p, ensuring p divides the term at a related period; the exceptional prime would make the same term vanish to at least second p-adic order., and type the carrier, state every parameter and convention in the definition, test that p is prime and the exact period or equivalent Fibonacci quotient congruence satisfies the squared-divisibility condition under the stated exceptional cases, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Wall–Sun–Sun prime Domain-specific

Parents (1) — more general patterns this builds on

Wall–Sun–Sun prime is a kind of Constraint Prime

The proposed strict upward parent is prime:constraint.

Hierarchy path (1) — routes to 1 parentless root

Wall–Sun–Sun prime → Constraint

Neighborhood in Abstraction Space¶

Wall–Sun–Sun prime sits in a moderately populated region (44^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