Wieferich-type Primes¶

Prime classes singled out when a theorem-backed congruence holds one prime-power modulus more strongly than it must.

Version: v1 · 2026-10-17 · History
Domain-specific #: 14183
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Number Theory, Prime Congruences → Mathematics
Aliases: Higher-power congruence primes

Core Idea¶

Wieferich-type primes, in this entry's editorial sense, are exceptional prime classes formed by asking one precise question: a theorem already makes an integer expression divisible by a power of a prime; for which primes is it divisible by one more power? Fix a prime-indexed integer expression Eₚ, an eligible prime range, and an exponent k for which the theorem guarantees pᵏ divides Eₚ. A prime belongs to the resulting subtype when pᵏ⁺¹ also divides that same Eₚ. The expression, range, theorem, and k belong to the question; there is no single unnamed congruence that all these classes share.^[1]^[2]^[3]

The selected Wolstenholme-prime case strengthens a binomial congruence from modulus p³ to p⁴. A base-2 Wieferich prime instead strengthens a Fermat congruence from p to p²; a Wilson prime does the same for a factorial congruence. Those three source-backed cases justify comparing the form of the test, not merging their distinct theorems or asserting a universal frequency law. “Wieferich-type” is a useful umbrella here, not an assertion that all cited authors use one established family name.^[1]^[2]^[3]

Structural Signature¶

Fixed expression and prime range. Name Eₚ and every parameter, including a fixed exponential base or Fibonacci index and any excluded small primes. Otherwise a change of formula silently changes the set being classified.^[1]^[2]^[4]
Theorem-backed lower power. State why pᵏ divides Eₚ throughout that range. In the principal cases k is 3 for Wolstenholme's binomial expression and 1 for the base-2 Fermat and Wilson expressions. A congruence at a single convenient prime is not this role.^[1]^[2]^[3]
One-power test and class verdict. Test whether pᵏ⁺¹ divides the same Eₚ for a particular p. Success defines membership in the corresponding exceptional-prime class; failure leaves that prime at the theorem-guaranteed level.^[1]^[2]^[3]

Because the lower divisibility is known, Eₚ/pᵏ is an integer. Its residue modulo p is zero if and only if the extra power divides Eₚ. A Fermat quotient or Wilson quotient is a subtype-specific version of this convenient diagnostic. Computing the quotient is not a fourth necessary stage: one can test the higher modulus directly.^[2]^[3] A short ledger for a proposed new subtype is therefore: expression and eligible primes → universal pᵏ fact → pᵏ⁺¹ test → class membership.

What It Is Not¶

This is not the mere fact that a congruence holds modulo p. That is the baseline theorem; the exceptional class is defined by the additional power. Nor is the quotient itself the rare prime class. Wilson's quotient, for example, is defined for every prime, while Wilson primes are exactly those whose quotient is zero modulo p.^[3]

It is not a theorem that such primes have probability 1/p, grow in number like log log x, or exist infinitely often. Crandall and coauthors model quotient residues as roughly uniform and tests at different primes as independent; Costa and coauthors discuss a similar Wilson-prime heuristic. Both are ways to estimate scarcity, not proofs or roles in the definition.^[2]^[3] A list of discovered primes is evidence about a particular bounded search, not the identity of the class.

Scope of Application¶

The literal home of this abstraction is number theory: prime-indexed integer expressions, congruences modulo powers of p, and theorem-relative exceptional subclasses. The inspected positive settings differ in mathematical carrier: a binomial coefficient, a fixed-base exponential, and a factorial product. The common question is legitimate across those carriers because each has its own proved lower-power statement and its own extra-power test.^[1]^[2]^[3]

A Fibonacci version can be defined too. For primes other than 2 and 5, set ε = (p/5), the Legendre symbol with value +1 or −1, and let F_{p−ε} be the Fibonacci number at the integer index p − ε. Bragman and Rowland state that p divides F_{p−ε}; the Wall–Sun–Sun condition asks whether p² also divides it. Their 2025 paper reports no known qualifying prime. It is a formal subtype and boundary comparison, not a fourth observed positive or evidence that every such class has a member.^[4]

Clarity¶

The extra-power question separates three statements often compressed into “special prime.” First, a theorem may cover every eligible p at its baseline modulus. Second, the normalized quotient exists for every such p. Third, only a zero quotient residue marks the strengthened congruence. In the base-2 case p = 3 illustrates the gap: 2² − 1 = 3 is divisible by 3 but not by 9, so it satisfies Fermat's baseline and fails the Wieferich test. The arithmetic follows directly from the definition.^[2]

The same distinction prevents a historical search result from becoming a timeless definition. Booker and coauthors prove that the Wolstenholme primes below 10¹¹ are 16843 and 2124679; the bound is part of what their theorem establishes. Costa and coauthors report no new Wilson prime below 2 × 10¹³ in their 2012 search. Neither sentence establishes a complete global census for all later work.^[1]^[3]

Manages Complexity¶

Different theorems, formulas, search algorithms, and named quotients can obscure the shared decision structure. Keeping four parameters visible—Eₚ, eligible p, baseline k, and tested k + 1—compresses that variety into a small comparison without discarding the conditions that make each subtype meaningful. For Wolstenholme primes Eₚ is binomial and k = 3; for base-2 Wieferich and Wilson primes k = 1 but the expressions and proofs differ.^[1]^[2]^[3]

This also keeps definition, proof, search, and prediction apart. The baseline theorem makes the quotient integral. A computation or theorem then resolves membership for a particular range. A heuristic about how often zero residues occur may motivate a search, but a successful or unsuccessful finite search neither proves nor refutes that heuristic for all primes.^[1]^[2]^[3]

Abstract Reasoning¶

Given another proposed class, first ask whether a lower congruence is actually universal on a stated prime range. If so, subtract its fixed residue to form Eₚ and identify k. Only then ask whether pᵏ⁺¹ divides Eₚ or, equivalently, whether Eₚ/pᵏ vanishes modulo p. This procedure distinguishes a true one-power exceptional subclass from an arbitrary rare-looking congruence or a theorem that already guarantees the higher modulus.^[1]^[2]^[3]

The procedure does not transfer the mathematical proof. Wolstenholme's binomial theorem does not prove Fermat's congruence; Wilson's factorial test needs its own baseline. Likewise, observing a few positives in one subtype cannot determine how many positives another subtype has. The transferable reasoning is the theorem-relative classification test.^[1]^[2]^[3]

Knowledge Transfer¶

Within number theory, the Wolstenholme case teaches a question that can be put to Fermat's and Wilson's theorems: what does the next p-adic digit do once the compulsory digits are fixed? The answer must be computed or proved afresh. In Crandall and coauthors' terminology the Fermat quotient supplies the next-digit test for base 2; in Costa and coauthors' treatment the Wilson quotient does so for the factorial. Their resemblance does not make the underlying expressions interchangeable.^[2]^[3]

The Fibonacci Wall–Sun–Sun condition shows the limit of transfer. Its theorem-backed baseline and formal p² test fit the schema, but the cited 2025 work knows no positive example. One can transfer the question and its quotient logic while honestly leaving the existence of a qualifying prime unresolved. Carrying the 1/p heuristic or a discovered-prime count into that new setting would require a separate argument.^[4]^[2]

Examples¶

Selected binomial case: Wolstenholme prime¶

For p ≥ 5, Wolstenholme's theorem makes Eₚ = C(2p − 1, p − 1) − 1 divisible by p³. A Wolstenholme prime makes the same Eₚ divisible by p⁴. Booker and coauthors identify 16843 and 2124679 as the only such primes below 10¹¹ in their proved search range. The first is a concrete positive, and the second is another prime within this same binomial setting rather than a distinct setting.^[1]

Mapped back: the expression and range are the binomial difference for p ≥ 5; the baseline is p³ divisibility; the one-power verdict is p⁴ divisibility, satisfied by 16843. The quotient Eₚ/p³ mod p is an equivalent diagnostic. The harmonic and Bernoulli characterizations in the cited paper are case-specific equivalences, not required formulas for the other examples.^[1]

Unlike exponential case: base-2 Wieferich prime¶

For odd p, Fermat's little theorem makes Eₚ = 2ᵖ⁻¹ − 1 divisible by p. A base-2 Wieferich prime makes it divisible by p². Crandall and coauthors report 1093 and 3511 as known positives in their 1997 search; 1093 supplies the unlike positive here. Their Fermat quotient (2ᵖ⁻¹ − 1)/p is zero modulo p exactly when the stronger congruence holds.^[2]

Mapped back: the expression and range are the fixed-base exponential difference for odd p; the baseline is p divisibility; the one-power verdict is p² divisibility, satisfied by 1093. The carrier is exponentiation and k is 1, unlike the binomial k = 3 case. A hypothetical first-case Fermat's Last Theorem counterexample with the paper's conditions would imply this congruence; the implication does not run backward.^[2]

Third setting: Wilson prime¶

Wilson's theorem makes Eₚ = (p − 1)! + 1 divisible by p for prime p. A Wilson prime makes it divisible by p², equivalently giving Wilson quotient Eₚ/p residue zero modulo p. Costa and coauthors list 5, 13, and 563 as known positives in 2012 and found no new Wilson prime below 2 × 10¹³. The dated search bound is evidence about that work, not an all-time count.^[3]

Mapped back: the expression and range are the factorial-plus-one for prime p; the baseline is Wilson's theorem modulo p; the one-power verdict is modulo p², satisfied by p = 5. This factorial setting confirms the shared schema without borrowing either the binomial or exponential proof.^[3]

Structural Tensions¶

No all-instance contest between opposed forces or goals defines these prime classes. A theorem-guaranteed lower modulus and an occasionally successful higher modulus are two levels of the same arithmetic test, not a trade-off that an actor balances. Likewise, a computational search's cost and a class's possible scarcity are issues in studying the class, not constitutive pressures inside it.

Crandall and coauthors and Costa and coauthors explicitly present frequency estimates as heuristics.^[2]^[3] A heuristic can suggest where and how long to search, while a theorem or checked computation establishes a bounded result. Treating the former as a proof would erase that evidential distinction. It is a contrast between kinds of knowledge, not a fabricated structural tension or a claim that the heuristic has been confirmed for every subtype.

Structural–Framed Character¶

Evaluative weight: a stronger congruence is mathematically notable and may matter to other results, but whether p belongs to the class is a fixed arithmetic truth once Eₚ and k are chosen. Human-practice dependence: mathematicians choose which theorem and expression to study and design searches; their choice does not cause the divisibility. Institutional origin: the individual names and recorded searches belong to a history of number theory, while the extra-power condition is not an institutional rule.

Vocabulary travel: “Wieferich-type” travels as a comparison label among prime-congruence subtypes; it does not license the claim that their proofs, distributions, or applications travel together. Import versus recognition: importing the test means specifying a new baseline theorem and next-power question; recognizing an instance means establishing its divisibility, not noticing that another rare prime looks similar. The known Wolstenholme, Wieferich, and Wilson positives make the comparison concrete without turning an unobserved Wall–Sun–Sun class into a discovered case.

Its character: a formal, theorem-relative category within number theory, structural in its repeatable one-power decision rule and framed by the chosen arithmetic expression and named subtype.

Structural Core vs. Domain Accent¶

The core is the three-role relation: fix Eₚ and eligible primes, prove pᵏ divisibility for them, and classify a particular p by pᵏ⁺¹ divisibility. The accents are substantial: the binomial, exponential, factorial, or Fibonacci expression; the relevant theorem and k; the named quotient; and any application or dated census. Remove those parameters and there is no testable subclass. Keep the parameters fixed and the same formal question can be asked across the inspected number-theoretic settings.^[1]^[2]^[3]^[4]

A future Prime for the one-power theorem-relative pattern itself would need source-backed cases outside number theory that preserve an independently guaranteed baseline and an exact one-level exceptional strengthening, rather than metaphorically calling an unusual event a “higher-power exception.” The present evidence is wholly arithmetic. The named family therefore remains domain-specific; its portability among formulas does not establish a cross-domain mechanism. Its existing generic parents describe the collection and its internal truth-valued membership test, not a cross-domain version of this arithmetic pattern.

This entry is part of Predicate and is a kind of Set and Membership.

The live Prime Set and Membership is the genus: each named subtype is a collection of individual primes defined by an exact inclusion rule. The live Prime Predicate is a distinct internal constituent: after the expression, eligible range, baseline theorem and exponent are fixed, the one-power divisibility test returns true or false for each prime. The class is the test's extension, not the truth-valued function itself. Predicate's Relation ancestor applies in its unary sense; no binary relational law is implied.

The live Prime Classification names a deliberate practice of assigning things to categories by explicit criteria. A mathematician may use a quotient-zero criterion to classify primes, but the arithmetic property of a prime does not depend on that practice, and Classification's human sorting work is not the genus of this entry. The live Prime Constraint has a feasible-set and binding/slack structure that the one-power congruence does not require. These are reasoning neighbors, not typed parents. The live domain-specific Wilson Quotient is a diagnostic defined for all primes.^[3] Wall–Sun–Sun Prime is a narrower Fibonacci test with no known positive in the inspected 2025 source.^[4] The latter is a narrower class, not a parent of this umbrella family.

Relationships to Other Abstractions¶

Current abstraction Wieferich-type Primes Domain-specific

Parents (2) — more general patterns this builds on

Wieferich-type Primes is a kind of Set and Membership Prime

Each subtype is a collection of individual primes defined by an exact theorem-relative one-power membership rule.
Wieferich-type Primes is part of Predicate Prime

The one-power yes-or-no predicate is an indispensable defining component inside each exceptional-prime class.

Hierarchy paths (2) — routes to 2 parentless roots

Wieferich-type Primes → Set and Membership

Show alternative path (1)

Neighborhood in Abstraction Space¶

Wieferich-type Primes sits in a sparse region of the domain-specific corpus (94^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2721 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-17

Not to Be Confused With¶

The baseline theorem: its lower-power statement covers every eligible prime, while the additional power selects a subtype.^[1]^[2]
A quotient: Fermat and Wilson quotients exist at primes failing the extra-power test; only residue zero identifies the corresponding exceptional class.^[2]^[3]
A proven distribution law: the 1/p and log-log statements in the inspected papers are heuristic, not proofs of infinitude or exact density.^[2]^[3]
Wall–Sun–Sun nonexistence: no example known in the inspected 2025 article does not mean none exists.^[4]
A first-case FLT solution: the cited classical implication makes a hypothetical solution force the Wieferich congruence; a Wieferich prime does not supply a solution.^[2]

References¶

[1] Andrew R. Booker, Shehzad Hathi, Michael J. Mossinghoff, and Timothy S. Trudgian, “Wolstenholme and Vandiver Primes,” arXiv:2101.11157v2 (2022), https://arxiv.org/html/2101.11157 . Full original author manuscript inspected: Introduction, Wolstenholme baseline congruences and equation (1.1), and Theorem 1.1 for the bounded search below 10¹¹. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o

[2] Richard Crandall, Karl Dilcher, and Carl Pomerance, “A Search for Wieferich and Wilson Primes,” Mathematics of Computation 66, no. 217 (1997): 433–449, https://math.dartmouth.edu/~carlp/PDF/paper111.pdf . Full original author-hosted PDF inspected: printed p. 433 for the Fermat and Wilson definitions, quotient, conditional FLT connection, and dated search results; printed p. 446, §3, for the explicitly heuristic 1/p and log-log reasoning. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w

[3] Edgar Costa, Robert Gerbicz, and David Harvey, “A Search for Wilson Primes,” arXiv:1209.3436v3 (2012), https://arxiv.org/html/1209.3436v3 . Full original author manuscript inspected: §1 for Wilson's theorem, quotient and known examples, the no-new-result below 2 × 10¹³, and the uniform-residue heuristic. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u

[4] Nicholas Bragman and Eric Rowland, “Limiting Density of the Fibonacci Sequence Modulo Powers of a Prime,” Research in Number Theory 11, article 88 (2025), https://link.springer.com/article/10.1007/s40993-025-00667-1 . Full original publisher article inspected: §1 for the integer index p − ε, where ε = (p/5) is the Legendre symbol, and baseline p-divisibility; §3, Theorem 10 and following paragraph, for the Wall–Sun–Sun condition, p = 2 and 5 exclusions, and the 2025 no-known-example qualification. registry ↩a ↩b ↩c ↩d ↩e ↩f