Wieferich-type Primes¶
Prime classes singled out when a theorem-backed congruence holds one prime-power modulus more strongly than it must.
Core Idea¶
Wieferich-type primes are an editorial family of exceptional prime classes. For each subtype, first fix a prime-indexed integer expression Eₚ and eligible primes. A theorem guarantees that pᵏ divides Eₚ; a prime enters the exceptional class when pᵏ⁺¹ also divides that same expression. The examples share this one-power question, not one formula or proof.[ref-d73037a8e101][ref-36c7b3a7c641][^ref-c0d65b48b2af]
Scope of Application¶
This is a number-theory comparison among theorem-backed congruence classes. Its parameters must be stated separately for each subtype.
- Wolstenholme primes: for p ≥ 5, a binomial expression has the p³ baseline and the class tests p⁴.[^ref-d73037a8e101]
- Base-2 Wieferich primes: for odd p, 2ᵖ⁻¹ − 1 has the p baseline and the class tests p².[^ref-36c7b3a7c641]
- Wilson primes: for prime p, (p − 1)! + 1 has the p baseline and the class tests p².[^ref-c0d65b48b2af]
- Wall–Sun–Sun condition: for p other than 2 or 5, a Fibonacci expression has a p baseline and a formal p² test; the inspected 2025 source reports no known positive.[^ref-6a1c3036fb7f]
Clarity¶
The baseline theorem covers every eligible prime. Only the extra-power result determines exceptional membership; the normalized quotient Eₚ/pᵏ is an equivalent way to test it, not a separate requirement. A bounded search result is evidence for its stated range, while estimates about how often a class occurs remain heuristic.[ref-d73037a8e101][ref-36c7b3a7c641][^ref-c0d65b48b2af]
Manages Complexity¶
Track four items: Eₚ, eligible primes, guaranteed exponent k, and tested exponent k + 1. This small ledger allows comparison among binomial, exponential, and factorial cases without borrowing one case's theorem or search result for another. The expression and range cannot be omitted without changing the class.[ref-d73037a8e101][ref-36c7b3a7c641][^ref-c0d65b48b2af]
Abstract Reasoning¶
Given a proposed subtype, establish a universal lower-power congruence on its declared prime range, then test the next power of the same p in the same expression. If the theorem already guarantees the higher power, or if only isolated primes satisfy the lower statement, the proposal is not this one-power exceptional-class pattern. A search count does not prove infinitude.[ref-d73037a8e101][ref-36c7b3a7c641][^ref-c0d65b48b2af]
Knowledge Transfer¶
The question transfers among the inspected number-theory settings; their proofs and distributions do not. Set and Membership describes each resulting collection, and Predicate describes its indispensable yes-or-no divisibility test. The class is the test's extension, not a deliberate human sorting process. Evidence outside number theory would be needed before treating the one-power pattern itself as a cross-domain Prime.
Example¶
Wolstenholme's theorem guarantees p³ divisibility of C(2p − 1, p − 1) − 1 for p ≥ 5. For p = 16843 the expression is also divisible by p⁴, so that prime belongs to the Wolstenholme class. Booker and coauthors prove that 16843 and 2124679 are the only members below 10¹¹; the bound is part of the result.[^ref-d73037a8e101]
Relationships to Other Abstractions¶
Current abstraction Wieferich-type Primes Domain-specific
Parents (2) — more general patterns this builds on
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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.
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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
Neighborhood in Abstraction Space¶
Wieferich-type Primes sits in a sparse region of the domain-specific corpus (94th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2721 abstractions)
Nearest neighbors
- Regular prime — 0.80
- Knuth–Eve Algorithm — 0.77
- Fermat's Little Theorem — 0.77
- Hensel's Lemma — 0.77
- Wall–Sun–Sun prime — 0.77
Computed from structural-signature embeddings · 2026-10-17
Not to Be Confused With¶
- A quotient defined for every eligible prime: only its zero residue marks the exceptional class.[ref-36c7b3a7c641][ref-c0d65b48b2af]
- A proven 1/p frequency or infinitude law: the inspected frequency arguments are heuristic.[ref-36c7b3a7c641][ref-c0d65b48b2af]
- An observed Wall–Sun–Sun prime: the cited 2025 paper reports none known.[^ref-6a1c3036fb7f]
References¶
[^ref-d73037a8e101]: 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.
[^ref-36c7b3a7c641]: 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.
[^ref-c0d65b48b2af]: 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.
[^ref-6a1c3036fb7f]: 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.