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Wagstaff Prime

A prime number of the form (2^p + 1)/3 for an odd prime exponent p.

Version
v1 · 2026-09-28 · History
Domain-specific #
12834
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomain
Number Theory → Mathematics
Aliases
Wagstaff primes

Core Idea

A Wagstaff prime is defined by two simultaneous conditions. An odd prime exponent p is inserted into (2^p+1)/3, and the resulting integer must itself be prime. Thus p=3, 5, and 7 yield 3, 11, and 43. The formula generates candidates, not automatic members: a prime exponent does not guarantee a prime quotient.

The construction extends to Q(b,n)=(b^n+1)/(b+1) for other bases and odd exponents. Those generalized numbers can be represented as negative-base repunits, and some base forms force algebraic factorization. These connections organize neighboring families but do not loosen the base-two definition. Probable-prime status, conjectured infinitude, and cryptographic use are properties or research contexts, not substitutes for the membership test.

Structural Signature

Sig role-phrases:

  • odd prime exponent p — selects the exponent admitted by the base-two class It is essential parameter. Counterfactual: A composite or even exponent does not meet the named definition.
  • power of two 2^p — supplies the exponential term It is essential. Counterfactual: Changing the base produces a generalized Wagstaff number, not the base-two class.
  • increment by one — forms the numerator whose divisibility by three follows for odd p It is essential operation. Counterfactual: A minus-one numerator belongs to a different exponential-number family.
  • division by three — forms the integer candidate from the base-two numerator It is essential operation. Counterfactual: Leaving the numerator undivided yields another number class.
  • primality of the quotient — distinguishes a Wagstaff prime from a composite Wagstaff-form number It is essential membership test. Counterfactual: The formula and prime exponent are necessary but not sufficient.
  • generalized base and exponent — extends the construction to Q(b,n) while keeping it distinct from the core class It is variation boundary. Counterfactual: Importing generalized identities into the base-two definition changes the carrier.

What It Is Not

  • It is not every integer having the quotient form; the quotient must be prime.
  • It is not guaranteed by choosing an odd prime exponent.
  • It is not a Mersenne prime, whose defining form uses 2^p−1.
  • It is not every generalized Wagstaff prime with an arbitrary base.
  • Closest near-miss. A Wagstaff-form probable prime is the closest near miss because it satisfies the formula and computational screens but lacks a completed primality warrant.

Scope of Application

  • Prime-number classification. The quotient formula and primality define a named subclass.
  • Computational number theory. Candidate generation, probable-prime testing, and certification are tracked separately.
  • Exponential Diophantine structure. Factorizations constrain which bases and exponents can yield primes.
  • Repunit theory. Generalized values correspond to odd-length repunits in a negative base.
  • Mersenne-related conjectures. The class appears in the New Mersenne conjecture and neighboring investigations.

Clarity

State the exponent p, verify that p is an odd prime, compute the exact integer quotient, and distinguish proven prime, probable prime, and composite status. For generalized claims state base b and exponent n, identify any algebraic factorization, and avoid transferring conjectures or records from one base to another.

Manages Complexity

One formula isolates a sparse prime-number family and links exponent sequences, generalized repunits, factorization, conjectures, and computation. The compact notation hides two independent tests—admissible exponent and quotient primality—as well as the difference between base two and generalized bases and between probable and certified primes.

Abstract Reasoning

  1. Choose p and verify that it is odd and prime.
  2. Form the exact numerator 2^p+1 and divide by three.
  3. Check integrality and retain the exact quotient rather than a floating approximation.
  4. Establish the quotient's status as composite, probable prime, or proven prime.
  5. Grant Wagstaff-prime membership only after a primality warrant.
  6. When generalizing, replace the base and divisor explicitly and inspect algebraic factors before search.
  7. Keep empirical records and infinitude conjectures separate from the class definition.

Knowledge Transfer

The recognition rule transfers exactly among candidate computations for the base-two sequence: odd-prime exponent, fixed quotient form, then primality. It stops at other bases, minus-sign families, or probable-prime screens unless the generalized label and changed proof obligations are stated. The cargo is formula-plus-primality classification, not a metaphor for rarity.

Examples

Applied / In Practice

For p=5, (2^5+1)/3=11, and 11 is prime, so 11 is a Wagstaff prime.

Mapped back: exponent → 5, an odd prime; form → (32+1)/3; membership → quotient 11 is prime.

Applied / In Practice

Q(10,5)=(10^5+1)/11 belongs to a generalized base-ten family, not to the base-two Wagstaff-prime class even if the quotient is prime.

Mapped back: variation → base 10; boundary → generalized rather than core class.

Applied / In Practice

A number has the required base-two form for an odd prime exponent but is known only as a probable prime.

Mapped back: boundary → uncompleted primality status.

Structural Tensions

T1 — Compact Formula versus Hard Primality Certification. The expression defines candidates succinctly, while quotient size grows exponentially and primality remains a separate demanding property.

Diagnostic: Keep formula generation, probable-prime screening, and proof of primality as distinct statuses.

T2 — Base-Two Identity versus Generalized Family. Generalization reveals repunit and factorization structure but can blur which theorems apply to the named Wagstaff primes.

Diagnostic: State b and n explicitly and label results as core or generalized before transferring claims.

Structural–Framed Character

Formula membership and primality are formal structure; naming, computational record status, and conjectural context frame the research literature. The identity is fully reproducible within number theory but not substrate-independent in the sense required of a prime abstraction.

Structural Core vs. Domain Accent

The skeleton is a constrained exponential expression filtered by primality. Number theory supplies powers, divisibility, prime exponents, proof of primality, negative-base repunits, and algebraic factorization. Those commitments make Wagstaff Prime a specialist number class rather than a general pattern.

  • Approved root. The frozen graph leaves Wagstaff Prime unparented; generic prime and pattern nodes do not entail the exact quotient construction.

  • Related — generalized repunit and Mersenne-prime families. They expose algebraic neighbors while preserving different defining formulas.

Neighborhood in Abstraction Space

Wagstaff Prime sits in a crowded region of the domain-specific corpus (36th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Polynomial Rings & Rational Approximants (15 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

  • Mersenne prime. Tell: Is the candidate 2^p−1, or the distinct quotient (2^p+1)/3?
  • Probable prime. Tell: Has primality been proved, or has the candidate only passed necessary computational tests?
  • Generalized Wagstaff prime. Tell: Is the base fixed at two with divisor three, or is an arbitrary base b used?
  • Repunit prime. Tell: Is the class defined by repeated unit digits in a positive base, or by the Wagstaff quotient and its negative-base relation?

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Wagstaff_prime (revision 1366161998).
  • Preserved source candidate: https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf
  • Preserved source candidate: https://mathworld.wolfram.com/Repunit.html
  • Preserved source candidate: http://primes.utm.edu/top20/page.php?id=67
  • Preserved source candidate: http://primenumbers.net/Documents/TestNP.pdf
  • Preserved source candidate: https://www.mersenneforum.org/attachment.php?attachmentid=24152&d=1610368418
  • Preserved source candidate: https://web.archive.org/web/20210330125451/https://www.mersenneforum.org/attachment.php?attachmentid=24152&d=1610368418
  • Preserved source candidate: http://www.primenumbers.net/Henri/us/MersFermus.htm
  • Preserved source candidate: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.