Wagstaff Prime¶
A prime number of the form (2^p + 1)/3 for an odd prime exponent p.
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.
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. Inclusion test: An integer is a Wagstaff prime exactly when some odd prime p gives the integer through (2^p+1)/3 and that resulting integer is prime. Exclusion test: A composite quotient, a quotient from a nonprime exponent, or a generalized Q(b,n) with b other than two is excluded from the base-two prime class. Nearest boundary: A Wagstaff-form probable prime is the closest near miss because it satisfies the formula and computational screens but lacks a completed primality warrant. Exit condition: The identity exits when the base, sign, divisor, exponent condition, or proven-prime condition changes. Common misclassifications: 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. Nearest named distinctions: Mersenne prime: Is the candidate 2^p−1, or the distinct quotient (2^p+1)/3? Probable prime: Has primality been proved, or has the candidate only passed necessary computational tests? Generalized Wagstaff prime: Is the base fixed at two with divisor three, or is an arbitrary base b used? Repunit prime: Is the class defined by repeated unit digits in a positive base, or by the Wagstaff quotient and its negative-base relation?
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¶
- Choose p and verify that it is odd and prime.
- Form the exact numerator 2^p+1 and divide by three.
- Check integrality and retain the exact quotient rather than a floating approximation.
- Establish the quotient's status as composite, probable prime, or proven prime.
- Grant Wagstaff-prime membership only after a primality warrant.
- When generalizing, replace the base and divisor explicitly and inspect algebraic factors before search.
- 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.
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
- Achilles Number — 0.92
- Sexy Primes — 0.90
- Power Residue Symbol — 0.88
- Dyadic Rational — 0.88
- Peters Polynomials — 0.87
Computed from structural-signature embeddings · 2026-10-08