Fortunate number¶

For each positive index n, select the least integer m greater than one for which the nth primorial plus m is prime, producing the sequence governed by Fortune's still-open primality conjecture.

Version: v2 · 2026-08-30 · History
Domain-specific #: 1869
Origin domain: number theory
Subdomain: prime gaps near primorials

Core Idea¶

The nth Fortunate number is \(F_n=\min\{m>1:p_n\#+m\text{ is prime}\}\).^[1] The primorial is divisible by every prime at most \(p_n\), so many small offsets are excluded by divisibility; the ordered search stops at the first offset producing a prime, and that least offset is the sequence value.

Its autonomous residual is the primorial-indexed least positive prime-producing offset, not all prime gaps, primorial primes, or the set of distinct values after duplicates are removed. The identity fails when one is admitted as an offset, an arbitrary prime above the primorial is chosen instead of the nearest eligible one, n indexes integers rather than primes, duplicates are silently removed, or conjectured primality is stated as theorem.

Recognition requires an analyst to fix indexing and primorial notation, calculate or certify the base, test offsets in increasing order beginning above one, prove primality of the selected sum, and retain minimality rather than recording an arbitrary later prime. Once established, it supports studying prime gaps immediately above primorials, examining Fortune's conjecture, comparing nearby-prime sequences, and deriving divisibility restrictions on any possible composite value without turning those uses into the definition.

Structural Signature¶

Carrier: a positive index \(n\), the first \(n\) primes, their primorial \(p_n\#\), and the ordered integers \(m>1\)
Inputs or antecedent state: prime enumeration, primorial multiplication, primality of successive offsets, the usual order on positive integers, and a minimality requirement
Constitutive operation: The primorial is divisible by every prime at most \(p_n\), so many small offsets are excluded by divisibility; the ordered search stops at the first offset producing a prime, and that least offset is the sequence value
Invariant: the base is exactly the nth primorial, the offset is strictly greater than one, the sum is prime, and every smaller admissible offset fails the primality predicate
Recognition test: fix indexing and primorial notation, calculate or certify the base, test offsets in increasing order beginning above one, prove primality of the selected sum, and retain minimality rather than recording an arbitrary later prime
Output or consequence: studying prime gaps immediately above primorials, examining Fortune's conjecture, comparing nearby-prime sequences, and deriving divisibility restrictions on any possible composite value
Failure boundary: one is admitted as an offset, an arbitrary prime above the primorial is chosen instead of the nearest eligible one, n indexes integers rather than primes, duplicates are silently removed, or conjectured primality is stated as theorem

What It Is Not¶

It is not the whole field of number theory; many objects in that field do not satisfy its constitutive rule.
It is not its canonical example. For \(n=7\), \(p_7\#=510510\), and the least \(m>1\) making the sum prime is 19, because smaller offsets fail and \(510529\) is prime. That is an instance, not a definition.
It is not Sparsely Totient Number. Sparsely totient numbers are records for inverse totient behavior; Fortunate numbers are least prime-producing offsets from primorials and use a different function, order, and witness obligation.
It is not an unrestricted metaphor. Fortunate numbers can repeat as n varies, and 'Fortunate primes' sometimes names either the same conjecturally prime sequence or the distinct values sorted without duplicates

Scope of Application¶

Fortunate number applies when the analyst can specify a positive index \(n\), the first \(n\) primes, their primorial \(p_n\#\), and the ordered integers \(m>1\) and establish that the base is exactly the nth primorial, the offset is strictly greater than one, the sum is prime, and every smaller admissible offset fails the primality predicate. The entry states the exact integer sequence and known elementary consequences; computation is evidence, not proof that every term is prime.^[2]

Recognition. fix indexing and primorial notation, calculate or certify the base, test offsets in increasing order beginning above one, prove primality of the selected sum, and retain minimality rather than recording an arbitrary later prime
Comparison. Compare legitimate instances through index origin, primorial convention, offset lower bound, ordering, primality certification, duplicate retention, nearby-prime direction, conjecture status, and computational range.
Boundary. Fortunate numbers can repeat as n varies, and 'Fortunate primes' sometimes names either the same conjecturally prime sequence or the distinct values sorted without duplicates
Use. Preserve every assumption when using the identity for studying prime gaps immediately above primorials, examining Fortune's conjecture, comparing nearby-prime sequences, and deriving divisibility restrictions on any possible composite value.

Clarity¶

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because fortunate number is sometimes written as if primality were definitional, but the value is defined by the primality of the shifted primorial and its own primality remains conjectural. The disciplined statement is that the object counts as Fortunate number exactly when the base is exactly the nth primorial, the offset is strictly greater than one, the sum is prime, and every smaller admissible offset fails the primality predicate

Identity and measurement remain separate. Large-index calculations require verified big-integer multiplication, primality certification, and an auditable minimality search; a probable-prime test alone weakens the claim. Approximation or noisy evidence may weaken a classification without changing its definition.

Manages Complexity¶

The abstraction compresses indexed sequence with duplicates, distinct sorted values, upper and lower nearby-prime offsets, primorial-prime special cases, and computational extensions into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Compression can hide assumptions. A responsible use therefore declares index origin, primorial convention, offset lower bound, ordering, primality certification, duplicate retention, nearby-prime direction, conjecture status, and computational range and returns to the full diagnostic whenever a convention or boundary case changes.

Abstract Reasoning¶

Type the carrier. Establish a positive index \(n\), the first \(n\) primes, their primorial \(p_n\#\), and the ordered integers \(m>1\) and reject examples from a different problem.
Lock the rule. Express that the base is exactly the nth primorial, the offset is strictly greater than one, the sum is prime, and every smaller admissible offset fails the primality predicate independently of one notation or implementation.
Derive carefully. Infer studying prime gaps immediately above primorials, examining Fortune's conjecture, comparing nearby-prime sequences, and deriving divisibility restrictions on any possible composite value only under the stated assumptions.
Stress-test. Contrast the legitimate boundary case—Fortunate numbers can repeat as n varies, and 'Fortunate primes' sometimes names either the same conjecturally prime sequence or the distinct values sorted without duplicates—with this counterexample: a prime q larger than p_n# yields an offset q-p_n#, but it is not F_n unless no smaller offset above one also yields a prime.

Knowledge Transfer¶

Transfer within number theory is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from For \(n=7\), \(p_7\#=510510\), and the least \(m>1\) making the sum prime is 19, because smaller offsets fail and \(510529\) is prime. to If a Fortunate number had a prime divisor at most \(p_n\), that divisor would divide both the primorial and the offset and hence the candidate sum, contradicting its primality. demonstrates that continuity.^[3]

Outside the domain, only the skeleton—scan an ordered candidate set above a structured landmark and retain the first member satisfying a hard predicate—travels automatically. The terms prime, primorial, offset, least element, prime gap, primality certificate, conjecture, duplicate, next prime, and Fortune's conjecture retain domain-specific meanings, so every role and inference must be revalidated.

Examples¶

Canonical¶

For \(n=7\), \(p_7\#=510510\), and the least \(m>1\) making the sum prime is 19, because smaller offsets fail and \(510529\) is prime. This establishes F_7=19 only after both primality and the exclusion of all smaller admissible offsets are shown. It is canonical because the carrier, rule, invariant, and consequence are all inspectable.^[1]

Mapped back: a positive index \(n\), the first \(n\) primes, their primorial \(p_n\#\), and the ordered integers \(m>1\) → The primorial is divisible by every prime at most \(p_n\), so many small offsets are excluded by divisibility; the ordered search stops at the first offset producing a prime, and that least offset is the sequence value → the base is exactly the nth primorial, the offset is strictly greater than one, the sum is prime, and every smaller admissible offset fails the primality predicate → studying prime gaps immediately above primorials, examining Fortune's conjecture, comparing nearby-prime sequences, and deriving divisibility restrictions on any possible composite value

Applied / In Practice¶

If a Fortunate number had a prime divisor at most \(p_n\), that divisor would divide both the primorial and the offset and hence the candidate sum, contradicting its primality. Therefore every prime divisor of a hypothetical composite \(F_n\) exceeds \(p_n\), so such a counterexample must be at least \(p_{n+1}^2\); this supports but does not prove the conjecture. It qualifies only after the same diagnostic and failure boundary are checked.^[2]

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Exact identity vs. practical recognition. The constitutive condition may be exact while evidence is indirect. Diagnostic: Can the reviewer state both the condition and the warrant?
T2: Canonical form vs. variants. indexed sequence with duplicates, distinct sorted values, upper and lower nearby-prime offsets, primorial-prime special cases, and computational extensions can preserve or change the identity. Diagnostic: Which named role is invariant across the variants?
T3: Compression vs. hidden assumptions. The label is useful only while prerequisites remain visible. Diagnostic: Can each downstream inference be traced to a declared assumption?
T4: Autonomy vs. reduction. The candidate uses broader structures but claims the primorial-indexed least positive prime-producing offset, not all prime gaps, primorial primes, or the set of distinct values after duplicates are removed. Diagnostic: Does that residual still support independent recognition after the parent and neighbors are subtracted?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is scan an ordered candidate set above a structured landmark and retain the first member satisfying a hard predicate; its identity-bearing terms are prime, primorial, offset, least element, prime gap, primality certificate, conjecture, duplicate, next prime, and Fortune's conjecture. Those terms determine admissible objects, evidence, and consequences inside number theory.

Structural Core vs. Domain Accent¶

The structural core is a carrier governed by The primorial is divisible by every prime at most \(p_n\), so many small offsets are excluded by divisibility; the ordered search stops at the first offset producing a prime, and that least offset is the sequence value and tested by fix indexing and primorial notation, calculate or certify the base, test offsets in increasing order beginning above one, prove primality of the selected sum, and retain minimality rather than recording an arbitrary later prime. The domain accent is constitutive rather than decorative, so an analogy that preserves only the skeleton is not another instance of Fortunate number.

The proposed strict upward parent is prime:selection. The definition literally selects one retained offset from an ordered eligible population by a primality criterion and a least-element rule; primorial arithmetic supplies the residual. The edge is proposal-only and points to a frozen prior-baseline Prime.

The entry does not collapse into the parent because the primorial-indexed least positive prime-producing offset, not all prime gaps, primorial primes, or the set of distinct values after duplicates are removed A thematic neighbor is declined whenever it does not literally subsume that rule.

The prospective workspace queue contains one strict upward edge to prime:selection. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Fortunate number Domain-specific

Parents (1) — more general patterns this builds on

Fortunate number is a kind of Selection Prime

The proposed strict upward parent is prime:selection.

Hierarchy path (1) — routes to 1 parentless root

Fortunate number → Selection

Neighborhood in Abstraction Space¶

Fortunate number sits in a sparse region of the domain-specific corpus (60^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Integer Functions & Special Numbers (13 abstractions)

Nearest neighbors

Euclid number — 0.89
Quasiperfect number — 0.87
Fermat number — 0.86
Prime signature — 0.86
Wilson quotient — 0.86

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Lucky number. Generated by a sieve process unrelated to primorial offsets.
Primorial prime. A prime of form p_n# plus or minus one, a special outcome rather than the least general offset.
Prime gap. The difference between consecutive primes broadly; F_n is tied to one composite landmark.
Less fortunate number. The corresponding offset to a prime below a primorial under a separate convention.

References¶

[1] Solomon W. Golomb, 'The Evidence for Fortune's Conjecture,' Mathematics Magazine 54(4), 209–210 (1981), DOI 10.1080/0025570X.1981.11976928. registry ↩a ↩b

[2] Richard K. Guy, Unsolved Problems in Number Theory, 3^rd ed., Springer, 2004, problem A2, DOI 10.1007/978-0-387-26677-0. registry ↩a ↩b

[3] OEIS Foundation, On-Line Encyclopedia of Integer Sequences, A005235, 'Fortunate numbers,' definition, examples, references, and status notes, accessed 2026-08-30, https://oeis.org/A005235. registry ↩