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.
Core Idea¶
The nth Fortunate number is \(F_n=\min\{m>1:p_n\#+m\text{ is prime}\}\). 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.
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.
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
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. 2. 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.
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.
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 (60th 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