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

v3 · 2026-09-19 · History

Domain-specific #

1869

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Number Theory, Prime Gaps Near Primorials → Mathematics

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.

How would you explain it like I'm…

The Lucky Add-On Number

Prime numbers are numbers you can't share into equal groups, except one big group or groups of one, like 2, 3, 5, and 7. Multiply the first few primes together to get a big number. Then add 2, then 3, then 4, and so on, and stop the first time you land on a prime. The number you added is a Fortunate number.

Smallest Step to a Prime

Primes are numbers like 2, 3, 5, 7, 11 that can only be divided evenly by 1 and themselves. Multiply the first few primes together, for example 2 × 3 × 5 = 30. Now add 2, 3, 4, … in order and stop at the first sum that is prime. 30 + 2 = 32 isn't prime, and neither are 33, 34, 35, or 36. But 30 + 7 = 37 is prime, so the Fortunate number for the first three primes is 7. Adding 1 is never allowed, and you must stop at the first prime you hit, not a later one.

Least Prime Offset Above Primorial

The primorial pₙ# is the product of the first n primes. The nth Fortunate number is the smallest m greater than 1 such that pₙ# + m is prime. Because pₙ# is divisible by every prime up to pₙ, many small values of m are ruled out quickly: adding any such prime, or a multiple of it, gives a number that prime divides. The rules are strict. Offsets start at 2 (m = 1 is excluded), you take the first prime found rather than any later prime, n counts primes rather than integers, and repeated values in the sequence are kept. Fortune's conjecture, that these numbers are always prime, is still unproven and should not be stated as fact.

 

Define Fₙ = min{m > 1 : pₙ# + m is prime}, where pₙ# is the product of the first n primes. Since every prime up to pₙ divides pₙ#, any offset m sharing a factor with pₙ# makes the sum composite. That divisibility excludes many small offsets. The value is found by an ordered search: test m = 2, 3, 4, … and stop at the first m whose sum is provably prime. The definition is fragile in specific ways. Admitting m = 1, choosing an arbitrary later prime instead of the least offset, indexing by integers instead of primes, or deduplicating the sequence all produce a different object. It is also distinct from prime gaps in general and from primorial primes. The sequence supports the study of prime gaps just above primorials and comparisons with other nearby-prime sequences. It also yields divisibility constraints on any hypothetical composite value, which bears on Fortune's conjecture that every Fₙ is prime. That conjecture remains unproven.

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

1. 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) — outgoing typed relations

• Fortunate number is a kind of Selection Prime

  The definition literally selects one retained offset from an ordered eligible population by a primality criterion and a least-element rule.

Hierarchy path (1) — routes to 1 parentless root

• Fortunate number → is a kind of Selection

Neighborhood in Abstraction Space

Fortunate number sits in a moderately populated region (45^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Integer Classifications & Arithmetic Functions (45 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 · 2026-10-25