fortunate number

Indexed prime product

A proposed occurrence must identify n and P_n unambiguously; prime-index notation must not be confused with a product over primes bounded by the integer n.

1. Does n count the primes in the product, with n ≥ 1? definition
2. Does the recorded P_n equal the product of exactly the first n primes? measurement

Strict offset boundary

Recognition requires m > 1 and q = P_n + m, followed by endpoint and minimality validation.

1. Is the recorded fortunate number m rather than q or P_n? boundary
2. Does the search exclude m = 1 even when P_n + 1 is prime? definition

Verified prime endpoint

The model should retain whether endpoint evidence is a proof or a probable-prime result, together with enough information for independent checking.

1. What method or certificate establishes that the exact integer P_n + m is prime? provenance
2. Does the evidence justify a certified occurrence or only a provisional candidate? action

Complete minimality coverage

Certification requires showing that P_n + k is composite for every integer k with 1 < k < m, through direct checks or justified collective exclusions.

1. Which witnesses or arguments exclude every smaller admissible offset without gaps? provenance
2. If a smaller prime endpoint is found, which candidate and dependent records must be corrected? action

Reproducible search state

A computation should preserve the exact primorial, tested or excluded offsets, stopping point and primality procedure so it can be resumed or checked.

1. What exact-arithmetic procedure searches offsets above one and justifies any skipped candidates? action
2. Through which offset has the computation completed its exclusions, and what remains untested? measurement

Preserved index witnesses

A distinct-value record should retain its known generating indices; absence from a searched finite range does not establish absence from the entire sequence.

1. Which verified indices witness that this integer is a fortunate-number value? provenance
2. Does the collection preserve index order, sort distinct values or report only a bounded search? boundary

Conjecture and evidence separation

Fortune's conjecture asserts that fortunate numbers are prime; finite checks must remain distinct from a universal proof. Reference: [Guy's discussion hosted by OEIS](https://oeis.org/A002572/a002572_3.pdf).

1. Is a claim about offset primality supported by an individual certificate, a bounded computation or a general proof? provenance
2. For a proposed counterexample, have both fortunate-number membership and compositeness of m been independently established? action

Construction boundary check

Subtraction defines lesser fortunate numbers, a neighboring construction with its own boundary conditions. Reference: [OEIS A055211](https://oeis.org/A055211).

1. Does the source use addition to the primorial, subtraction from it or a different number-generating process? boundary
2. Should a record involving offset one, a sieve or subtraction be linked to a neighboring model instead? action

Check these first

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these check these first hold for the sense of fortunate number this model covers, and on what evidence? provenance

Real-world use

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these real-world use hold for the sense of fortunate number this model covers, and on what evidence? provenance

Failure modes and hazards

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these failure modes and hazards hold for the sense of fortunate number this model covers, and on what evidence? provenance

Neighbouring kinds and how to tell them apart

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these neighbouring kinds and how to tell them apart hold for the sense of fortunate number this model covers, and on what evidence? provenance