{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.fortunate-number","code":"thing-q5151604","url":"https://ver.cy/models/thing/q5151604/","name":"fortunate number","alternateNames":[],"kind":"thing","status":"published","version":"0.1.0-research.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QTY"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"For a positive integer n, the nth Fortunate number is the least integer m greater than 1 such that the product of the first n prime numbers plus m is prime.","purpose":"Enable an agent to recognize, compute and validate fortunate numbers while separating verified terms from conjectures about their arithmetic properties.","scope":{"in":["The defining relationship between a prime index, its primorial and the least qualifying additive offset","Indexed fortunate-number occurrences and their distinct integer values","Exact computation and verification of the resulting prime","Evidence excluding every smaller admissible offset","The distinction between established membership and claims about the offset's own primality"],"out":["General theories and implementations of primality testing","Primorials as independently modeled integer products","Lesser fortunate numbers, which use subtraction from a primorial","Lucky numbers generated by a sieve","Primorial primes defined by offsets of plus or minus one","Physical quantities, measurement units and interpretations of personal good fortune"],"boundaries":[]},"distinguishingFeatures":["Its reference integer is the product of the first n primes, not n factorial or an arbitrary product of primes.","Its value is an additive offset m, not the resulting prime P_n + m.","The offset must exceed one; a prime at P_n + 1 does not qualify.","The offset must be minimal: primality of P_n + m alone does not establish membership at index n.","Primality of the offset itself is not a defining requirement and must be evaluated separately."],"structure":{"bundles":[{"id":"defining-offset","name":"Defining offset","description":"Establishes precisely which integer is being classified and the reference product that gives it meaning.","layers":[{"id":"primorial-reference","name":"Primorial reference","description":"Fixes the positive index and the associated product of consecutive primes beginning with two.","findings":[{"id":"indexed-prime-product","name":"Indexed prime product","description":"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.","questions":[{"text":"Does n count the primes in the product, with n ≥ 1?","id":"indexed-prime-product-q01","kind":"definition"},{"text":"Does the recorded P_n equal the product of exactly the first n primes?","id":"indexed-prime-product-q02","kind":"measurement"}]}]},{"id":"admissible-offset","name":"Admissible offset","description":"Separates the candidate offset, its endpoint and the strict lower bound on eligibility.","findings":[{"id":"strict-offset-boundary","name":"Strict offset boundary","description":"Recognition requires m > 1 and q = P_n + m, followed by endpoint and minimality validation.","questions":[{"text":"Is the recorded fortunate number m rather than q or P_n?","id":"strict-offset-boundary-q01","kind":"boundary"},{"text":"Does the search exclude m = 1 even when P_n + 1 is prime?","id":"strict-offset-boundary-q02","kind":"definition"}]}]}]},{"id":"membership-certification","name":"Membership certification","description":"Organizes the two independent obligations needed to certify a fortunate-number occurrence.","layers":[{"id":"endpoint-primality","name":"Endpoint primality","description":"Records the arithmetic and evidence supporting primality of P_n + m.","findings":[{"id":"verified-prime-endpoint","name":"Verified prime endpoint","description":"The model should retain whether endpoint evidence is a proof or a probable-prime result, together with enough information for independent checking.","questions":[{"text":"What method or certificate establishes that the exact integer P_n + m is prime?","id":"verified-prime-endpoint-q01","kind":"provenance"},{"text":"Does the evidence justify a certified occurrence or only a provisional candidate?","id":"verified-prime-endpoint-q02","kind":"action"}]}]},{"id":"smaller-offset-exclusion","name":"Smaller-offset exclusion","description":"Accounts for every admissible offset preceding the proposed value.","findings":[{"id":"complete-minimality-coverage","name":"Complete minimality coverage","description":"Certification requires showing that P_n + k is composite for every integer k with 1 < k < m, through direct checks or justified collective exclusions.","questions":[{"text":"Which witnesses or arguments exclude every smaller admissible offset without gaps?","id":"complete-minimality-coverage-q01","kind":"provenance"},{"text":"If a smaller prime endpoint is found, which candidate and dependent records must be corrected?","id":"complete-minimality-coverage-q02","kind":"action"}]}]}]},{"id":"computation-and-occurrences","name":"Computation and occurrences","description":"Supports reproducible searches and preserves the distinction between sequence positions and integer values.","layers":[{"id":"exact-offset-search","name":"Exact offset search","description":"Captures the search convention, arithmetic precision and completed range.","findings":[{"id":"reproducible-search-state","name":"Reproducible search state","description":"A computation should preserve the exact primorial, tested or excluded offsets, stopping point and primality procedure so it can be resumed or checked.","questions":[{"text":"What exact-arithmetic procedure searches offsets above one and justifies any skipped candidates?","id":"reproducible-search-state-q01","kind":"action"},{"text":"Through which offset has the computation completed its exclusions, and what remains untested?","id":"reproducible-search-state-q02","kind":"measurement"}]}]},{"id":"occurrence-value-mapping","name":"Occurrence-to-value mapping","description":"Maintains generating indices when terms are grouped or presented as distinct values.","findings":[{"id":"preserved-index-witnesses","name":"Preserved index witnesses","description":"A distinct-value record should retain its known generating indices; absence from a searched finite range does not establish absence from the entire sequence.","questions":[{"text":"Which verified indices witness that this integer is a fortunate-number value?","id":"preserved-index-witnesses-q01","kind":"provenance"},{"text":"Does the collection preserve index order, sort distinct values or report only a bounded search?","id":"preserved-index-witnesses-q02","kind":"boundary"}]}]}]},{"id":"arithmetic-claims-and-boundaries","name":"Arithmetic claims and boundaries","description":"Separates claims about fortunate offsets from their definition and distinguishes nearby constructions.","layers":[{"id":"offset-primality-claims","name":"Offset primality claims","description":"Handles tests and general assertions about m independently of primality of P_n + m.","findings":[{"id":"conjecture-evidence-separation","name":"Conjecture and evidence separation","description":"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).","questions":[{"text":"Is a claim about offset primality supported by an individual certificate, a bounded computation or a general proof?","id":"conjecture-evidence-separation-q01","kind":"provenance"},{"text":"For a proposed counterexample, have both fortunate-number membership and compositeness of m been independently established?","id":"conjecture-evidence-separation-q02","kind":"action"}]}]},{"id":"neighboring-number-constructions","name":"Neighboring number constructions","description":"Prevents similar names or altered endpoint formulas from being treated as equivalent definitions.","findings":[{"id":"construction-boundary-check","name":"Construction boundary check","description":"Subtraction defines lesser fortunate numbers, a neighboring construction with its own boundary conditions. Reference: [OEIS A055211](https://oeis.org/A055211).","questions":[{"text":"Does the source use addition to the primorial, subtraction from it or a different number-generating process?","id":"construction-boundary-check-q01","kind":"boundary"},{"text":"Should a record involving offset one, a sieve or subtraction be linked to a neighboring model instead?","id":"construction-boundary-check-q02","kind":"action"}]}]}]},{"id":"evidence-and-external-alignment","name":"Evidence and external alignment","description":"What the world already says about this thing, gathered so the model can be checked against it.","layers":[{"id":"reported-evidence","name":"Reported evidence","description":"Findings from the breadth pass, kept separate from the structural claims.","findings":[{"id":"evidence-confidence-notes","name":"Check these first","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these check these first hold for the sense of fortunate number this model covers, and on what evidence?","id":"evidence-confidence-notes-q01","kind":"provenance"}]},{"id":"evidence-real-world-use","name":"Real-world use","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these real-world use hold for the sense of fortunate number this model covers, and on what evidence?","id":"evidence-real-world-use-q01","kind":"provenance"}]},{"id":"evidence-failure-modes-and-hazards","name":"Failure modes and hazards","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these failure modes and hazards hold for the sense of fortunate number this model covers, and on what evidence?","id":"evidence-failure-modes-and-hazards-q01","kind":"provenance"}]},{"id":"evidence-neighbours","name":"Neighbouring kinds and how to tell them apart","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"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?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Compute and explain Fortunate numbers, showing the primality checks.","Report the status of the conjecture that all Fortunate numbers are prime."],"mustNot":["Present Fortune's conjecture as proven.","Confuse primorials with factorials in the definition.","Report values from unchecked primality tests as certain."],"requiresHuman":["Publishing new results under someone else's name."]},"ethics":{"considerations":["Open conjectures must be stated honestly so learners know what is and is not known."],"affectedParties":["Students","Mathematicians"]},"owners":{"steward":"Nobody owns them; they are held in common as mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q16317911","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.fortunate-number","Wikidata: Q5151604 (https://www.wikidata.org/wiki/Q5151604)"]},"properties":{"applicability":"not-applicable","items":["Prime index: Exact dimensionless integer n ≥ 1","Reference primorial: P_n = product of the first n primes","Fortunate offset: Exact dimensionless integer m > 1","Resulting prime: q = P_n + m","Endpoint evidence: Untested | probable prime | proven prime | composite","Minimality evidence: Unchecked | partially excluded | fully excluded | smaller qualifying offset found","Representation role: Indexed occurrence | distinct value | finite sequence segment","Offset primality assessment: Unknown | probable prime | proven prime | composite"]},"recognition":{"applicability":"not-applicable","items":[]},"capabilities":{"applicability":"required","items":["Compute the least qualifying offset for a specified positive prime index using exact arithmetic.","Validate a proposed occurrence by checking its primorial, endpoint primality and exclusion of smaller offsets.","Reject a candidate when its endpoint is composite or a smaller qualifying offset exists.","Aggregate certified occurrences into a sequence segment or a distinct-value collection with explicit ordering.","Recheck imported terms against reproducible computation and source evidence.","Assess a proposed composite fortunate number by independently verifying membership and offset compositeness."]},"hazards":{"applicability":"required","items":["Errors from probable-prime tests treated as proofs.","Mixing definitions with factorial-based variants."]},"interfaces":{"applicability":"not-applicable","items":[]},"context":{"applicability":"required","items":["Fortunate numbers are studied in number theory through the relationship between primorials and nearby prime numbers.","They are used in computational investigations that test primality, verify minimal offsets and explore conjectures about the resulting sequence."]}},"sources":[{"title":"Wikidata item Q5151604: fortunate number","url":"https://www.wikidata.org/wiki/Q5151604","note":"identity and sense of the item"},{"title":"Wikipedia: Fortunate number","url":"https://en.wikipedia.org/wiki/Fortunate_number","note":"general description of the item"}],"openQuestions":["What is the current published status of Fortune's conjecture, and which primary references establish that status?","What is the largest independently reproducible range of certified fortunate-number occurrences, with both endpoint and minimality evidence available?","Do intended source datasets consistently distinguish prime indices, prime bounds and any empty-product extension at index zero?","Which imported tables provide proven primality rather than probable-prime results for their endpoints and offsets?","Does the existing Vercy catalogue already cover this concept in a world model that vr.tr.fortunate-number should reference?"],"resources":{"spec":"/models/things/publications/thing-q5151604/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q5151604/spec.json"],"providers":["Codex"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T09:51:09Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"not-applicable","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"not-applicable","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:31+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties.","interaction.interfaces":"No specific technical standard, law or official classification applies to Fortunate numbers as mathematical objects.","interaction.recognition":"A Fortunate number is an abstract mathematical object that cannot be distinguished by visual appearance."},"score":1.0}}