Primorials and prime-producing offsets

Fortune’s conjecture

Collaboration beta

After multiplying the first n ordinary primes, find the first offset greater than one that makes another prime. Must that offset always be prime? The packet develops arithmetic and sieve reductions but leaves the conjecture open.

P_{n} = \prod_{j = 1}^{n} p_{j}, F_{n} = \min {m \in ℤ : m > 1, P_{n} + m is prime}; F_{n} prime for every n \geq 1.

Known results and sources

Illustration: A worked example scans offsets above 1 from P = 30, reaching the prime 37 at offset 7.

Research problem

Exact mathematical statement

Let $p_1=2<p_2=3<\cdots$ enumerate all ordinary primes. For each $n\ge1$ , put $P_n=\prod_{j=1}^{n}p_j$ and

F_n=\min\{m\in\mathbb Z:m>1,\ P_n+m\text{ is prime}\}.

Fortune’s conjecture asserts

F_n\text{ is prime for every }n\ge1.

The offset $m=1$ is excluded even when $P_n+1$ is prime. The source reports no proof or counterexample to this full statement.

Problem infographic

Problem at a glance

Illustration: The exact m greater than one rule is explained with P = 30, composite sums 32 through 36, and the first allowed prime sum 37. — Illustration: Primality of the sum selects the first allowed offset. The open conjecture asks whether the selected offset is always prime.

Current mathematical picture

Where work on Fortune’s conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureRoughness and pairwise coprimality in a linear window

The source’s p=137 example has 47 composite candidates in the full linear window, including 42 that survive through 274 and are pairwise coprime. This is a counterexample to that inference, not to Fortune’s conjecture. A larger-window or original-minimality argument using additional arithmetic may still work; it must distinguish proper factors from prime values.

Route status · Narrowed route

Main reductionCurrent reduction

For $p=p_n$ , next prime $q=p_{n+1}$ , $P=P_n$ , $H=q^2-1$ , $X=P+H$ , and $L=\sqrt X$ , prime offsets $p<r\le H$ give the exact screen $T_p=\Phi_p(L)$ . Positivity is sufficient for the target, with no converse established. Separately, the original conjecture is equivalent to every successful composite offset having an earlier successful prime offset.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve an actual prime-producing lower bound at every sufficiently large ordinary primorial, with an explicit account of any initial range left outside the asymptotic theorem.Task status · Ready to work on

Work mapped so far

Fortune’s conjecture in numbers

1.1kretained lines of mathematical investigation1,096 in the current working snapshot

Argument development: 784 · 72%
Explored or eliminated routes: 82 · 7%
Computational analysis: 74 · 7%
Open obligations: 58 · 5%
Definitions and setup: 98 · 9%

9selected mapped statements2routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byReo FortuneStatus checkedSep 7, 2026

Source recordedSep 7, 2026

Recommended next task

Prove an actual prime-producing lower bound at every sufficiently large ordinary primorial, with an explicit account of any initial range left outside the asymptotic theorem.

Suggested move: Formulate a cofactor or prime-value estimate that excludes composite survivors, including candidates whose proper factors are unique to them. Show why it reaches the exact prime screen rather than merely another polynomial large-factor cutoff.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeRoughness and pairwise coprimality in a linear window

Route status · Narrowed route

Narrowed routeReplacing a polynomial factor cutoff by the primality threshold

At $X/q=\exp((1+o(1))p)/p$ , the available prime-base supply exceeds the candidate count and even $T_1$ is of order $1/(p\log p)$ , outside the required height range. A new cofactor estimate or the exact minimality route could bypass this failure. Improving the early-sieve numerical constant alone does not supply that information.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove an actual prime-producing lower bound at every sufficiently large ordinary primorial, with an explicit account of any initial range left outside the asymptotic theorem.Suggested move: Formulate a cofactor or prime-value estimate that excludes composite survivors, including candidates whose proper factors are unique to them. Show why it reaches the exact prime screen rather than merely another polynomial large-factor cutoff.

Ready to work on

Prove that every successful composite offset has an earlier successful prime offset without assuming the stronger quadratic window.Suggested move: Specify a well-founded decreasing quantity for a proposed descent and prove that each branch preserves a success or produces an earlier one. Test the argument against the source’s successful composite offsets; descent to one of their factors is invalid.

Ready to work on

Establish the actual signed first-elimination estimate $E_2(L)\le7S_0/8$ in the eventual regime $1<y<L$ , or supply another independently estimated margin implying $T_p>0$ .Suggested move: Estimate how the early survivors occupy later residue classes, including isolated cofactor intervals shorter than one. Retain the signed compensation between stages and do not apply a prime-progression bound to a sifted subset by changing its interval length.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026

Current statusOpen conjecture

For each primorial P, let m be the least integer greater than one for which P + m is prime. The assertion that every such m is prime remains open.

[1]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026-08-27
Computational resultAn OEIS comment by Brock Falfas reports that the first 4,600 Fortunate numbers are prime; this is a finite reported check.[2]

Mathematical neighborhood

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions2 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction3 of 93
lemma3 of 93
equivalence1 of 91
computational claim1 of 91

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.25 displayed rows · 2 routes included

retained route statementIs the first successful offset after a primorial always prime?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementA strict quadratic window is sufficientintermediate
retained route statementMinimality beyond the short windowintermediate
retained route statementMany large factors, with primality still missingintermediate
retained route statementA positive early-sieve fractionintermediate
retained route statementA signed late-elimination condition remains openintermediate
retained route statementThe p=137 linear window blocks a shortcutintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureRoughness and pairwise coprimality in a linear windowreported failure
Useful failureReplacing a polynomial factor cutoff by the primality thresholdreported failure
Research targetProve an actual prime-producing lower bound at every sufficiently large ordinary primorial, with an explicit account of any initial range left outside the asymptotic theorem.open
Research targetProve that every successful composite offset has an earlier successful prime offset without assuming the stronger quadratic window.open
Research targetEstablish the actual signed first-elimination estimate $E_2(L)\le7S_0/8$ in the eventual regime $1<y<L$ , or supply another independently estimated margin implying $T_p>0$ .open
ComputationThe source reports complete factorizations for 1,938 candidates in the ten rows p=13,17,19,23,29,31,37,41,43,47, with 5,657 recursive prime-certificate nodes. It separately reports exact compositeness witnesses for all 47 candidates in the p=137 linear window. These attachments were not rerun here.These finite records cover specified rows and examples only. At p=137, all 47 prime offsets in (137,411] give composite values; 42 survive the cutoff 274 and are pairwise coprime. This refutes a linear-window inference, while the relevant composite-offset threshold is 139²=19321. · reported unreproduced
Narrowed routeRoughness and pairwise coprimality in a linear windowThe source’s p=137 example has 47 composite candidates in the full linear window, including 42 that survive through 274 and are pairwise coprime. This is a counterexample to that inference, not to Fortune’s conjecture. A larger-window or original-minimality argument using additional arithmetic may still work; it must distinguish proper factors from prime values.
Narrowed routeReplacing a polynomial factor cutoff by the primality thresholdAt $X/q=\exp((1+o(1))p)/p$ , the available prime-base supply exceeds the candidate count and even $T_1$ is of order $1/(p\log p)$ , outside the required height range. A new cofactor estimate or the exact minimality route could bypass this failure. Improving the early-sieve numerical constant alone does not supply that information.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve an actual prime-producing lower bound at every sufficiently large ordinary primorial, with an explicit account of any initial range left outside the asymptotic theorem.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointProve an actual prime-producing lower bound at every sufficiently large ordinary primorial, with an explicit account of any initial range left outside the asymptotic theorem.

Fortune’s conjecture · ready to start

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Research contextPrepared context for any AI agent

After multiplying the first n ordinary primes, find the first offset greater than one that makes another prime. Must that offset always be prime? The current work develops arithmetic and sieve reductions but leaves the conjecture open.

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

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Sources and references3 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

1
The Prime Glossary: Fortunate numberauthoritative webpage · Prime Pages · accessed Sep 7, 2026
2
A005235 — Fortunate numberssoftware or dataset · The OEIS Foundation Inc. · OEIS A005235 · accessed Sep 7, 2026
3
Fortune.Defs — least Fortunate offsets and conjecture statementformalization · accessed Sep 7, 2026

Important qualifications

The original proposal date and original publication were not established in this bounded search.
Finite computations and public Lean statements were inspected as external metadata; no program, theorem build, or proof audit was run.
This private draft grants no mathematical acceptance, rights, authorship, contributor credit, or publication authority.

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Fortune’s conjecture

Exact mathematical statement

Problem at a glance

Where work on Fortune’s conjecture stands

Fortune’s conjecture in numbers

Prove an actual prime-producing lower bound at every sufficiently large ordinary primorial, with an explicit account of any initial range left outside the asymptotic theorem.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formal and computational footholds

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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