IMO 2026 · Problem 6

Frozen · pending review·Day 2·domain pending·Evidence I0 - Unscoped

Official-solution dependence: None·Official solution viewed: NO

Proof audit: NOT-STARTED·Internal prior-solution exposure: YES

Legacy solution manuscript

A complete solution write-up produced earlier in the project. Preserved here in full; not an official IMO solution and not yet independently audited.

IMO2026_Day2_Problem6.pdfDownload PDF

created 2026-07-16level: L4-candidate (human-checkable write-up)audit: AUDIT-PENDINGsha256:06b2172ce636…

Independent HIVE-IMO X manuscript. Not an official IMO solution, and not independently verified under the new pipeline.

Official problem

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers greater than $1$ . Suppose that for all positive integers $n$ , the number $a_{n+1}$ is the smallest positive integer greater than $a_n$ such that $\gcd(a_{n+1},a_i)>1$ for every $i=1,2,\ldots,n$ . Prove that there exist positive integers $T$ and $L$ such that $a_{n+T}=a_n+L$ for every positive integer $n$ .

(Note that $\gcd(x,y)$ denotes the greatest common divisor of positive integers $x$ and $y$ .)

official source ↗frozen 2026-07-21sha256:cbe3015874…

Vietnamese translation

Faithful literal draft by Fable, pending Human review (proper names kept in original form).

Cho $a_1,a_2,a_3,\ldots$ là một dãy vô hạn các số nguyên dương lớn hơn $1$ . Giả sử rằng với mọi số nguyên dương $n$ , số $a_{n+1}$ là số nguyên dương nhỏ nhất lớn hơn $a_n$ sao cho $\gcd(a_{n+1},a_i)>1$ với mọi $i=1,2,\ldots,n$ . Chứng minh rằng tồn tại các số nguyên dương $T$ và $L$ sao cho $a_{n+T}=a_n+L$ với mọi số nguyên dương $n$ .

(Ở đây $\gcd(x,y)$ ký hiệu ước chung lớn nhất của các số nguyên dương $x$ và $y$ .)

What the problem asks

A short plain-language explanation is written during S1–S3, after independent human and AI reads. S0 freezes the statement only - no interpretation yet.

Human × AI contribution matrix

Attribution per stage. Final approval is always human-led and verified independently.

Contribution	Human	Sol	Fable	Independent verifier
Problem interpretation	-	-	-	-
Core idea	-	-	-	-
Lemma discovery	-	-	-	-
Counterexample search	-	-	-	-
Proof writing	-	-	-	-
Formal checking	-	-	-	-
Final approval	-	-	-	-

Contribution attribution is recorded stage by stage; no roles are assigned until work begins.

Swipe horizontally to see all columns →

Audit preparation (S0-R)

The manuscript has been reconciled into a verifier-ready dossier: statement binding, section map, and a proof-obligation skeleton. No obligation has been verified.

state: RECONCILEDstatement: MATCHproof obligations: 6 (4 load-bearing)audited: 0/6

Every obligation is marked CLAIMED (asserted by the manuscript), never VERIFIED. Independent audit by an unexposed verifier has not started.

proof_obligations.json ↗verifier_input_manifest.json ↗sha256:79b35f7a49bb…

Timeline · S0–S14

S0Official problem freeze
HUMAN · Pending: Human V3 translation review + contamination attestation.AI · Fable froze the official statement, produced normalized EN + VI draft, source record, contamination ledger and SHA-256 manifest. Sol V2 semantic review pending. No solving.
problem_en.mdsha256:c551d0c72d…16f29b problem_en.normalized.jsonsha256:a685f23f76…fa45c7 problem_vi.reviewed.mdsha256:06096ebcb2…e89683 source_record.jsonsha256:2a31ac7916…dc7846 contamination_ledger.jsonsha256:c64b88c762…7d3199
S1Human blind read
Not started.
S2AI independent read
Not started.
S3Contrastive merge
Not started.
S4Formal decomposition
Not started.
S5Small-domain reconnaissance
Not started.
S6Route tournament
Not started.
S7Lemma forge
Not started.
S8Candidate proof assembly
Not started.
S9Adversarial review
Not started.
S10Independent reconstruction
Not started.
S11Human proof rewrite
Not started.
S12Independent / formal verification
Not started.
S13Official-solution comparison
Not started.
S14Publication and postmortem
Not started.

Route map

Up to three main routes survive the tournament at S6.

No routes proposed yet.

Failed attempts

Failed routes and rejected proofs are kept permanently, never deleted.

None recorded yet.

Key lemmas

Lemmas appear here once forged at S7, each with an exact statement and an evidence class.

Proof - compact

Written at S11 in natural olympiad style, scorable 0–7 per step. Not yet available.

Proof - annotated

The annotated proof with justifications and dependencies is not yet available.

Independent verification

Required before the status VERIFIED-INDEPENDENT may be shown.

No independent verification has been performed. Status will not reach VERIFIED-INDEPENDENT until an olympiad expert, an independent model family, or a formal proof (Lean/Isabelle/Coq) confirms the proof.

Official-solution comparison

Official solution viewed after proof freeze: NO
Official solution dependence: NONE

A full comparison (core idea, length, naturalness, differences) runs at S13, only after the proof is frozen.

Artifact downloads

Every file is published with a SHA-256 checksum.

problem_en.mdsha256:c551d0c72d…16f29b problem_en.normalized.jsonsha256:a685f23f76…fa45c7 problem_vi.reviewed.mdsha256:06096ebcb2…e89683 source_record.jsonsha256:2a31ac7916…dc7846 contamination_ledger.jsonsha256:c64b88c762…7d3199

MANIFEST sha256: ce54e4063c4c4a837fc8c7defc6b3ad65850b6c0a5795dfc84d51ff95dffd063

Revision history

21/07/2026Record initialized
Ledger created at status FROZEN-PENDING-REVIEW. No mathematical progress recorded yet.