certified
Erdős problem 10 - grechuk
Grechuk variant formalized
Infinitely many even natural numbers cannot be written as a prime plus at most three powers of two. This formalizes the target using the earlier Crocker construction; it is not a claim of a new number-theoretic construction.
Submitted by 5GeGrY…uLUScV
- Lean verification:
- Passed
- Conjectures review:
- Approved
- Reward:
- Paid
Certified 6 Aug 2026, published by conjectures.io.
Review decision
Approved in reviewREVIEW_APPROVED
Lean verified the exact published Grechuk task. The proof faithfully establishes infinitely many even natural numbers that cannot be written as a prime plus at most three powers of two, using the Crocker covering-congruence construction. Although the underlying number-theoretic construction predates this submission, the result was not already available in the pinned Lean environment and no copying, misattribution, or other v1 disqualification was established. The submission earns the displayed conjecture bounty.
Approved · decided 6 Aug 2026
Formal statement
({n | Even n} \ Erdos10.sumPrimeAndTwoPows 3).InfiniteVerification report
Every box below had to hold before the proof counted. They are grouped in the order the verifier reaches them.
The task it was checked against
Manifest valid — Passed
The task bundle held together: the exact file set, a strict manifest, and every trusted hash matching the bytes on disk.
Task commitment matches — Passed
The bundle digest the submission committed to is the digest of the bundle that was actually verified.
Production task — Passed
The task came from the production pool rather than a test fixture.
Trusted file hashes match — Passed
The pinned dependencies agree across the manifest, the lockfile and the checkout, down to the same Formal Conjectures commit.
The submission and the sandbox
Submission policy respected — Passed
The submitted source passed the static scan: no imports, no axiom declarations, no sorry, no native_decide, no unsafe options.
Production sandbox — Passed
The run happened under real isolation, Landrun with seccomp, and the sandbox passed its own live self-test before the proof was touched.
The trusted build
Challenge built — Passed
The trusted Challenge.lean, which contains no miner code, compiled on its own.
Source type hash matches — Passed
The source theorem in the compiled environment still hashes to the type recorded in the task, so the statement has not drifted upstream.
The kernel's verdict
Solution built — Passed
The submitted Solution.lean compiled.
Statement unchanged — Passed
The theorem the proof establishes has exactly the same canonical type as the task's target - it was not weakened or restated.
Only permitted axioms — Passed
The transitive axiom closure of the proof stays inside the axioms this task permits.
Lean kernel accepted — Passed
The Lean kernel replayed the proof and accepted it.
Nanoda accepted — Not run
A second kernel, written independently of Lean's, also accepted the proof.
One kernel, not two
This task does not require a second, independent kernel, so Nanoda was not run. The verdict rests on a single kernel implementation.
Theorems established
- Bounty.target
Stage COMPLETED · Axioms permitted: propext, Quot.sound, Classical.choice