google-deepmind / google-deepmind/formal-conjectures

In-repo resolutions: A001359 proof and A078590 counterexample

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Description

Planned contribution: two kernel-checked resolutions of research-open OEIS conjectures, proved in-repo (no external proof links, no sorry).

1. **A001359** (`FormalConjectures/OEIS/1359.lean`, `OeisA1359.conjecture`): full proof via a Wilson-theorem argument with a fixed-divisor trial-division sieve and a verified length-7840 certificate. `lake --wfail build` passes; `#print axioms` gives `[propext, Classical.choice, Quot.sound]`.

2. **A078590** (`FormalConjectures/OEIS/78590.lean`, new `OeisA78590.conjecture_counterexample`): the integrality conjecture is false at n = 7 — `a 5 = 171` does not divide `2 ^ a 6 + 1` (remainder 9), so `a(7)` is not an integer. Proof: `a 6 = (2^171+1)/9 ≡ 3 [MOD 18]`, hence `2^a6 + 1 ≡ 9 [MOD 19]`, and `19 ∣ 171`. `lake --wfail build` passes; `#print axioms` gives `[propext, Quot.sound]`. The original `conjecture` statement is left untouched.

PR to follow, linking back here.

Contributor guide

Open the contributing guide

Research direction

Start with FormalConjectures/OEIS/1359.lean and OeisA1359.conjecture, then inspect the new FormalConjectures/OEIS/78590.lean counterexample theorem. Run `lake --wfail build` and check `#print axioms`; done means both kernel-checked resolutions compile without `sorry`, while the original A078590 conjecture remains untouched.

Written by the indexing model from the issue text.

Assessment

Domain
tooling
Issue type
Feature
Difficulty
4/5
Estimated time
3-5 days
Activity status
Active
Clarity
Clearly specified
Newbie friendliness
28/100

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