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feat(ErdosProblems/399): prove erdos_399.variants.cambie - #5425
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Formalise the mod-8 argument already described in the docstring: coprimality forces x, y not both even, so x^4 + y^4 ≡ 1 or 2 (mod 8) while 8 ∣ n! for n ≥ 4; the n ≤ 3 cases are bounded directly. Core Mathlib only, no new imports.
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Fills the
sorryinerdos_399.variants.cambie. The docstring already spells outCambie's argument ("considerations modulo 8 rule out any solutions"), so this just
formalises it directly.
Sketch:
z^4 % 8 = z % 2, so coprimality (⇒x, ynot both even) givesx^4 + y^4 ≡ 1 or 2 (mod 8), never0; but8 ∣ n!forn ≥ 4(4! ∣ n!,8 ∣ 24).The
n ≤ 3cases are justn! ≤ 6 < 16 ≤ x^4 + y^4.Only core Mathlib, no new imports; the statement and
@[category]tag are unchanged.lake buildis clean — no errors or warnings, nosorry— and#print axiomsgives theusual
{propext, Classical.choice, Quot.sound}.It's ~40 lines, a bit longer than the shortest fills here but elementary throughout. Happy
to move it to a linked
@[formal_proof]instead if you'd rather keep the file lean.AI usage (per CONTRIBUTING → Mathlib's policy): the proof was developed with Claude (Anthropic); I've been through it, and as with any Lean proof the trust is in the kernel —
#print axiomsgives the standard trio with nosorryAx.