Erdős 699: complete exclusion at seven and nine

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This is a retained earlier research note. The current reading guide gives the latest status and distinguishes completed proofs from remaining cases.

P699 research · Erdős problem 699 · Complete cutoff nine

Seven and nine are completely excluded, for the original threshold $p\ge i$. Their seven and nine combined proofs join fully numerical infinite tails to completed finite ranges. Neither index has a remaining exceptional set.

The current classification is:

Only four fixed indices remain among $i\ge4$. Their finiteness is known; effective row bounds are still missing here.

How the two ranges at seven meet

Put $h=n-6$. All five initial cofactor alternatives and all 258 joint inequalities now hold for $h\ge10^{100000000}$. The exact relaxed polynomial forest needs only $h\ge10^{99990923}$. The completed finite proof covers $n\le10^{100000001}$, so every row is excluded.

The main additional step is the block adjustment lemma. When the Padé index is chosen below its nominal value, the residual prime-power denominator cost decreases. Quantifying that saving lets one approximant certify a whole interval of logarithmic heights. Chains of such intervals reach the already numerical uniform tails. The source clauses and their exact polynomial coverage are retained.

The initial-source chains contain 122868 witnesses. Among the joint sources, 43 hold directly by uniform estimates and 215 are covered by 9811438 exact witnesses in 187 direction groups. Every direction is covered without a gap.

The finite proof has 18000 certified intervals, 28 residual rows, and 41740 exact column checks. Its extension to $10^{100000001}$ reuses the lower stages and adds eight exponent anchors for each of the three prime pairs. No interpolation rank is recomputed by the combined checker.

How the two ranges at nine meet

Put $h=n-8$. All initial and joint sources, and the exact polynomial coverage, apply for $h\ge10^{241547627}$. The finite proof covers $n\le10^{241547629}$, so the ranges overlap and all rows are excluded.

The initial-source covers contain 299324 retained witnesses. Eleven pair sources imply the nineteen initial pair alternatives; four ordinary product sources provide the other initial constraints. Among the 790 joint clauses, 440 hold directly at the target height and 350 are covered by 11599987 exact witnesses in 321 direction groups. Every direction is covered without a gap.

The relaxed polynomial proof checks all 104976 labeled position and cofactor cases with 24322 exact rational Farkas leaves. The finite proof retains 607975 certified intervals, thirteen residual rows, and 6762 exact column checks. The six extended prime-pair certificates add no exceptions. All pieces are bound by the combined checker and proof record.

Improvements shared by the two calculations

The numerical proofs use several complementary changes:

• Explicit Padé constants account for the integral prefactors and residual denominators.
• Sharper Chebyshev estimates improve the uniform coefficient-gcd estimates.
• The initial endpoints are weakened by $10^{-4}$ and joint thresholds by individually certified amounts; exact replacement forests check all these changes together.
• Integer Newton bases and actual Farkas margins reduce the polynomial coefficient requirements.
• Arithmetic nonvanishing at seven and nine removes the largest common-zero height bounds.
• The prime-power gcd lemma includes powers of primes in the approximation divisor and removes an unnecessary floor-stability restriction.

For comparison, the original index-seven polynomial exponent was $19813540001$. The sharper coefficient and nonvanishing argument lowers it to $44245906$ for the original clauses, or $99990923$ for the relaxed clauses used in the complete proof. At nine, the corresponding final polynomial exponent is $241547627$, replacing $5109634000001$.

The four indices still lacking effective bounds

At six, a degree-twelve product and a column valuation identity force all three selected primes into their high-residue branches in a specified cofactor box, for $n-5\ge10^{425}$. At eight, an exact column valuation argument excludes one Lucas residue branch in a difficult configuration. These are conditional restrictions, not whole-index effectivity results.

Exact line coverings at six and eight prove that no polynomial of any degree can give the required strict weighted degree gain at the specified hard profiles. Controlling common curve zeros alone cannot remove that obstruction; stronger arithmetic input is needed.

The Runge criterion makes bad pairs on a fixed curve effectively determinable when enough numerator fibers contribute poles. For four and five, three finite pole fibers suffice. It does not put all bad pairs on finitely many such curves. The cubic results remain unchanged.