Erdős 699: index 21 is effective, and seven numerical stopping bounds

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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 · Cutoff 26 and the preceding reduction

Completed follow-up. The cutoff-15 proof includes complete exclusions for every index in the table below, as well as $10,13,15,18,20$. This note retains the earlier stopping-bound derivation.

Index $21$ is effective, and all six previously effective but numerically uninstantiated cases have explicit stopping bounds. Every bad triple with one of the following indices satisfies the stated inequality:

No unspecified constant occurs in this table. At this stage these were stopping bounds; their finite ranges had not yet been exhausted. The existing complete exclusions $i=1,2,17,23$ and $i\ge26$ remain in force. At this stage the indices at least four still lacking effective stopping bounds were

$$\boxed{\{4,5,6,7,8,9,10,12,13,14,15,18,20\}.}$$

Thus this step decreased the previous list of fourteen to thirteen. The separate cubic finiteness question is unchanged. All assertions use the original prime threshold $p\ge i$.

Why index 21 becomes effective

For each small prime, let $P_p$ be the largest power of $p$ dividing a numerator factor $n-r_p$, and write $n-r_p=P_pu_p$. The bounded differences between these factors allow Bennett–Filaseta–Trifonov (BFT) to bound the cofactors $u_p$. Two new certified applications of their Theorem 2.2 give the additional exponent constraints

$$\max(z_2,z_{17})\ge.3560,\qquad \max(z_{17},z_{19})\ge.2176,\qquad z_p=\frac{\log u_p}{\log(n-20)},$$

once the effective source thresholds hold. The prime-pair note records the parameters and interval certificates.

Use the weighted Kummer polynomial

$$F_{14}(X,Y)=\prod_{a=0}^{13}((X-a)(Y-a))^{14-a}.$$

Its degree is $210$ and its multiplicity on row $t\le20$ is at least $28-t$. Combining these multiplicities with the individual cofactor constraints, rather than just their sum, gives the exact positive gain

$$G\ge\frac{7683}{5000}=1.5366.$$

The weighted-polynomial proof covers every assignment of the selected positions, including collisions. It first gives $n<\max\{10^{232},X_{21}+20\}$. For the fully numerical version, weaker but explicitly instantiated pair estimates still give $G\ge1711/10000>0$. Their algebraic bound is $10^{2075}$, and their largest source threshold is below $10^{45\,411\,830}$. The next power of ten covers the numerator shifts and proves the table's index-21 bound.

Evaluating the source thresholds

BFT Theorem 2.4 gives a completely explicit threshold if one supplies a lower bound for the gcd of its Padé coefficients. There are two useful ways to do this here.

First, the published bounds in BFT Proposition 5.1 suffice for three new pairs:

More precisely, each source threshold is strictly below the number in the last column. The estimates apply for differences at most $100$. The exact polynomial integrals in Theorem 2.4 are evaluated as rational numbers, and the remaining operations use directed interval arithmetic.

The $(2,17)$ pair with the original explicit table closes $i=19$ through the weighted polynomial. All three pairs close $i=24$. For $i=22,25$, even the unweighted argument suffices: the disjoint pair $(17,19)$ adds $.1855$ to the old six-prime saving $.913$, giving

$$\gamma=1.0985>1.$$

The complete derivation and weighted certificates give the four bounds with exponents $957$ or $976$.

Second, at $i=11,16,21$ a new elementary lower bound for the coefficient gcd improves the published numerical inputs. BFT Lemma 5.2 supplies disjoint prime intervals whose products divide that gcd. Combining their Lemma 5.4 estimates for $\theta(x)$ gives a finite rational sum bounding the gcd for every parameter value $m\ge m_0$. Integer square roots and downward rounding certify this sum. The numerical-threshold proof derives the uniform estimate and retains all parameters. This is the step that removes the unevaluated source constants.

Smaller algebraic terms with the original thresholds

The same work also decreases the earlier computable bounds while keeping their original source thresholds. Let $X_{11}$ and $X_{24}$ denote exactly the thresholds in the preceding index-11 and index-24 notes, and let $X=x_0(33)$ be the original BFT Corollary 2.3 threshold. Then:

For example, the index-11 algebraic term decreases from $10^{1525}$ to $10^{484}$, and the index-24 term from $10^{10000}$ to $10^{228}$. The improvement keeps the valuation of each individual factor away from its selected maximum, instead of replacing all of them by the same factorial bound. The index-11 proof checks all 63,360 position/vector patterns with outward rounding; the weighted-polynomial checker handles the other cases with exact rational and integer arithmetic.

Each numerical bound in the first table and each corresponding bound in this table holds independently, so their minimum may be used.

Retained certificates

• Weighted polynomial and collision checker and output.
• Explicit BFT threshold evaluator and output.
• Uniform coefficient-gcd checker and output.
• Refined index-11 constant checker and output.

These computations certify analytic parameters and finite systems of proof inequalities. They do not enumerate binomial rows, and do not claim to exclude the finite ranges left in the first table.