Erdős 699: further product constraints for selected small-prime positions

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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 · Quadratic constant-gap identities · Asymmetric cofactor bounds

The quadratic-product argument gives additional useful cofactor restrictions involving $2,3,5,7,11$. This note records eleven AP estimates, five estimates for two pairs of positions with equal sums, and one cubic product estimate. The underlying inequalities are proved; combining them into a complete index-12 argument remains separate work.

Write

$$n-r_p=p^{a_p}u_p,\qquad0\le r_p\le11,\qquad h=n-11.$$

In each table below, the listed targets mean that at least one of the inequalities $u_p>h^{\gamma_p}$ holds beyond the corresponding numerical threshold. The other primes impose no condition in that row.

Arithmetic-progression positions

These rows apply when the indicated middle prime occupies the middle of three distinct AP positions. The endpoint primes may occur in either order.

For example, the middle-$2$ row on $2,3,7$ uses $64-63=1$, with middle base $2^6$ and endpoint base $3^2\cdot7$. The middle-$7$ row on $2,5,7$ uses $2401-3\cdot800=1$, with bases $7^4$ and $2^5\cdot5^2$. The three-way alternative in the AP note explains why each endpoint prime contributes twice its seed logarithm, while the middle contributes once.

Equal sums of two pairs of positions

These rows require four distinct selected positions and the stated equality of their sums.

The seed decompositions are

$$\begin{gathered} 22=2\cdot11,\quad21=3\cdot7,\quad 56=2^3\cdot7,\quad55=5\cdot11,\\ 3025=5^2\cdot11^2,\quad432=2^4\cdot3^3,\quad 99=3^2\cdot11,\quad98=2\cdot7^2,\\ 3267=3^3\cdot11^2,\qquad1225=5^2\cdot7^2. \end{gathered}$$

As in the earlier four-factor lemma, equality of the position sums makes the difference of the corresponding quadratic products constant. Distinctness of the four positions makes this constant nonzero.

A cubic constant-gap identity

There is also the exact identity

$$Z(Z-3d)^2-(Z-d)^2(Z-4d)=4d^3.$$

Suppose the selected positions of primes $11,7,3,2$, in that order, are $a,a+d,a+3d,a+4d$, with integer $d\ne0$. The two products above then have degree three, with factor multiplicities $(1,2)$ and $(2,1)$. Use the seed $99-98=1$, where $99=3^2\cdot11$ and $98=2\cdot7^2$, and the Padé parameter $s=5/4$.

Matching all four prime-power blocks gives limiting cofactor exponents

$$1-\frac{3b_p\log p}{w_pK},$$

where $b_p$ is the prime exponent in the seed and $w_p$ is its factor multiplicity. This follows from the same determinant inequality as the quadratic argument: the two original products now have size at most $n^3$, so its degree factor is three. The checker proves the additional alternative

$$u_2>h^{.736}\quad\text{or}\quad u_3>h^{.5818}\quad\text{or}\quad u_7>h^{.2593}\quad\text{or}\quad u_{11}>h^{.0873}.$$

The position hypothesis allows either sign of $d$. The product gap is nonzero and at most $4\cdot11^3$, which is the bound used in the threshold calculation.

Certificates and thresholds

The checker evaluates every fixed parameter choice with the retained explicit coefficient-gcd estimate, using $m_0=10^8$ and 4000 prime intervals. It verifies $\Omega_3,\Omega_4>1$ and a margin of $10^{-4}$ between each target and its limiting exponent. The Padé parameters and all interval values are in the recorded certificate.

It also applies the general numerical product-threshold formula with maximum position $11$. The quadratic rows use degree $2$ and product gap $121$; the cubic row uses degree $3$ and product gap $4\cdot11^3$. Consequently every table entry holds uniformly, under its position condition, whenever

$$h\ge10^{9\,897\,968\,998}.$$

This is a convenient common threshold, not an optimized one. Individual thresholds are retained in the certificate. The calculation verifies fixed analytic parameters and does not enumerate binomial rows.

The position conditions are essential. In particular, these estimates must not be applied to arbitrary assignments of the selected maxima without checking the AP or equal-sum relation.