Erdős 699: numerical bounds for three and four cofactors

P699 research · Erdős problem 699 · AP and equal-sum cofactor bounds

The three- and four-factor estimates have fully numerical versions. For selected factors among $n,n-1,\ldots,n-9$, the five conditional constraints in the table below all apply once $n\ge10^{557368}$. This is a common threshold for these product constraints; it is not by itself a stopping bound for every bad triple at index ten. A full index-ten argument must also instantiate the other prime-pair estimates used in its interpolation certificates.

1. A numerical product lemma

Write selected numerator factors as

where the primes $p$ are distinct, $u_p$ are positive integers, and $R\ge1$. Partition these primes into two nonempty sets $I_0,I_1$ and choose positive integer multiplicities $b_p$ with

Suppose the corresponding products have bounded difference:

Choose positive integers $A_p$ and a BFT seed

The symbols $b$ and $b_p$ have different roles: $b$ is the fixed seed coefficient, while $b_p$ counts occurrences of a numerator factor. Take any explicit BFT parameters $s=c/d,L_1,m_0$ for which the coefficient gcd estimate holds and $\Omega_3,\Omega_4>1$. Use the constants $\kappa_1,\kappa_2$ from BFT Theorem 2.4 for the gap $D$.

For target exponents $0<t_p<1$, put

Assume $\nu L<K$. Define

Then

Proof. If every $u_p\le h^{t_p}$, then $p^{k_p}\ge h^{1-t_p}$. Match the seed blocks by choosing

Each product in (1) is respectively $P^{cm}U$ or $Q^{cm}V$, with positive integral residual cofactors. The absolute-gap integer Padé argument in BFT Section 7, before its final cofactor maximum, gives

This is the same step used in the asymmetric two-factor lemma. It uses the common block $m$ and does not require each residual cofactor to be a power of one original cofactor.

On the other hand, $h>R$ implies $Q^{cm}V\le(h+R)^\nu<2^\nu h^\nu$. Therefore

contradicting (2). This proves (3).

The limiting target at the coordinate $p$ is

For an AP identity, the middle factor has multiplicity two and both endpoint factors have multiplicity one. For the equal-sum identity, all four multiplicities are one. Either product may correspond to the larger seed; the absolute-gap argument handles both orientations.

2. Numerical certificates for the index-ten range

Put $h=n-9$, so $R=9$, and use $D=100$. The following table lists conditional alternatives: at least one of the listed cofactor inequalities holds whenever the condition on the positions is satisfied.

The AP conditions are used with distinct positions, giving a square minus a product equal to a nonzero fixed squared gap. The last condition is used with four distinct positions and gives

With all positions in $[0,9]$, these gaps are at most $100$. The source parameters are:

The first three gcd estimates are BFT Proposition 5.1 entries. The $7-6$ entry uses the Dusart refinement of the gcd bound. The last uses the explicit prime-interval gcd lemma with $T=1500$. It is a finite rational certificate valid for every $m\ge3000$, not a check of finitely many Padé degrees.

All five entries are evaluated by the direct product-threshold checker. Its recorded output contains the prime components of each seed, multiplicities, exact gcd certificate, directed intervals for (2), and the verified integer powers of ten. A floating approximation proposes each integer exponent; the final inequality $H<B\log10$ is checked with directed intervals.

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_product_thresholds.py

3. Numerical pair inputs for the same index

The near-limiting pair constraints used alongside these product bounds also have evaluated thresholds. The finite prime-interval bootstrap reduces the valid coefficient-gcd starting values while preserving their exact lower exponents. Re-evaluating the direct asymmetric threshold gives:

The input height is the minimum of the two selected factors, and the gap bound is $100$. Each table entry is an asymmetric disjunction. Its two endpoints dominate both the symmetric and asymmetric source requirements after the indicated weakening by $.001$ in the index-ten construction.

The pair-threshold checker regenerates the analytic tails, matches their exact exponents to the retained finite bootstrap, and then evaluates the smaller numerical thresholds. Its output records all six certificates. Hence every pair constraint in this table and every product constraint above applies whenever

This is the common evaluated source threshold; the interpolation argument has its own algebraic cutoff.

4. A conditional constraint relevant to index fourteen

The same construction gives another useful source estimate. Suppose the four distinct selected positions for $2,3,11,13$ lie in $[0,13]$ and satisfy

Use the seed

with $s=115/94$ and the exact Dusart coefficient-gcd bound at $m_0=10^8$. The checker and its certificate prove that, for $h=n-13\ge10^{21602776816}$,

The equal-sum product gap is nonzero and at most $42$, so the checked bound $D=100$ applies. This is a conditional restriction on labeled positions. It does not establish a whole-index result at fourteen; other position patterns and every branch of the disjunction still require coverage.