Erdős 699: two additional prime-power seeds for index 12

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

The rational-block inequality can use coefficients other than one in a Padé seed. Two useful choices are

Theorem 2.4 of BFT permits arbitrary positive integer coefficients $a,b$. They need not be powers of either prime. Their contribution is already included in the definitions of $\Omega_3$ and $\Omega_4$.

Fix a gap bound $D\ge1$, and write

For a seed $P=p^{k_0},Q=q^{l_0}$ in either order, put

Above an effective threshold, the rational-block inequality requires

The following three choices give strict contradictions throughout the indicated closed boxes. Here $\ell=\log L_1$ is an exact rational number, and the displayed margin is $C$ minus the sum of the two positive parts at the upper corner of the box. Monotonicity then handles every smaller cofactor exponent.

One may take $\epsilon=.24,.32,.48$, respectively. The directed interval calculation verifies $\Omega_3,\Omega_4>1$, $J>0$, $1<s<1/z_0$, and $\ell<\log L(s)$ in each case. The strict final inequality supplies an effective coefficient-gcd starting point by BFT Theorem 2.2. Thus these are effective restrictions, even though their numerical height thresholds have not been evaluated here.

Ordinary cofactor alternatives

The same two seeds also give the simpler alternatives from the asymmetric matched-block argument. For each prime power $p^{k_0}$ in a seed, its limiting endpoint is

A stronger ordinary alternative comes from the first seed with $s=53/31$ and $\ell=255299/500000=.510598$. Its endpoints are

Thus, above an effective height threshold,

The directed interval calculation gives $\log\Omega_3>.0007025768$ and $\log L(53/31)>.5105995243>\ell$, so all required inequalities remain strict despite the small gap in $\Omega_3>1$.

For comparison, the first seed with $s=61/33$ and $\ell=.577$ gives

Consequently, above an effective height threshold,

For the second seed, taking $s=80/39$ and $\ell=347257/500000=.694514$ gives the stronger endpoints

Thus one also has the effective alternative

The checker certifies $\log\Omega_3>0$ and $\ell<\log L(80/39)$; the strict rational-block margin at this box exceeds $.00000922$.

The earlier parameters $s=37/16$ and $\ell=.605$ give

It therefore gives the effective alternative

Here $u_p$ denotes the positive integer cofactor of the relevant prime power in two positive integers whose difference is bounded by $D$; $x$ is their minimum. The retained checker certifies all exact rational endpoint targets. These ordinary alternatives suffice wherever only an OR condition on individual cofactor exponents is needed.

These boxes arose from the remaining index-12 interpolation profiles, in prime order $2,3,5,7,11$:

The calculation in this note excludes those boxes. A proof for a whole index additionally needs complete coverage of all cofactor and position cases and the corresponding interpolation certificates.

The retained checker and interval output contain the exact seeds and parameters. They can be reproduced with

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_twelve_rational_pairs.py