Erdős 699: closing a terminal prime-power orthant analytically

P699 research · Prime-power strips · Finite index seven · Finite index nine

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A weaker global cofactor estimate can finish a stronger finite prime-pair calculation. It is enough to show that the remaining exponent orthant contains no positive solution at the proposed height.

Lemma. Fix distinct primes $p,q$, a difference bound $D$, and an independently proved numerical estimate

$$|p^a u-q^b v|\le D,\qquad h=\min(p^a u,q^b v)\ge T \quad\Longrightarrow\quad u\ge h^\alpha\ \text{or}\ v\ge h^\beta, \tag{1}$$

where $u,v$ are positive integers and $0<\alpha,\beta<1$. Suppose the unhandled exponents satisfy $a\ge A,b\ge B$, and write $P_0=p^A,Q_0=q^B$. If

$$\min(P_0,Q_0)\ge\max(T,D),\qquad P_0>2H^{1-\alpha},\qquad Q_0>2H^{1-\beta}, \tag{2}$$

then there is no positive solution in this orthant with both heights at most $H$.

Proof. Every such solution has $h\ge\min(P_0,Q_0)\ge T,D$. Both heights are at most $h+D\le2h$, while $h\le H$. Therefore

$$u\le\frac{2h}{P_0}<\frac h{H^{1-\alpha}}\le h^\alpha, \qquad v\le\frac{2h}{Q_0}<\frac h{H^{1-\beta}}\le h^\beta,$$

contradicting (1). $\square$

The global exponents in (1) need not match or exceed the exponent being certified by the finite strip calculation. The entire remaining orthant is impossible, so it contributes no exception to that finite calculation.

To apply the lemma after a retained strip computation, its already removed slabs and boundary lines must cover their unbounded transverse ranges. The continuation checks establish this: a potentially bad minimal solution on a boundary with fixed power $P$ has height less than $2P^{1+s/(s-r)}$. Verifying that this is below the old height cap shows that neither the old exponent cutoff nor its height filter truncated a boundary. The saved final exponents then specify exactly the remaining orthant.

There is no circular use of the binomial conjecture. Estimate (1) must be a numerical theorem about prime powers and arbitrary positive cofactors. The retained numerical pair estimates provide such statements; their filename refers to an earlier application, not to an assumption that index ten or any other binomial index is true. Their difference bound is $100$, covering the bounds six and eight needed here. Coincident heights are also harmless: coprimality gives $uv\ge h$, which supplies the relevant alternative when $\alpha+\beta<1$.

For $H=10^K$, conditions (2) require only short directed logarithms:

$$A\log p>\log2+K(1-\alpha)\log10,\qquad B\log q>\log2+K(1-\beta)\log10.$$

The checker and output verify these conditions against every saved final orthant and every numerical source threshold. They perform no new modular inversions. The resulting common extensions of the pair ranges are:

The original exception lists remain sufficient. No new lower-height gap is introduced: entering the terminal orthant already forces the analytic source threshold. These additional pair ranges are available for the same weighted reductions if a larger finite bridge is needed; the already completed index-seven and index-nine finite aggregate certificates are not modified here.