Erdős 699: the four-cofactor comparison 225−224

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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 · Rational product bounds

Let $0\le r_p\le13$ for $p\in\{2,3,5,7\}$, and write

$$n-r_p=p^{a_p}u_p,\qquad h=n-13.$$

Suppose

$$r_2+r_7=r_3+r_5,\qquad \{r_2,r_7\}\ne\{r_3,r_5\},$$

where the unordered pairs include multiplicity. For $h\ge10^{720\,587\,641}$, the following alternative holds:

$$\boxed{u_2>h^{.35}\quad\text{or}\quad u_3>h^{.35}\quad\text{or}\quad u_5>h^{.03}\quad\text{or}\quad u_7>h^{.35}.} \tag{1}$$

This is a restriction on one position relation, not an effective bound for an entire index.

Proof and numerical data

Put $X=(n-r_3)(n-r_5)$ and $Y=(n-r_2)(n-r_7)$. The position assumptions give

$$0<|X-Y|=|r_3r_5-r_2r_7|\le169.$$

Apply the rational product estimate with

$$P=225=3^2 5^2,\qquad Q=224=2^5 7,\qquad P-Q=1, \qquad c/d=5/3.$$

The exact coefficient-gcd calculation, valid for every $m\ge10^8$, gives

$$\log L_1=481050309501/10^{12}.$$

Using the notation of the rational product lemma, its directed interval evaluation gives

$$C=1.62240418258714\ldots,\qquad K=8.05108979481149\ldots,\qquad J=K-C=6.42868561222434\ldots.$$

For the upper cofactor box complementary to (1), choose the rational block parameter

$$t=18287447141/60685589176.$$

The denominator cost $\Delta$ satisfies

$$tC-\Delta>.08238,\qquad tK-\Delta-2>.01965.$$

Both terms in the Padé upper bound therefore tend to zero. The explicit threshold formula in the rational product lemma yields the displayed power of ten, proving (1).

The checker and certificate retain the coefficient-gcd bound, rational block parameter, directed margins, and threshold terms. They enumerate no binomial rows.

For example, the labeled positions $(r_2,r_3,r_5,r_7)=(1,3,0,2)$ satisfy the required relation. The cofactor profile $(.2757,.2921,0,.2667)$ left by the earlier separate endpoint bounds lies strictly inside the excluded box. Other profiles remain, so this observation alone does not settle index nine.