Erdős 699: a three-cofactor alternative from 49−48

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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 · Unequal Padé blocks

Fix $D\ge1$. Suppose $0\le r_2,r_3,r_7\le D$ satisfy

$$r_2+r_3=2r_7,\qquad r_2\ne r_3,$$

and write $n-r_p=p^{a_p}u_p$ for $p=2,3,7$. There is an effectively computable $H(D)$ such that, for $h=n-D>H(D)$,

$$\boxed{u_2>h^{.36}\quad\text{or}\quad u_3>h^{.19}\quad\text{or}\quad u_7>h^{.15}.} \tag{1}$$

The starting height has not been numerically evaluated. This is a conditional cofactor estimate, not a whole-index assertion.

Unequal blocks at the AP identity

The two products

$$X=(n-r_7)^2,\qquad Y=(n-r_2)(n-r_3)$$

satisfy $0<X-Y=(r_2-r_7)^2\le D^2$. Use the seed

$$P=49=7^2,\quad Q=48=2^4\cdot3,\quad P-Q=1,$$

with Padé directions $(A,B,C)=(49,12,20)$. Its two block lengths are $A+B=61$ and $B+C=32$. The finite coefficient-gcd sum through $t=200$ is greater than $5.5496$. Thus the strictly smaller logarithmic gcd rate $\ell=5$ has an effective starting index.

The directed growth calculation gives

$$C_0>9.9872414455,\qquad K_0>285.7745674357.$$

Set $\tau=29/4000$. For the cofactor box complementary to (1), the denominator loss is

$$\Delta=\bigl(122\tau\log7-2(1-.15)\bigr)_+ +\bigl(128\tau\log2-(1-.36)\bigr)_+ +\bigl(32\tau\log3-(1-.19)\bigr)_+.$$

The retained interval calculation certifies

$$\tau C_0-\Delta>.04800,\qquad \tau K_0-\Delta-2>.04746.$$

Choose the Padé index proportional to $\tau\log h$. The nonzero integer from the unequal-block lemma is then bounded by two terms tending effectively to zero, with the displayed positive powers. This is a contradiction for effectively large $h$, proving (1).

The integer and interval checker and certificate record the exact finite gcd sum, strict gcd margin, rational block parameter, and directed bounds. No binomial rows are enumerated.

At the labeled positions $(r_2,r_3,r_5,r_7)=(2,0,3,1)$, the required AP is present. In particular (1) excludes the cofactor box containing $(.355407,.174272,.463949,.141301)$ in prime order $2,3,5,7$; the cofactor $u_5$ is unrestricted by this source.