Erdős 699: a column restriction in the difficult index-eight region

P699 research · The remaining cofactor profile · Actual common-prime valuations

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The difficult cofactor configuration can be restricted using the actual column $j$. The following result is uniform in the row; it does not search through finitely many rows.

Proposition. Suppose $(n,8,j)$ is bad, put $h=n-7$, and choose maximal small-prime positions with

$$(r_2,r_3,r_5,r_7)=(1,3,0,2).$$

Write $P_p=p^{v_p(n-r_p)}$ and $u_p=(n-r_p)/P_p$. Assume

$$u_2,u_3,u_5\ge h^{3/10}. \tag{1}$$

There is no hypothesis on $u_7$. If $h\ge10^{850001}$, then

$$\boxed{j\bmod P_7>2.} \tag{2}$$

Moreover, writing $c=n-j$ and using $[a]_p=p^{v_p(a)}$,

$$\left[\binom nj\right]_7>\sqrt h, \qquad [j(j-1)(j-2)]_7<\sqrt h, \qquad [c(c-1)(c-2)]_7<\sqrt h. \tag{3}$$

Thus the low-residue Lucas branch cannot realize this cofactor region at large height. The alternative in (2) remains possible; this proposition does not make the whole index effective.

An exact column valuation identity

Here is a general lemma independent of badness. Let $p$ be prime, $0\le r<p$, $r<j\le n$, and write

$$n-r=p^A u,\qquad A\ge1,\quad p\nmid u.$$

Then

$$\boxed{ p^{v_p\binom nj}\,[j(j-1)\cdots(j-r)]_p\le n-r. } \tag{4}$$

To prove it, use

$$(j)_{r+1}\binom nj =(n)_{r+1}\binom{n-r-1}{j-r-1}.$$

Exactly one numerator factor in $(n)_{r+1}$ is divisible by $p$, so its valuation is $A$. The bottom $A$ base-$p$ digits of $n-r-1=p^Au-1$ are all $p-1$. Kummer therefore gives

$$v_p\binom{n-r-1}{j-r-1} =v_p\binom{u-1}{\lfloor(j-r-1)/p^A\rfloor} \le\lfloor\log_pu\rfloor.$$

For $u=1$ the middle valuation is zero, as required. Taking valuations in the displayed binomial identity proves (4). The same bound holds with $j$ replaced by $n-j$ whenever that column exceeds $r$.

In particular, if $j\bmod p^A\le r$, the falling factorial on the left contains a multiple of $p^A$. Hence $p^{v_p\binom nj}\le u$. For

$$E_p=p^{\min\{A,v_p\binom nj\}},$$

this implies

$$E_p\le\min(p^A,u)\le\sqrt{n-r}. \tag{5}$$

Equation (4) also retains information in the other Lucas branch, where (5) need not hold.

The interpolation certificate

Use coordinate factors

$$H(X,Y)=\prod_{a=0}^7((X-a)(Y-a))^{h_a},$$

with integer vectors

$$(h_a)=(39,29,21,10,0,0,0,0),$$ $$(m_t)=(102,81,73,56,56,46,40,33).$$

At a grid point $x+y=t<8$, prescribe residual multiplicity

$$R_{xy}=\max\{0,m_t-h_x-h_y\}.$$

The degree-84 symmetric jet matrix is $1845\times1849$ and has full row rank modulo $251$. The nonvanishing interpolation lemma therefore provides a finite integer family $G$ of degree at most 85 with these residual multiplicities, at least one of which is nonzero at every physical point $(j,c)$ with $j+c>7$. Coordinate factors cannot vanish there, since $j,c\ge9$.

Consequently one of the corresponding $F=HG$ is nonzero, has degree at most

$$D=2\sum h_a+85=283,$$

and has the row multiplicities $m_t$. The integer data and the exact rank calculation are retained in the checker and certificate.

Put $B=\binom nj$ and

$$E_7=7^{\min\{v_7(n-2),v_7(B)\}}.$$

The exact localization inequality, retaining this actual overlap, gives

$$\prod_{t=0}^7(n-t)^{m_t} \le (7!)^{102}|F(j,c)|\, P_2^{81}P_3^{56}P_5^{102}E_7^{73}. \tag{6}$$

The contribution of every nonmaximal small-prime power is covered by the factor $(7!)^{102}$. The prime threshold remains the original $p\ge8$, which here means primes at least 11.

The sum of the row multiplicities is 487. By (1), each of $P_2,P_3,P_5$ is at most $2h^{7/10}$. Also $n\le2h$. A deliberately generous coefficient bound obtained from a nonzero maximal jet minor gives

$$h^{367/10}\le10^{169000}E_7^{73}. \tag{7}$$

For completeness, a jet entry is smaller than $2\cdot8^{84}$. Hadamard, adjugates, and the generator construction bound the coefficient norm of $G$ by $10^{1845(84+6)+50}$. The logarithmic bounds for $H$, $(7!)^{102}$, and the replacements $n\le2h$ add respectively $198$, $408$, and $283+239$. Their total is $167228<169000$. The remaining height exponent in (6) is exactly

$$487-283-\frac7{10}(81+56+102)=\frac{367}{10}.$$

If the low-residue branch in (2) held, (5) would give $E_7\le\sqrt{n-2}\le\sqrt{2h}$. Then (7) would imply

$$h^{1/5}\le10^{169000}2^{73/2}<10^{170000},$$

contrary to $h\ge10^{850001}$. This proves (2).

Finally, (7) gives

$$E_7\ge10^{-169000/73}h^{367/730}.$$

Combine this with (4), first at $j$ and then at $c$. Since $7^{v_7(B)}\ge E_7$ and $n-2\le2h$,

$$[(j)_3]_7,[(c)_3]_7 \le2\cdot10^{169000/73}h^{363/730}.$$

At the stated height these quantities are strictly below $\sqrt h$; the preceding lower bound for $E_7$ is strictly above $\sqrt h$. This proves (3).