Erdős 699: effective exclusion of small cofactors at indices four and five

P699 research · The effectivity reduction

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For $i\in\{4,5\}$, choose positions $r_2,r_3\in\{0,\ldots,i-1\}$ maximizing the respective valuations, and write

$$n-r_2=2^a u,\qquad n-r_3=3^b v.$$

There are effectively computable absolute constants $K>0$ and $N_i$ such that every bad triple at index $i$ with $n\ge N_i$ satisfies

$$\boxed{\min(u,v)> \frac14\exp\!\left(\frac{\delta_i\log n} {2K\log(3\log n)}\right),\qquad \delta_4=\frac16,\quad\delta_5=\frac25.}\tag{1}$$

In particular, for every fixed real $A\ge0$, all bad triples with $\min(u,v)\le(\log n)^A$ have an effectively computable upper row bound. The exponents $a,b$ are unrestricted. This is an effective exclusion of an infinite family of rows; the range of larger cofactors remains, so neither entire index becomes effective by this argument.

A uniform two-prime valuation estimate

We use Yu's theorem in the form stated as Theorem 3.2 in Bacik, arXiv:2409.01221v5, §3.1; the source provenance section below records what is held. For $q\in\{2,3\}$ there is an effectively computable $C_q>0$ such that, for nonzero rational $q$-adic units $\alpha_1,\alpha_2$ and integers $e_1,e_2$, if $\Xi=\alpha_1^{e_1}\alpha_2^{e_2}-1\ne0$, then

$$v_q(\Xi)\le C_q A_1A_2\log B,\qquad A_j\ge\max\{h(\alpha_j),\log q\},\qquad B\ge\max\{|e_1|,|e_2|,3\}.\tag{2}$$

Here $h(m/d)=\log\max(|m|,|d|)$ for a reduced nonzero rational number. The theorem requires nonvanishing of $\Xi$, not multiplicative independence of its bases. Thus rational dependencies cause no missing case below.

Suppose $\{p,q\}=\{2,3\}$ and

$$p^a u-q^b v=d,\qquad 0<|d|\le4,\qquad u,v\ge1.$$

Put $s=v_q(d)$. If $b\le s$, then $b\log q\le\log4$. Otherwise $v_q(u)=s$, since $p$ is a $q$-adic unit. Set $u_0=u/q^s$ and $d_0=d/q^s$. Applying (2) to

$$\alpha_1=p,\quad\alpha_2=u_0/d_0,\quad e_1=a,\quad e_2=1,\quad \Xi=p^a u_0/d_0-1=q^{b-s}v/d_0$$

is legitimate: both bases are $q$-adic units and $\Xi\ne0$ because $v>0$. Its valuation is at least $b-s$; equality holds when $q\nmid v$. We may take $A_1=\log3$ and $A_2=\log(4u)$. Consequently, with

$$K=4\bigl(1+\max(C_2,C_3)\bigr),$$

both branches give the convenient uniform bound

$$\boxed{\log(q^b)\le K\log(4u)\log\max(a,3).}\tag{3}$$

This includes the cases $q\mid d$, negative $d$, and bases equal to $1$ or $-1$. If one prefers to omit a base equal to $1$, the one-base instance of Yu's theorem gives the same statement after increasing $K$ effectively.

Applying the quartic and quintic certificates

The retained polynomial estimates imply that badness forces

$$2^a3^b>\kappa_i n^{1+\delta_i},\tag{4}$$

where one may take

$$\begin{array}{c|c|c|c} i&\delta_i&\kappa_i&\text{range}\\ \hline 4&1/6&4^{5/6}/12&n\ge12\\ 5&2/5&4^{6/5}/240&n\ge20. \end{array}$$

For $i=4$, these are exactly inequalities (11) and (12) of the effectivity note. For $i=5$, its inequality (5) has $m=5$, $A=6$, $d=18$, so $V_5<n^{18/5}/4^{6/5}$. Also $\binom n5\ge n^5/240$ for $n\ge20$, whence $U_5>(4^{6/5}/240)n^{7/5}$. The off-maximum valuation estimate in that note gives $U_5\le2^a3^b$: the off-maximum factors contribute at most $[4!]_{\{2,3\}}=[5!]_{\{2,3\}}=24$ before division by $5!$. This proves (4) in both cases.

If $r_2=r_3$, then $2^a3^b\mid n-r_2\le n$, contradicting (4) as soon as $\kappa_i n^{\delta_i}\ge1$. Hence the positions are distinct in the range below, and $d=r_3-r_2$ is nonzero with $|d|\le4$.

Let

$$N_i=\max\{20,\ \kappa_i^{-2/\delta_i}\}.$$

For $n\ge N_i$, use $2^a=(n-r_2)/u\le n/u$ in (4). Then

$$\log(3^b)> \delta_i\log n+\log u+\log\kappa_i \ge\frac{\delta_i}{2}\log n+\log u. \tag{5}$$

On the other hand, $a\le\log n/\log2$ gives $\log\max(a,3)\le\log(3\log n)$. Combining (3) and (5), and dropping the nonnegative $\log u$, proves the lower bound in (1) for $u$. Interchanging the primes and replacing $d$ by $-d$ proves the same bound for $v$.

Effectivity of the logarithmic cofactor exclusion

Given $A\ge0$, choose an integer $T\ge4$ large enough that

$$e^T>\frac{2K}{\delta_i} (T+\log3)(\log4+AT).\tag{6}$$

Such a $T$ can be found effectively. The ratio of the two sides of (6) is increasing for $T\ge4$, since the logarithmic derivative of $e^T/((T+\log3)(\log4+AT))$ is positive there. For

$$n\ge\max\{N_i,\exp(e^T)\},$$

equation (1) therefore gives $u,v>(\log n)^A$. No numerical value of the Yu constants, and hence no numerical stopping bound for this subfamily, is evaluated here.

Source provenance

The form of Yu's theorem used above is Theorem 3.2 of Bacik, arXiv:2409.01221v5, §3.1, which cites Kunrui Yu, p-adic logarithmic forms and group varieties II, Acta Arithmetica 89 (1999), 337–378, Theorem 1, DOI. Neither source is held in the library: the repository has no card for Bacik's preprint or for Yu's paper, the reading copy of the preprint from which the statement was taken is not retained here, and the statement has not been checked against Yu's paper. Only this established logarithmic-form theorem is used here; no claim about the Skolem problem is needed.