Erdős 699: rational Padé blocks and a cofactor tradeoff

P699 research · The asymmetric BFT bounds

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The Padé block in BFT Section 7 can exceed one of the available prime-power blocks. The remaining cofactor is then rational; clearing its denominator retains a useful integer inequality. This yields a piecewise linear restriction beyond the earlier endpoint alternatives.

One concrete consequence is the following effective improvement:

For every fixed $D\ge1$, there is an effectively computable $x_0(D)$ such that

$$|2^a u-3^b v|\le D,\qquad x=\min(2^a u,3^b v)\ge x_0(D)$$

implies

$$\boxed{\max(u,v)>x^{.2926}.}$$

Here $u,v$ are positive integers and $a,b$ are nonnegative integers.

The numerical value of $x_0$ has not been evaluated. Since $u,v\ge1$, the same conclusion gives $uv>x^{.2926}$. This is a small improvement over the retained exponent $.2921$; it is far from the product exponent $5/6$ that would directly settle the remaining quartic configurations. No additional whole index follows from this note.

An integer inequality with rational residual cofactors

Fix valid BFT Theorem 2.4 parameters, with seed

$$P=p^{k_0},\qquad Q=q^{l_0},\qquad aP-bQ=D_0>0,\qquad s=c/d.$$

Here $p,q$ are distinct primes. Suppose the coefficient-gcd estimate holds for every $m>m_0$, with constants $L_1,\kappa_1,\kappa_2$ and $\Omega_3,\Omega_4>1$. Write

$$A=\log P,\qquad B=\log Q,\qquad K=\max(A,B)+\frac{\log\Omega_4}{s}.$$

Let $X=p^k u$, $Y=q^l v$ be positive integers with $|X-Y|\le D$. For any integer $m>m_0$, define

$$U_m=X/P^{cm},\qquad V_m=Y/Q^{cm},$$ $$Z_m=p^{(k_0cm-k)_+}q^{(l_0cm-l)_+},\qquad \Delta_m=\log Z_m =(cmA-k\log p)_++(cmB-l\log q)_+. \tag{1}$$

The notation $z_+=\max(z,0)$ denotes the positive part. The factor $Z_m$ clears both residual denominators.

The two BFT Padé choices have integral coefficients $\mathcal P_m,\mathcal Q_m,\mathcal E_m$ satisfying

$$(aP)^{cm}\mathcal P_m-(bQ)^{cm}\mathcal Q_m=\mathcal E_m.$$

Their determinant identity ensures that at least one choice has

$$T_m=b^{cm}\mathcal Q_m U_m-a^{cm}\mathcal P_m V_m\ne0.$$

This nonvanishing argument works for positive rational $U_m,V_m$ too. Since $Z_mT_m$ is a nonzero integer,

$$1\le Z_m|T_m| \le Z_m\left( \frac{Db^{cm}|\mathcal Q_m|}{P^{cm}} +\frac{|\mathcal E_m|Y}{(PQ)^{cm}} \right). \tag{2}$$

The same estimates used in BFT (7.4)--(7.6) give

$$\frac{Db^{cm}|\mathcal Q_m|}{P^{cm}} \le\frac{\kappa_1}{2}\Omega_3^{-dm},\qquad \frac{|\mathcal E_m|}{(PQ)^{cm}} \le\frac{1}{2\kappa_2}e^{-cmK}.$$

Consequently

$$\boxed{ 1\le\frac{\kappa_1}{2} e^{\Delta_m-dm\log\Omega_3} +\frac{Y}{2\kappa_2}e^{\Delta_m-cmK} \quad\text{for every integer }m>m_0. } \tag{3}$$

The source argument corresponds to choosing $m$ so that $Z_m=1$. Equation (3) also permits larger blocks, with their exact denominator cost.

A directly usable positive-part restriction

Put

$$C=\frac{\log\Omega_3}{s},\qquad J=K-C,$$

and suppose $J>0$. For $x=\min(X,Y)>1$, define

$$e_p=\frac{\log u}{\log x},\qquad e_q=\frac{\log v}{\log x},$$ $$T(e_p,e_q)=(A-J(1-e_p))_++(B-J(1-e_q))_+.$$

For every $\epsilon>0$, sufficiently large inputs necessarily satisfy

$$\boxed{T(e_p,e_q)>C-\epsilon.} \tag{4}$$

The threshold is effective. In fact it suffices to take

$$x>\max\left\{ D,\ e^{cJ(m_0+1)},\ \left(\frac{\kappa_1}{2}e^{cC} +\frac{e^{cK}}{\kappa_2}\right)^{J/\epsilon} \right\}. \tag{5}$$

Proof. Suppose $T(e_p,e_q)\le C-\epsilon$ and choose $m=\lfloor\log x/(cJ)\rfloor$. Then $m>m_0$ by (5). Because $X,Y\ge x$, equation (1) gives

$$\Delta_m\le \frac{T(e_p,e_q)}J\log x.$$

Also $cm>\log x/J-c$, and $Y\le x+D<2x$. The right side of (3) is therefore at most

$$\left(\frac{\kappa_1}{2}e^{cC} +\frac{e^{cK}}{\kappa_2}\right)x^{-\epsilon/J}<1,$$

a contradiction. $\square$

This is a sufficient contradiction criterion obtained from one particular block size. It is not asserted to describe the complete optimum over all $m$: the function $Kt-\Delta(t)$ can decrease after both available blocks have been exceeded.

When $J>\max(A,B)$, the two coordinates in (4) have positive transition values $1-A/J$ and $1-B/J$. For example, if $e_q\le1-B/J$, then (4) requires

$$e_p>\frac{K-A-\epsilon}{J}. \tag{6}$$

When both positive parts occur, (4) instead gives

$$e_p+e_q>2-\frac{A+B-C+\epsilon}{J}. \tag{7}$$

These are conditional parts of one restriction; neither is a simultaneous unconditional lower bound on both cofactors.

The seed $9-8=1$

Take $P=9$, $Q=8$, $a=b=1$, $c/d=3/2$, and

$$\log L_1=\frac{308}{625}=.4928.$$

The retained coefficient-gcd formula gives

$$\log L(3/2)>.49283363527>.4928.$$

Thus BFT Theorem 2.2 supplies an effective $m_0$ for this strictly smaller $L_1$. The directed interval checker certifies

$$\begin{aligned} C&=.1178365156647256\ldots,\\ K&=3.0576959900410894\ldots,\\ J&=2.9398594743763638\ldots>\log9>\log8. \end{aligned}$$

In prime order $3,2$, the transition values are

$$1-\frac{\log9}{J}=.2526089779164293\ldots, \qquad 1-\frac{\log8}{J}=.2926731499229395\ldots.$$

If both cofactors have exponent at most $.2926$, the second positive part in (4) vanishes. The first falls below $C$ by more than $.0002685$. Taking, for example, $\epsilon=.0002$ in (4) proves the displayed maximum-cofactor improvement.

The limiting intercept when the $2$-cofactor lies below its transition value is

$$\frac{K-\log9}{J}=.2926913412714745\ldots.$$

This also gives a product bound directly: for nonnegative $e_3,e_2$, the sum of the two positive parts is at most $J(e_3+e_2-(1-\log9/J))_+$. Thus every fixed exponent smaller than $.29269134127$ is an effective lower exponent for $uv$.

The proof uses the identities and coefficient bounds in BFT Section 7. The checker and directed interval output verify the strict coefficient-gcd and exponent margins. They do not assign a numerical value to the effective threshold.