Erdős 699: numerical thresholds for stronger prime-pair bounds

P699 research · Erdős problem 699 · Small-index effectivity

The stronger prime-pair estimates need not retain an unevaluated source constant. A finite sum of explicit prime-counting estimates supplies a numerical lower bound for the coefficient gcd in the Padé construction of Bennett, Filaseta and Trifonov (BFT). Their Theorem 2.4 then gives a numerical threshold. In particular, the argument below supplies the unconditional row bound

This is a very large stopping bound. Its significance is that every constant has been evaluated. Smaller numerical prime-pair thresholds below also apply to the weighted certificates at other indices.

The input is the BFT author preprint, dated 26 February 2007, associated with the 2009 article On the factorization of consecutive integers. References to theorem numbers below concern that preprint.

1. An explicit coefficient-gcd bound

Take relatively prime integers $c>d\ge1$, put $H=c+d$, and fix positive integers $T,m_0$ with $m_0\ge2T$. BFT's coefficient gcd is

For each $1\le t\le T$ satisfying $\{dt/H\}>1/2$, define

The denominators are positive. BFT Lemma 5.2 and the floor identities (5.7)--(5.8) show that every prime in

divides $G(c,d,dm-\delta)$ for both values of $\delta$, whenever $m\ge m_0$. Indeed, the actual lower endpoint from that lemma is at most $mv_t$, and its actual upper endpoint is at least $(Hm-2)/t$.

These prime intervals are disjoint. The identity $2(k_t+1)+(t-2k_t-1)=t+1$ gives $v_t\ge H/(t+1)$, so (3) lies inside $(Hm/(t+1),Hm/t]$.

Write $\eta=213/10^6$. BFT Lemma 5.4 gives

Consequently $\theta(y)\le(1+\eta)y$ for every $y\ge1$. Set $X_t=(Hm_0-2)/t$ and choose any rational number

Then $\theta(x)\ge(1-e_t)x$ for every $x\ge X_t$: the square-root estimate handles $x<10^8$, and the relative estimate handles larger $x$. The ranges in (4) overlap; there is no upper restriction on $m$.

Summing the prime products in the disjoint intervals proves

where any rational lower bound for

can be used as $\ell$. To see this directly, the logarithm of the prime product in (3), divided by $m$, is at least its summand in (6), before the factor $1/d$. Terms with a negative lower bound may be discarded.

The checker uses $T=4000$ and forms $e_t$ using integer square roots:

Thus $w_t/10^6\le\sqrt{X_t}$ exactly. Each positive normalized summand in (6) is rounded down to a multiple of $10^{-12}$. The resulting $\ell$ is rational, with total downward rounding less than $4\cdot10^{-9}$. No numerical prime enumeration is needed.

2. Applying the explicit Padé theorem

Use BFT Theorem 2.4 with $L_1(c/d)=\exp\ell$ and the $m_0$ from (5). The three choices below use $m_0=100000$ and gap bound $D=100$. The oriented seed is $aP-bQ=D_0>0$, where $P,Q$ are powers of the indicated primes.

For each row, whenever $p^k u,q^l v\ge10^B$ and $|p^k u-q^l v|\le100$, the conclusion is

The theorem is evaluated with $\epsilon=1/20000$. The certificate verifies $\Omega_3>1$, $\Omega_4>1$, and $\lambda_3-\epsilon>w$ by directed interval arithmetic. The two polynomial integrals defining each $C_{1,\delta},C_{2,\delta}$ are evaluated as exact rational numbers. The remaining logarithms and square roots use 80-digit interval arithmetic. For the reported integer $B$, the final check is $\log x_0<B\log10$; a floating estimate merely proposes $B$.

The same checker instantiates fourteen further choices directly from BFT Proposition 5.1. In particular, using $c/d=25/17$, $L_1=1.5540$, $m_0=582$ at the pair $2,7$ yields exponent $.27346$, stronger than the $.259 printed in BFT Theorem 2.1. All parameters, exact gcd certificates, exponent intervals, and thresholds are retained in the output.

For the weaker improvements sufficient at index sixteen, much smaller thresholds are possible. At $5,7$, use $c/d=79/51$, $m_0=100$, $T=50$, and $\epsilon=1/1000$. Only three summands in (6) are positive, and the exact certificate is

It gives exponent $.23237$ with $x_0<10^{163784}$. At $2,7$, the Proposition 5.1 choice $25/17$, $L_1=1.5540$, $m_0=582$ with $\epsilon=1/2500$ gives exponent $.27307$ and $x_0<10^{70024}$. Both choices are retained in the index_16_pairs section of the certificate. They can be combined with the original numerical Theorem 2.1 table for all other pairs.

3. A fully numerical bound at index eleven

For $i=11$, use (5) with $m_0=10^7$ and these six choices:

The seed parameters are the corresponding BFT seeds, retained in the checker; here $\epsilon=1/10000$ and again $D=100$.

The weighted certificate for index eleven applies with these exponents. Its exact minimum over the 64 choices of witnessing endpoints is now

where $v_{(0)}\le\cdots\le v_{(3)}$ are the sorted vertex bounds. This finite optimization is checked with integers. The argument therefore gives, beyond the numerical prime-pair thresholds,

The second inequality is verified by an exact integer comparison. If the selected maximal positions coincide, the earlier argument already gives $n<10^{22}$. Finally, $n\ge10^{3\,639\,160\,902}$ implies that all $n-r_p$ exceed the six numerical thresholds, and contradicts (9). This proves (1), including the bounded shifts in the numerator factors.

Reproduction

The gcd and parameter checker and its recorded output use the explicit BFT threshold evaluator. These are finite checks of analytic parameters and small maximum-constraint systems. They do not enumerate rows of Pascal's triangle.

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_bft_gcd_bounds.py