Erdős 699: finite prime-interval bounds for the BFT coefficient gcd

P699 research · The asymmetric BFT argument

The six coefficient-gcd bounds needed by the index-ten cofactor argument hold at much smaller numerical thresholds. No target cofactor exponent is weakened.

Write $G(c,d,dm-\delta)$ for the coefficient gcd in BFT, with $\delta\in\{0,1\}$. The following bounds are proved for every integer $m\ge m_{\mathrm{start}}$:

The 231306 certified intervals cover each finite range from its new start to one less than its previous analytic start. The existing analytic bound then covers the infinite tail. All interval margins are positive; the smallest is greater than $2.9\cdot10^{-8}$ after dividing by $dm$. The finite calculations concern primes and Padé parameters, not binomial rows or columns.

The checker, input parameters, and certificate retain all constants, exact margins, and partition hashes.

Prime intervals that divide the coefficient gcd

Put $h=c+d$. For an integer $t\ge1$, write

Retain $t$ when $2r>h$. For $m\ge2t$, define

The denominators in (2) are positive. BFT Lemma 5.2 and equations (5.7)--(5.8) show that every prime

divides both coefficient gcds in (1). The upper endpoint uses the smaller value for $\delta=0,1$, and the lower endpoint uses upper bounds valid for both. These prime intervals are disjoint for different $t$.

Consequently, for any finite retained set of eligible $t\le m/2$,

BFT describes this finite prime-interval method and its extension across ranges of $m$ in Section 5, following Lemma 5.4.

Certifying a whole parameter interval

For an integer interval $L\le m\le R$, keep eligible $t\le\min(4000,L/2)$. Both endpoints in (2) increase with $m$, so

The sufficient check is simply

All arguments of $\theta$ are rounded down to integers, exactly matching its definition. The program evaluates a rigorous integer-scaled lower bound for (4) and checks (5) by integer multiplication. It subdivides an interval when this lower bound is insufficient.

The traversal runs from the right endpoint downwards. Once it encounters an unsupported singleton, all larger integers have already been covered by certified intervals. The chosen new start is one greater than that singleton. This does not claim that (1) fails at the singleton: the particular lower bound (4) was merely insufficient there.

Rigorous logarithms and the sieve

The sieve retains every prime at most $10^8$, a total of $5761455$ primes. No numerical approximation is used to decide primality.

For logarithmic weights, let $Q=2^{32}$ and $M=2^{14}$. Directed interval arithmetic gives integer lower and upper bounds for

For each prime $p$, calculate exactly

Then

The program obtains integer bounds for $Q\log p$ by adding the corresponding table entries. Both prefix sums are integer arrays, with explicit bounds preventing overflow. When both endpoints in a prime interval lie below the sieve limit, subtracting two entries of the lower prefix sums the lower weights of exactly the primes in that interval. This avoids losing the errors from primes below its lower endpoint.

The sieve, both logarithm tables, and both prefix sums have retained SHA-256 hashes. The real arithmetic used to construct the logarithm table is directed interval arithmetic; every subsequent certificate comparison is an integer calculation.

Larger prime arguments and the infinite parameter tail

Above the sieve limit, the checker combines

from BFT Lemma 5.4 with the following retained bounds from Dusart's Theorem 5.2:

The retained Dusart reading copy records the theorem and its exact thresholds. Integer square roots and the lower logarithmic table bounds make these inequalities rigorous at every requested argument. All arguments used in the finite certificate are below $10^{11}$, where the simpler upper bound $\theta(x)<x$ applies.

Finally, the checker regenerates each original infinite-tail certificate using the rational Dusart gcd calculation and checks that it supplies exactly the displayed $\ell$. Thus the finite interval calculations and the existing analytic tail together prove (1) for every integer $m$ at or above the new start.

To reproduce all six finite certificates:

mkdir -p tmp/erdos699
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_bft_prime_bootstrap.py \
  wiki/research/erdos_699/computations/erdos_699_bft_prime_bootstrap_parameters.json \
  --output tmp/erdos699/erdos699_prime_bootstrap.json

The data include elapsed times, which can differ between runs. The exact bounds, interval counts and partition hashes are deterministic.