Erdős 699: complete proofs at further small indices

P699 research · Erdős problem 699 · Numerical stopping bounds · Finite prime-pair proof

There are no bad triples at $i=10,13$ or at any $i\ge15$, with the original threshold $p\ge i$. The new certificates here cover $10,13,15,18,19,20,21,22,24,25$. The other required indices $16,17,23$ and every $i\ge26$ were already excluded.

The numerical stopping bounds already excluded $n\ge10^{976}$ for $i=19,24$ and $n\ge10^{957}$ for $i=22,25$. The asymmetric cofactor bound also gives $n<10^{1046}$ at $i=13$ and $n<10^{1732}$ at $i=21$; their weighted constants are retained in the exact checker. The numerical index-18 proof gives $n<10^{700\,000}$. The fully numerical interpolation bounds at index 10, index 15, and index 20 give $n<10^{15\,000\,000}$, $n<10^{15\,170\,264}$, and $n<10^{27\,027\,659}$, respectively. Interval certificates handle these finite ranges. They leave two small rows at index 10 and forty at index 13, each verified by exact binomial gcd calculations.

Bridging the bounded range

The finite prime-pair certificates prove

for the retained pairs, bounded differences at most 24, and all heights up to $10^{976}$, above explicit small thresholds. A second table extends the required pairs to $10^{1732}$ with differences at most 20 for index 21. A further table completes the five-prime graph needed at index 13. For index 18, the strip argument certifies all 15 pairs among $2,3,5,7,11,13$ through $10^{1\,200\,000}$. The same graph is certified through $10^{15\,170\,264}$ for index 15. The earlier checker enumerates prime-power exponents; the strip checker certifies entire regions of exponents using modular coefficient bounds. Both reduce their remaining cases to minimal solutions of a linear congruence. Every other positive solution is handled by a uniform inequality.

For selected maximal positions $n-r_p=P_pu_p$, these estimates give cofactor constraints of exponent $3333/10000<1/3$. The common row threshold is $39\,059\,469$ at index 13, $73\,516\,020$ at index 15, $73\,516\,023$ at index 18, and $1\,771\,595$ at $19,21,22,24,25$. Combine them with the original explicit BFT constraints. The interval proof takes the maximum over all minimal cofactor vectors, exactly as in the preceding interval proof.

The sum of the vector coordinates also yields

At $i=13,15,18,19,21,24$, the new savings are $\gamma=1.3332,1.6665,1.6665,1.3549,1.5922,1.6462$, respectively. Comparing with the previous fat-point lower bound excludes the ranges above $10^{27},10^{27},10^{37},10^{43},10^{59},10^{61}$, respectively, up to the already established stopping bounds. These comparisons do not apply beyond the finite prime-pair cap: the earlier infinite-tail proof supplies that part of the argument.

Stronger edges and their exceptional row windows

Indices 10 and 20 use the exponent $2/5$. The generalized strip argument certifies $\max(u_p,u_q)^5\ge h^2$, where $h$ is the smaller numerator factor. At index 20, use the two triangles $\{2,3,5\}$ and $\{7,11,13\}$, and the pair $17,19$. All seven edges are certified through $10^{27\,027\,659}$. Together with the original explicit BFT edges, their minimum cofactor sum is $\gamma=2.038$, reducing the remaining cap to $10^{39}$.

At index 10, five edges of the complete graph on $\{2,3,5,7\}$ are certified at least through $10^{15\,000\,000}$. The omitted edge $2,7$ retains its original explicit BFT exponent $.259$. Their minimum sum is $\gamma=1.059$, reducing the cap to $10^{80}$. The stronger $2,7$ edge of exponent $2/5$ had already been certified through $10^{1732}$, so it applies throughout this remaining range. Thus the interval proof can use the full four-prime graph without repeating the large-height computation for that sixth edge.

The window checker keeps the exact exceptional solutions from each finite pair certificate. An exception with numerator factors $x,y$ can occur in an index-$i$ row only when

Above the elementary small-power threshold, an edge is therefore valid throughout any interval avoiding its exceptional row windows. For each interval, the checker uses exactly the available edges and recomputes the finite set of minimal cofactor vectors. It also checks the height cap of each source, so certificates with different caps can be combined. The completed index-20 proof uses 43 such graphs and 124 edge-specific windows; index 10 uses 35 graphs and 142 edge-specific windows.

Complete partitions

All 2,810,557 accepted intervals have strictly positive directed-rounding margins. The exact margins, chosen prime pairs, vector counts, and interval-partition hashes are in the four-case certificate and the index-21 and index-13 certificates, and the index-15, index-18, index-20, and index-10 certificates.

At index 10, the two remaining rows are $n=1029,252$. All 620 admissible columns have a common prime at least 10, checked by exact binomial gcds after removing the primes $2,3,5,7$.

At index 13, the 40 residual rows are all at most 2060 and are listed in the JSON. For each one and every $13<j\le n/2$, the checker computes $\gcd(\binom n{13},\binom nj)$ and removes all factors $2,3,5,7,11$. The remaining integer exceeds one in all 8969 column checks. Thus each pair has a common prime at least 13. The certificate retains the count of these exact calculations. The other eight indices require no individual row or column checks.

The interval valuation upper bound is unchanged mathematically. Its evaluation now skips all powers $p^a\le R-L$ by binary search, since every interval $[L,R]$ necessarily crosses a multiple of each such power. Only the remaining powers require the original division and residue checks. This makes very large caps practical without weakening the certificate.

Reproduce the ten partitions, writing the named outputs under the ignored repository tmp/, with:

mkdir -p tmp/erdos699
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_further_intervals.py --indices 19 22 24 25 --max-nodes 2000000
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_further_intervals.py \
  --indices 21 --max-nodes 2000000 \
  --pair-certificate wiki/research/erdos_699/computations/erdos_699_finite_prime_pairs_21.json \
  --output tmp/erdos699/erdos_699_further_intervals_21.json
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_further_intervals.py \
  --indices 13 --max-nodes 2000000 --exact-singleton-limit 2100 \
  --pair-certificate wiki/research/erdos_699/computations/erdos_699_finite_prime_pairs_21.json \
  --extra-pair-certificate wiki/research/erdos_699/computations/erdos_699_finite_prime_pairs_13_extra.json \
  --output tmp/erdos699/erdos_699_further_intervals_13.json
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_further_intervals.py \
  --indices 18 --max-nodes 2000000 --exact-singleton-limit 10000 \
  --pair-certificate wiki/research/erdos_699/computations/erdos_699_prime_pair_strips_k6.json \
  --output tmp/erdos699/erdos_699_further_intervals_18.json
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_further_intervals.py \
  --indices 15 --max-nodes 2000000 --exact-singleton-limit 10000 \
  --pair-certificate wiki/research/erdos_699/computations/erdos_699_prime_pair_strips_15.json \
  --output tmp/erdos699/erdos_699_further_intervals_15.json
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_window_intervals.py \
  --index 20 --tail-power 27027659 --max-nodes 2000000 \
  --pair-certificate wiki/research/erdos_699/computations/erdos_699_prime_pair_strips_20.json \
  --output tmp/erdos699/erdos_699_window_intervals_20.json
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_window_intervals.py \
  --index 10 --tail-power 15000000 --max-nodes 5000000 \
  --pair-certificate wiki/research/erdos_699/computations/erdos_699_prime_pair_strips_20.json \
    wiki/research/erdos_699/computations/erdos_699_prime_pair_strips_10_global.json \
    wiki/research/erdos_699/computations/erdos_699_prime_pair_strips_two_fifths.json \
  --output tmp/erdos699/erdos_699_window_intervals_10.json

Compare each regenerated partition with the retained file of the same name under computations/ on its mathematical fields; elapsed times differ between runs. The source prime-pair JSON must first be reproduced using the command in its proof note. The interval output names that file as the run's own record.