Erdős 699: finite exclusion at index seven through 10^100000001

P699 research · Index-seven effectivity · Three-prime CRT strips

Theorem. There is no admissible bad triple with $i=7$ and $n\le10^{100\,000\,001}$, for the original threshold $p\ge7$.

This unconditional finite-range exclusion now meets the completed numerical infinite tail, settling index seven entirely. The calculation here supplies the finite part of that combined proof.

The combination checker and its record retain the following reductions.

The pair reductions

The first pair certificate works through $10^{100\,000\,001}$; the second works through $10^{132427}$. Both allow differences up to six. Their finitely many exceptional pairs have explicit heights below the row caps subsequently obtained.

For a complete three-prime graph of exponent $\delta$, the cofactor witnesses are permutations of $(0,\delta,\delta)$. Apply the already verified polynomial families from the index-seven effectivity proof. Only the families with uniform nonvanishing are used here; the coprime-kernel certificate is unnecessary. The central column is already excluded by the elementary argument in that proof.

The fixed-slope lemma forces at least one selected maximal prime-power position to be zero. The checker therefore considers all $90$ distinct position assignments and all three witnesses: $270$ exact gain comparisons at each step. Repeated positions use the separate collision certificate. The explicit coefficient bounds give the caps in the table, including the shift from $h=n-6$ to $n$. No large modular-rank calculations are repeated.

Three primes and the final intervals

The $.61$ certificate and $.66$ certificate cover heights through $10^{3000}$. They include all $127$ selected-position patterns with minimum zero, including coincidences.

Combining the $.61$ alternative with the pair exponent $.47$ gives a cofactor sum of at least $1.08$. The ordinary fat-point lower bound then reduces the remaining range to $n\le10^{40}$. The larger $.66$ alternative improves the interval comparisons where it is available. The interval checker disables a triple alternative precisely on its listed exceptional rows, and handles the pair exceptions by their corresponding row windows.

The interval driver and output record:

• 18,000 certified intervals, including 145 intervals containing no row divisible by eight, nine, or twenty-five;
• 28 residual rows, all at most $26,250$;
• 41,740 exact column checks, each producing a common factor with a prime divisor at least seven.

The smallest directed logarithmic margin is greater than $0.00000808393925427$. The terminal-prime lemma covers the prefix below row $89$. The output records the interval partition hash, the source file names, and the count of residual column gcd checks.

Large pair certificates without large auxiliary powers

The optimized strip generator uses the same strip inequalities as the earlier integer checker. It avoids forming high powers of integers with tens of millions of digits.

If $A=2^s a+e$, with $0\le e<2^s$, then

Leading integer bits therefore give rigorous short intervals for $\log A$. Directed arithmetic certifies the required power comparisons. Ambiguous small comparisons are evaluated with integers; higher precision handles large comparisons, with perfect-power equality treated exactly.

One extended gcd supplies both modular coefficient progressions. Thus the loop does not repeatedly multiply and divide the large prime powers for every small difference. At cap $1000$, the new implementation reproduces all anchors and exceptions of the original checker for each of the three pairs. The extended certificates contain 176, 171, and 165 anchors.

The original pair certificate through $10^{44\,245\,906}$ is preserved. Its certified slabs are valid without an upper restriction on the transverse exponent. At a boundary with first power $P$, any potentially bad second power satisfies $Q-D<P^{s/(s-r)}$, so every minimal CRT height is less than $2P^{1+s/(s-r)}$. The horizontal boundary has the symmetric bound. These bounds verify that neither the old height filter nor the old prime-power caps truncated any boundary check.

The extension therefore starts at the retained final orthant and adds eight anchors for each pair. It does not repeat the earlier large modular inversions. The records name the original file and each retained prefix, and the combination checker verifies that the continuation reproduces the prefix's anchors, boundary checks and height cap. A small continuation from cap $50$ to cap $1000$ reproduces the fresh calculation's anchors, exceptions, and boundary hash for all three pairs.

To reproduce the final interval and combination steps using the already retained source certificates, with the pinned dependencies installed in the repository's .venv/:

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_7_finite_intervals.py
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_7_finite.py