Erdős 699: complete exclusion at index 16

P699 research · Erdős problem 699

For every $16<j\le n/2$, some prime $p\ge16$ divides both $\binom n{16}$ and $\binom nj$. All rows are covered by an analytic tail and certified intervals; no singleton row requires direct testing.

The asymmetric cofactor estimate for $5,7$, with targets $(.29,.20)$, is valid above input height $10^{913}$. With the original explicit BFT table, the weighted polynomial $F_{10}$ has minimum gain $23/100$, covering all 36 cofactor vectors and every collision pattern. The exact off-maximum constant is

For $n\ge2\,960\,000$, the weighted inequality gives

An exact integer comparison yields the algebraic cap $10^{686}$. Including the source threshold and shifts proves that every bad row satisfies $n<10^{914}$. The weighted certificate retains this calculation.

On that finite range, use the prime-power method for the seven pairs

The retained pair certificates cover heights at least through $10^{914}$ and differences at least through $15$. Every pair gives a maximum-cofactor exponent $1/3$ above its displayed small exceptional range. Using the conservative weight $.3333$ and the original BFT table leaves thirteen minimal cofactor vectors, whose minimum sum is $1.2589$. All estimates apply to the selected numerator factors whenever $n\ge1\,771\,595$.

Thus, on the remaining range,

The retained fat-point lower bound has exponent $5$. A directed-rounding comparison reduces the possible row range further to $n<10^{41}$. This reduction is applied only below the already proved cap $10^{914}$.

The terminal-prime argument excludes all admissible $n<523$. On $[523,10^{41}]$, the interval method accepts 100,204 intervals, with 200,407 visited nodes. Its minimum certified logarithmic margin is

There are no unresolved singletons. The complete output records the partition hash and, as its own run record, the names of its inputs. Together with the analytic tail, it proves the stated full-index result.

Reproduce the last two stages with:

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_asymmetric_ramp_caps.py
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_16_complete.py

The finite pair records can be reproduced by the generic prime-pair checker, using their recorded primes, height cap and difference bound.