Erdős 699: an explicit bound at index 13

P699 research · Erdős problem 699 · Surjective interpolation lemma

Complete exclusion. A subsequent asymmetric bound and interval proof excludes every bad triple with $i=13$. The interpolation construction below remains useful in the proofs for indices $15,20$.

Every bad triple with $i=13$ satisfies $n<10^{56000}$. The bound uses the original threshold $p\ge13$. It covers all assignments of the small-prime maxima, including coinciding positions.

The proof combines an exact finite-field rank calculation with a position-sensitive polynomial argument. No binomial rows are enumerated. The earlier interpolation construction applied to one position pattern; the nonincreasing row multiplicities below allow rearrangement to handle every pattern, while surjectivity supplies nonvanishing throughout the physical region.

1. The finite interpolation system

Use coordinate-line exponents and row multiplicities

At each integer point $x,y\ge0$, $x+y\le12$, require residual multiplicity

The symmetric jet map on polynomials of total degree at most 106 has 2911 rows and 2916 columns. Its rank over $\mathbb F_{65521}$ is exactly 2911. This proves surjectivity over $\mathbb Q$. The exact rank checker and retained certificate give the matrix conventions and verification. Its additional polynomial gcd calculation is not required for the present proof.

The surjective interpolation lemma now supplies finitely many integer polynomials $G_a$ of degree at most 107, all having the residual multiplicities, that cannot vanish simultaneously when $X+Y>12$. Put

Then $\deg H=534$, $\deg F_a\le641$, and every $F_a$ has multiplicity at least $m_t$ at each Kummer point on $x+y=t$. In the admissible region $13<j\le n-j$, the factor $H(j,n-j)$ is positive. Hence at least one $F_a(j,n-j)$ is nonzero at every admissible pair.

2. Every position assignment has a positive gain

For $p\in\{2,3,5,7,11\}$, choose a position $r_p$ maximizing $v_p(n-r_p)$, and write $n-r_p=P_pu_p$ as before. Use the original, fully explicit BFT Theorem 2.1 table, together with the certified asymmetric estimate

The asymmetric source calculation proves (1) when the smaller numerator factor is at least $10^{1731}$. The other pair estimates hold beyond the much smaller published threshold. In particular, every source estimate applies for $n\ge10^{1732}$.

The finite maximum/disjunction system has 18 componentwise minimal cofactor vectors $v$, one of which satisfies $u_p\ge(n-12)^{v_p}$ for every small prime. Its coordinates are less than $1/2$.

A singleton maximal position has exponent capacity $1-v_p$; a position occupied by two or more primes has capacity one. As proved in the weighted-certificate note, choose the singleton subset and give the other primes as many paired positions as possible. This finite set of capacity lists includes every collision pattern. Because $m_t$ is nonincreasing, the worst assignment places the largest capacities at the smallest positions.

Writing such a list as $c_0\ge\cdots\ge c_{g-1}$, the exact minimum is

A minimizing vector in prime order $2,3,5,7,11$ is $(0,.329,.258,.259,.059)$, with all positions distinct. The arithmetic checker verifies (2) using rational arithmetic and all 18 vectors. No selection of a favorable position pattern is made.

The collision argument extends unchanged from linear weights to the present nonincreasing sequence. Inserting a capacity one before a decreasing list increases its weighted contribution by at least the last new weight: for $0\le c_r\le1$,

3. A coefficient bound without computing rational interpolants

Let $r=2911$, $D=106$, and put

Every entry of the integer symmetric jet matrix has magnitude at most $2\cdot13^{106}$. Choose a nonsingular maximal minor with determinant $\Delta$. Adjugates give integer polynomials $q_j$ of degree at most 106 whose jets are $\Delta$ times the coordinate vectors, with

In the invariant Taylor-coordinate basis, multiplication by $X+Y$ has column norm at most 15; use the convenient bound 20. The construction in the surjectivity lemma can therefore take the integer generators

where $a$ is the jet vector of 1 and $T$ is the multiplication matrix. There are at most 49 nonzero entries of $a$, each equal to one. Hence all these generators have coefficient norm less than $10^6C$.

The elementary exact comparisons

give

Consequently every resulting $F_a$ satisfies

4. The stopping bound

Let $W=\max m_t=114$ and $L=\sum t m_t$. The weighted divisibility argument, retaining the maximum-position constants as in the weighted note, gives for each nonzero evaluation

provided $n\ge1000(L+114\cdot5\cdot12)$ and the source estimates apply. The factor $13^{114}$ accounts for the rough denominator prime in $13!$. The elementary threshold is below $10^8$.

Using $13^{114}<10^{128}$, $12!<10^9$, and (3), the right side of (4) is less than $10^{356876}$. But

Thus $n\ge10^{56000}$ is impossible. This final bound also exceeds the elementary and source thresholds, proving the stated explicit last row.

Reproduce the arithmetic and rank checks separately:

python3 wiki/research/erdos_699/computations/erdos_699_index_13_effective.py
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_13_interpolation_gcd.py

The rank command uses NumPy and python-flint. Versions used for these certificates are pinned in the interpolation requirements. The arithmetic output records the exact gain, source threshold, and complete row bound.