Erdős 699: index 14 is effective

P699 research · Erdős problem 699 · Off-diagonal Padé estimates

Theorem. There is an effectively computable integer $N_{14}$ such that every bad triple $(n,14,j)$ satisfies $n<N_{14}$, for the original threshold $p\ge14$.

One polynomial certificate works for every cofactor alternative, every selected position, and every collision between selected positions. Its symmetric interpolation matrix has only $831$ rows and $841$ columns. The pair-source starting heights are effective but have not all been numerically evaluated. Thus this proves effectivity, rather than a complete exclusion of the finite range below $N_{14}$.

Subsequent complete result. The numerical tail and finite certificates exclude this entire index. The effective proof below is retained as an earlier step.

Unequal Padé blocks give stronger pair alternatives

Let $\mathcal P=\{2,3,5,7,11,13\}$. For each $p\in\mathcal P$, choose $r_p\in\{0,\ldots,13\}$ maximizing $v_p(n-r_p)$, and put

The off-diagonal argument allows different prime-power block lengths on the two sides of a seed $aP-bQ=\delta>0$. For positive integer directions $(A,B,C)$, write $L_0=A+B$, $L_1=B+C$, and let $C_0,K_0$ be the two positive decay rates in that argument. If $P=p^k$ and $Q=q^l$, then every pair of targets strictly below

gives an effective alternative

The source checker and source records retain $294$ such alternatives. Each has an exact finite rational coefficient-gcd sum, a strictly smaller rational gcd growth rate, directed interval bounds on $C_0,K_0$, and an endpoint margin greater than $1/10000$. All endpoints are exact rationals with denominator $10000$. The coefficient-gcd proof has an effective threshold because it uses finitely many prime intervals and a computable prime-number estimate; no infinite sum or ineffective approximation theorem enters.

The usual application assumes $r_p\ne r_q$, so that the factor gap is nonzero and at most $13$. There is also a direct treatment of the zero-gap case. If $r_p=r_q$, the selected prime powers are coprime and both divide $n-r_p$, hence $u_pu_q\ge n-r_p\ge h$. Every retained pair satisfies $\gamma_p+\gamma_q<1$. Thus (1) also holds when the positions coincide.

Together with $18$ retained earlier pair alternatives, these give $312$ disjunctions. Their exact, componentwise minimal witnessing vectors number $1598$. This means that for every sufficiently large row there is a listed vector $v=(v_p)_{p\in\mathcal P}$ with $u_p\ge h^{v_p}$ for every prime. The whole-index checker regenerates the vectors by induction over the alternatives and matches every source against its retained certificate.

One polynomial family handles every position

Use coordinate-factor exponents

and row multiplicities

Define

The symmetric Hasse-jet map on polynomials of degree at most $56$ has $831$ prescribed coefficients and $841$ monomials. Its exact rank modulo $65521$ is $831$, proving surjectivity over $\mathbb Q$. The nonvanishing lemma therefore supplies a finite integer polynomial family of residual degree at most $57$, with no common zero in $X+Y>13$. After multiplication by $H$, its members have degree at most

For $14<j\le n-j$, the coordinate factor does not vanish. At least one member $F$ consequently has $F(j,n-j)\ne0$.

For a given cofactor vector, a selected position occupied by exactly one prime $p$ has capacity $1-v_p$. A position occupied by several primes has capacity at most one for the whole group, since the product of its selected prime powers divides the corresponding factor $n-r$. The multiplicities $m_r$ are nonincreasing. Rearrangement puts the largest capacities at the earliest rows, covering all assignments of the selected positions, not just consecutive assignments.

For every subset of singleton primes, the checker assigns capacity one to the maximum possible number of remaining groups, namely half the number of remaining primes rounded down. A single remaining prime is omitted from this construction because it would itself be a singleton. This yields an upper bound for every possible collision pattern. All these capacity vectors are checked against all $1598$ cofactor vectors using exact integers. The minimum gain is

No position-dependent AP identity or further branch of a product alternative is needed in this argument.

Effective stopping bound

Badness and Kummer's formula imply that the large-prime part of $\prod_{r=0}^{13}(n-r)^{m_r}$ divides the nonzero integer $F(j,n-j)$. Away from the selected position for a prime $p<14$,

The omitted small-prime contribution away from all selected positions is thus bounded by $(13!)^{66}$. Together with (2), polynomial growth gives

above an elementary computable correction threshold. The constant factor absorbs the bounded shifts between $n-r$, $h$, and $n$.

The explicit adjugate and Hadamard estimate from the interpolation extension proof bounds this right side by $10^{100814}$. The checker verifies

Consequently every bad row for which all the source estimates hold satisfies $n<10^{3\,400\,000}$. Taking the maximum of this evaluated algebraic bound, the elementary correction threshold, and the finitely many effective pair-source thresholds defines $N_{14}$ effectively. The numerical values of the latter thresholds remain to be evaluated.

The certificate data retain the full alternatives, the minimal vectors, the rank parameters, and the exact worst capacity comparison. Reproduce it with the pinned interpolation dependencies:

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_off_diagonal_pairs.py
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_14_effective.py

The second command accepts --arithmetic-only to repeat the source and finite cofactor comparisons without repeating the modular rank.