Erdős 699: a conditional 64/65 bound at index 14

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This is a retained earlier research note. The current reading guide gives the latest status and distinguishes completed proofs from remaining cases.

P699 research · Erdős problem 699

For $p\in\{2,5,13\}$, write $n-r_p=p^{a_p}u_p$, with $0\le r_p\le13$ and $u_p$ a positive integer. Suppose the selected positions form a nonconstant arithmetic progression with middle prime 2:

$$r_5+r_{13}=2r_2,\qquad r_5\ne r_{13}.$$

Put $h=n-13$. The certificate proves that for $h\ge10^{1\,956\,927\,531}$,

$$\boxed{u_2>h^{.400}\quad\text{or}\quad u_5>h^{.535}\quad\text{or}\quad u_{13}>h^{.260}.}$$

Indeed,

$$(n-r_2)^2-(n-r_5)(n-r_{13})=(r_2-r_5)^2$$

is a nonzero integer of absolute value at most 36. Apply the retained bounded-product argument with seed $65=5\cdot13$, $64=2^6$, auxiliary coefficients $a=b=1$, and Padé parameter $s=13/10$. The GCD estimate uses $m_0=10^8$ and 4000 interval terms. It gives the certified value

$$K>6.94367847019434.$$

The resulting limiting exponents for primes $2,5,13$ exceed respectively $0.40105477$, $0.53643074$, and $0.26121309$, leaving positive margins above the displayed targets. The checker recomputes the exact rational GCD lower bound and the directed-interval product estimate, including a numerical threshold for $h$.

For the previously examined position assignment $(r_2,r_3,r_5,r_7,r_{11},r_{13})=(2,5,3,4,0,1)$ with cofactor lower profile $(.1399,.3453,.2711,.3218,0,.0729)$, the new disjunction forces at least one of the three relevant exponent coordinates to increase. This alone does not exclude the entire position case. It does not apply after exchanging the positions of 11 and 13, giving $(2,5,3,4,1,0)$. That latter profile remains compatible with every applicable retained conditional clause in the bounded exploration. The numerical interpolation objective there remained approximately $1.131143$, above the value required by that construction.

This is an exploratory limitation of the current approach, not a counterexample or an impossibility theorem. Index 14 remains unresolved.