Smooth integers in every short block

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literature: Each located theorem on smooth integers in every short interval, read at its exact interval length and smoothness, with the diagonal exponent it gives for k(n) in Problem 962, including the classical 2/5 exponent, the parameter mismatch in recent results, and Bourgain's analytic input for the project bound 38/97+epsilon.

proof_sketch: Proof sketch for upper bounds on k(n) in Problem 962: the near-square construction gives 2/5 with one multiplier, 11/28 explicitly and 38/97+epsilon with Bourgain's exponent pair (L250, L251, L253); exact limits of these estimates and the nonresidue reduction locate the gaps.

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This folder's results that are not drawn from a cited source are Plasma AI's, credited to Plasma AI.

The question is how long a block $m+1,\ldots,m+k$ with starting parameter $m\le n$ can be when every term has a prime factor larger than the block's length. This defines the function $k(n)$ of Problem 962, whose assessment records both of its questions as open; this research does not change that status. Equivalently, $k(n)+1$ is the least positive integer $K$ such that every block $(m,m+K]$ with $0\le m\le n$ contains a $K$-smooth integer. The quantifier is over every allowed block.

The proof sketch carries the argument and its standing. Erdős and Sárközy's 1979 theorem gives $k(n)\ll n^{2/5}$, recorded with explicit constants as L250; averaging their multiplier and bounding the resulting sawtooth sum by the third-derivative test gives L251, the bound k(n) < max(64, (2^15 n)^(11/28)). Using Bourgain's exponent pair in the same construction gives L253, the bound k(n) <<_epsilon n^(38/97+epsilon) for every epsilon > 0. These project deductions are author-recorded at tier 0. The sketch also records the $3/8$ limit of its exponent-pair estimate and the exact $1/3$ limit imposed by producing a product of three factors at most $K$, together with the difficulty reduction of L252: an exponent below $1/(4\sqrt e)$ would improve the least-nonresidue bound for primes $3\bmod4$. The literature comparison matches the located every-interval theorems at their exact parameters. The displayed question of Problem 962, $\log k(n)\le(\log n)^{1/2+o(1)}$, is not implied by any fixed power bound.