Every block of K integers below K^(5/2)/128 contains a product of three integers at most K

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_proof: Scale m by c so that m/c is at most K^2/16, take the next square q^2 above m/c, and subtract the largest admissible t^2; the three factors stay below K and the overshoot c(2t+1) stays below 3K/4.

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This result is recorded by Plasma AI as an explicit specialization of a construction due to Erdős and Sárközy. It bears on the upper bound of Problem 962, whose function k(n) it bounds by an explicit multiple of n^(2/5).

Meaning and source

A block $m+1,\ldots,m+K$ in which every term has a prime factor greater than $K$ is exactly a block with no $K$-smooth member. The statement exhibits a $K$-smooth member of every such block with $m\le K^{5/2}/128$, as a product $c(q-t)(q+t)$ of three factors at most $K$: the multiplier $c$ brings $m/c$ below $K^2/16$, the next square $q^2$ above $m/c$ lies within $2q$ of it, and subtracting the largest $t^2$ that keeps $c(q^2-t^2)$ above $m$ leaves a gap of at most $c(2t+1)<K$.

The construction is that of Erdős and Sárközy's Theorem 2, On the prime factors of $\binom nk$ and of consecutive integers, Utilitas Math. 16 (1979), 197--215, section 3, the proof of their Theorem 2 (printed p. 198): "If $k>k_0$, then $n_k>k^{5/2}/16$", where $n_k$ is the smallest $n$ with $P(n+i)>k$ for $1\le i\le k$ and $P$ is the greatest prime factor (printed p. 197). A public scan is at the Rényi Institute's Erdős archive. Friedlander and Lagarias (J. Number Theory 25 (1987), 249--273, Theorems 3 and 4) give the same exponent, with an $\varepsilon$ of slack, from the case $r=2$ of their construction, which they credit independently to Balog and Sárközy. This card adds what the asymptotic statements leave implicit: explicit constants valid for every $K\ge64$, every $m\ge0$ with no exceptional small range, and the translation to $k(n)$ with its endpoint convention.

Standing and verification

Tier 0, author-recorded, with a complete elementary proof; reading it is the verification procedure. Nothing remains open within the stated scope. The exponent $2/5$ is the known waypoint: it is the 1979 theorem with explicit constants, not a new exponent.

Uses and context

The claim L251 consumes this statement for the blocks with m at most K^(5/2)/128 and covers the larger range up to $2^{-15}K^{28/11}$ by averaging the multiplier $c$. The research sketch on smooth integers in every short interval places both exponents against the full question of Problem 962 and records why this construction stops at $2/5$ when the multiplier is fixed.