Exact short-block extrema over prime sets

../

---

For fixed $2<\alpha<3$, let $G_\alpha(u)$ minimize the number of distinct multiples over sets of $u$ primes and blocks of $\lfloor\alpha P\rfloor$ consecutive positive integers, with $P$ the largest chosen prime. The author-recorded L37 proves $G_\alpha(u)=\lceil2\sqrt u\rceil$ for every $u\ge1$.

This is a rectangular version of the recorded Erdős–Selfridge square construction, not a novelty claim. The lower bound counts prime incidences in the two halves and transfers distinct multiples across their midpoint. The upper bound finds $r$ repeated patterns of $s$ primes, with $rs\ge u$ and $r+s=\lceil2\sqrt u\rceil$, and uses congruences to place every multiple at one of the row or column positions. The proof checks the floor in the block length and makes the whole block positive.

For Problem 1143, minimization over prime sets is essential. The result is not a formula for each prescribed set and does not address $\alpha\ge3$. Extending the interval past three times the largest prime introduces further multiple positions; the displayed two-position construction gives no sharp bound there. The original claim and this author argument retain their separate standing; no catalog acceptance is recorded by this account.