Erdős Problem #18: independent continuation
Date: 23 September 2026. Source versions read in full:
18.tex, 18_v2.tex in
gpt_pro_5.4.
1) OBJECTIVE AND ALL-VERSION AUDIT
For practical $m$, let $h(m)$ be the maximum minimum number of distinct divisors needed to represent an integer in $[0,m)$. Both supplied versions concern lower bounds and a factorial extension lemma. They do not prove either the infinitely-many-practical-numbers polylogarithmic target or the factorial upper-bound targets.
2) A PRODUCT INEQUALITY
If $a,b$ are practical, then $ab$ is practical and \[
\boxed{h(ab)\le h(a)+h(b).}
\] Indeed write $x=aq+r$ with $0\le q<b$ and $0\le r<a$. Represent $q$ by at most $h(b)$ distinct divisors of $b$, and multiply them by $a$. Represent $r$ by at most $h(a)$ distinct divisors of $a$. Every resulting term divides $ab$. Terms in the first group are at least $a$; terms in the second group are smaller than $a$, so the groups cannot collide. Within each group distinctness is preserved. This proves the assertion including $q=0$ or $r=0$. Consequently $h(a^j)\le jh(a)$ and, for any factorization into practical integers, $h(\prod_i a_i)\le\sum_i h(a_i)$. Unlike the source's interval-divisibility lemma, this does not require every small integer to divide one factor.
3) WHAT THIS CONSTRUCTION DOES NOT ACHIEVE
For fixed $a>1$, the bound for $a^j$ is linear in $j$, whereas $\log\log(a^j)=O(\log j)$. Thus repeatedly multiplying a fixed practical seed does not yield the required polylogarithmic upper bound. The product inequality also cannot be applied to an arbitrary factorial quotient without proving that quotient practical.
4) INDEPENDENT FINITE VERIFICATION
An exact bounded-cardinality subset-sum computation was rerun for $n=2,\ldots,9$, giving \[
h(n!)=1,2,3,4,5,5,6,7.
\] The saved checker updates cardinality bitsets in descending order, so each divisor is used at most once, and checks every target below $n!$. I have not rerun the supplied $12!$ computation and therefore do not claim its stated witness or exhaustive coverage as newly certified.
7) FINAL
UNRESOLVED. The product construction is proved, but no subpolynomial factorial upper bound follows. Completion measures progress on the original upper-bound questions, rather than on auxiliary lower bounds. Statement checked:
https://www.erdosproblems.com/18. Verification:
verification/root_checks.py.
8) COMPLETION ESTIMATE
COMPLETION: 12% This is a subjective estimate of progress toward the entire stated problem, not a probability that the conjecture or an unverified proof is correct.