number-theory / Number Theory, Additive Bases
Let $k\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\geq c\log n$ for all large $n$ (where $r(n)$ counts representations of $n$ as a sum of at most $k$ elements of $A$) then $A$ must contain a minimal basis of order $k$? The claimed answer is no, for every $k\geq 3$.
Temporal state
No reconciled state yet.
Append-only history
A total refutation is claimed for all k>=3, building on the Larsen-Larsen resolution of problem #868; erdosproblems.com still lists the problem open
Research memory
Let $k\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\geq c\log n$ for all large $n$ (where $r(n)$ counts representations of $n$ as a sum of at most $k$ elements of $A$) then $A$ must contain a minimal basis of order $k$? The claimed answer is no, for every $k\geq 3$.
A total refutation is claimed for all k>=3, building on the Larsen-Larsen resolution of problem #868; erdosproblems.com still lists the problem open
Evidence graph
No public relationships recorded yet.