In this note, we prove that the covering radius problem for Euclidean lattices is complete for the second level of the polynomial hierarchy. The note also documents the author's first experiment with generative AI as a tool for mathematical research.
The discussion of space-bounded nondeterminism is discussed and the theorem that NSPACE is closed under complement is shown, followed by introducing polynomial-time hierarchy and proving some of its basic properties.
We show that the problem of deciding whether a simplicial complex is partitionable is NP-complete in dimensions two and higher, answering a 1995 question of Kleinschmidt and Onn.
We prove that the set of numbers of type $n = 2m^2$ below $X$ which cannot be written as a sum of two primes has cardinality at most $X^{19/24 + o(1)}$.
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