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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
9
votes
Accepted
Comparing two numbers given their factorization
According to this paper from 2013 by Etessami, Stewart, and Yannakakis, the time complexity of the [a priori] harder problem where the $p_k$ are not required to be prime is still open. In that paper …
7
votes
on the set of numbers generated by integer linear combination of two real numbers.
Yes, the set $X = a\mathbb{N} + b\mathbb{N}$ is order-isomorphic to $\mathbb{N}$.
For any positive real $k$ the set $X\cap [0,k]$ is finite. Indeed, to have $ap+bq \leq k$ for $p,q\in\mathbb{N}$ we …