Verifying that no prime factor smaller than m exists requires either exhaustive search or trust in the completeness of the presented factorization, reintroducing the original computational difficulty in disguised form.

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Completeness (in logic/mathematics)(mathematical/logical property discussed in the statement): The quality of being thorough and having nothing left out—in this case, whether a list of factors actually includes all the factors that exist.
Computational difficulty(the main problem being discussed in the statement): How hard or time-consuming it is for a computer to solve a problem, especially when the problem requires many steps or calculations.
Exhaustive search(computational method referenced in the statement): Checking every single possible option or answer one by one until you find what you're looking for, rather than using a shortcut.
FACTORIZATION(Used as an example of a problem in NP ∩ coNP not currently known to be in P.): The computational problem: given ⟨n,m⟩, does n have a factor d with 1 < d ≤ m?

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