The intersection NP ∩ coNP is a syntactic complexity class defined relative to a fixed computational model; FACTORIZATION as a decision problem (does n have a factor ≤ k?) differs structurally from the search problem, and membership claims must specify which formulation is at issue.

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Computational model(as used in computer science and logic): A theoretical set of rules describing how a computer solves problems—basically the 'machine' we're imagining when we think about difficulty.
Decision problem(as used in computational complexity theory): A question you can only answer with 'yes' or 'no'—like 'does this number have a factor smaller than 5?'
NP (nondeterministic polynomial time)(Major complexity class based on nondeterministic model): The union over all natural numbers k of NTIME(n^k); the class of languages decidable by a nondeterministic Turing machine in polynomial time.
Search problem(as used in computational complexity theory): A task where you need to actually find something that satisfies conditions, not just answer whether it exists—like 'find me all the factors of this number.'
Syntactic complexity class(as used in computational complexity theory): A way of grouping computer problems based on the rules and structure of how they're written down, rather than what they mean.
coNP (Complement of NP)(as used in computational complexity theory): A category of computer problems where disproving an answer is fast—basically the 'opposite' version of NP problems.
∩ (intersection symbol)(as used in set theory and logic): A mathematical symbol meaning 'overlap' or 'things that belong to both groups at the same time.'

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