Uniqueness of prime factorization (the Fundamental Theorem of Arithmetic) guarantees a single valid certificate exists, but does not guarantee that certificate is polynomially sized relative to the binary encoding of n.

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Certificate(Used within the formal definition of brute force search; e.g., a satisfying valuation is a certificate for SAT membership.): A witness y ∈ S_x whose existence establishes that x is a member of problem X, verified by the relation R_X(x, y).
Fundamental Theorem of Arithmetic(as a mathematical principle): A mathematical rule stating that every whole number greater than 1 can be broken into prime numbers in exactly one way, and no other way works.
Polynomially sized(as used in computer science complexity): Growing at a reasonable, manageable rate as numbers get bigger—roughly speaking, if the input is twice as big, the size grows by a fixed multiple rather than exploding exponentially.
Prime factorization(as used in the Fundamental Theorem of Arithmetic): Breaking a number down into its prime number building blocks—for example, 12 breaks down into 2 × 2 × 3. Every number has exactly one way to do this.

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