A prime factorization of n with no factor less than m serves as a polynomial-size certificate for non-membership in FACTORIZATION, and primality of factors is verifiable in polynomial time via AKS
A number that divides evenly into another number; for example, 3 is a factor of 12 because 12 ÷ 3 = 4 with no remainder.
Non-membership(as used in set theory and logic)
Not belonging to or not being part of a particular group or category.
Polynomial-size certificate(as used in computer science and computational complexity)
A piece of evidence or proof that can be written down in a reasonably short way (the length grows at a manageable rate as the problem gets bigger), making it easy to verify.
Primality(as used in mathematics and logic)
The quality of being a prime number—a number that can only be divided evenly by 1 and itself, like 2, 3, 5, or 7.
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.
polynomial time(Used to characterize feasible computation)
Computational time complexity expressed as t(x)=x^c, where c is a constant and x is the length of the input
verifiable(The second category of meaningful sentences under verificationism; the text notes the precise meaning of 'verifiable' is contested)
A sentence that can in principle be confirmed or disconfirmed through empirical observation
The graph \(G_{\phi}\) for the formula \((p_1 \vee p_2 \vee p_3) \wedge (\neg p_1 \vee p_2 \vee \neg p_3) \wedge (p_1 \vee \neg p_2 \vee \neg p_3)\). A reduction of \(3\text{-}\sc{SAT}\) to \(\sc{INDEPENDENT}\ \sc{SET}\) can now be described as follows: Let \(\phi\) be a \(3\text{-}\sc{CNF}\) formula consisting of \(n\) clauses as depicted above. We construct a graph \(G_{\phi} = \langle V,E \rangle\) consisting of \(n\)-triangles \(T_1,\ldots,T_n\) such that the nodes of \(T_i\) are respect