Yes, I would tell everybody because I'm a computational number theorist and (1) such an algorithm would also be very useful for many other algorithms in algebraic number theory, e.g., computation of rings of integers of number fields which requires factoring discriminants, and (2) whatever amazing insight led to the algorithm would probably also open up many new research directions in computational number theory, just as past progress on integer factorization has done (my thesis advisor Hendrik Lenstra introduced the groundbreaking elliptic curve factorization technique, telling the world in research papers and beautiful lectures). That said, sadly I'm not optimistic that there exists a fast non-quantum algorithm to factor large composite integers. It may simply be a hard problem.
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Yes, I would tell everybody because I'm a computational number theorist and (1) such an algorithm would also be very useful for many other algorithms in algebraic number theory, e.g., computation of rings of integers of number fields which requires factoring discriminants, and (2) whatever amazing insight led to the algorithm would probably also open up many new research directions in computational number theory, just as past progress on integer factorization has done (my thesis advisor Hendrik Lenstra introduced the groundbreaking elliptic curve factorization technique, telling the world in research papers and beautiful lectures). That said, sadly I'm not optimistic that there exists a fast non-quantum algorithm to factor large composite integers. It may simply be a hard problem.