## Fourth Edition: Final Results on P vs NP Via Integer Factorization and Optimization

**Authors:** Yuly Shipilevsky

We develop two different polynomial-time integer factorization algorithms.
We reduce integer factorization problem to equivalent problem of minimizing
a quadratic polynomial with integer coefficients over the integer points
in a quadratically constrained two-dimensional region.
Next, we reduce those minimization problem to the polynomial-time minimizing
a quadratic polynomial with integer coefficients over the integer
points in a special two-dimensional rational polyhedron.
Next, we reduce integer factorization problem to the problem of enumeration
of vertices of integer hull of a special two-dimensional rational polyhedron,
solvable in time polynomial by Hartmann's algorithm.
Finally, as we show that there exists an NP-hard minimization problem,
equivalent to the original minimization problem, we conclude that P = NP.

**Comments:** 20 Pages.

**Download:** **PDF**

### Submission history

[v1] 2018-09-10 17:59:12

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