Choose a polynomial's coefficients randomly and independently from your favorite nontrivial distribution. Then it should be irreducible with high probability for polynomials of high enough degree. This was previously conjectured for the uniform distribution on {0,1} by Odlyzko and Poonen; now Breuillard and Varjú ( have proven that it follows from a form of the Riemann hypothesis.
Just like with prime numbers, the most basic thing to know about prime equations is: How often do they occur? Mathematicians have now proven that the answer is “almost never.” The new work has wide-reaching implications for fields from online encryption to the mathematics of randomness.
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