A253280 - OEIS

%I #24 Nov 22 2020 12:20:44

%S 3795,8925840,56446139763,568059199631352,4114789794835622912,

%T 75005556404194608192050,1744054672674891153663590400,

%U 49598666989151226098104244512918,1754638089240473418053140582402752512

%N Greatest k such that a polynomial f(x) with integer coefficients between 0 and k is irreducible if f(n) is prime.

%C This is an extension of Cohn's irreducibility theorem, which is a(10) >= 9.

%C Brillhart, Filaseta, & Odlyzko show that a(n) >= n-1; Filaseta shows that 10^30 < a(10) < 62 * 10^30.

%C a(10) is due to Filaseta & Gross, a(8)-a(9) and a(11)-a(20) to Cole, and a(3)-a(7) to Dunn. Dunn proves that 7 <= a(2) <= 9, but its value is not known at present.

%D J. Alexander. Irreducibility criteria for polynomials with non-negative integer coefficients. Master's Thesis, University of South Carolina. 1987. Cited in Dunn 2014.

%H Charles R Greathouse IV, <a href="/A253280/b253280.txt">Table of n, a(n) for n = 3..20</a>

%H J. Brillhart, M. Filaseta, and A. Odlyzko, <a href="http://dx.doi.org/10.4153/CJM-1981-080-0">On an irreducibility theorem of A. Cohn</a>, Canad. J. Math. 33 (1981), pp. 1055-1059.

%H Morgan Cole, <a href="https://scholarcommons.sc.edu/etd/1590">Sharp Bounds Associated With An Irreducibility Theorem For Polynomials Having Non-Negative Coefficients</a> (2013).

%H Scott Michael Dunn, <a href="https://scholarcommons.sc.edu/etd/2809">Explorations in Elementary and Analytic Number Theory</a> (2014).

%H Michael Filaseta, <a href="http://dx.doi.org/10.4153/CJM-1988-013-6">Irreducibility criteria for polynomials with nonnegative coefficients</a>, Canad. J. Math. 40 (1988), pp. 339-351.

%H Michael Filaseta and Samuel Gross, <a href="http://dx.doi.org/10.1016/j.jnt.2013.11.001">49598666989151226098104244512918</a>, J. Number Theory 137 (2014), pp. 16-49.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cohn%27s_irreducibility_criterion">Cohn's irreducibility criterion</a>

%K nonn,nice

%O 3,1

%A _Charles R Greathouse IV_, Sep 30 2015