A253280 Greatest k such that a polynomial f(x) with integer coefficients between 0 and k is irreducible if f(n) is prime.

3795, 8925840, 56446139763, 568059199631352, 4114789794835622912, 75005556404194608192050, 1744054672674891153663590400, 49598666989151226098104244512918, 1754638089240473418053140582402752512

Offset: 3

Author

Charles R Greathouse IV, Sep 30 2015

Keywords

• nonn
• nice

Comments

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

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

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.

References

• J. Alexander. Irreducibility criteria for polynomials with nonnegative integer coefficients. Master's Thesis, University of South Carolina. 1987. Cited in Dunn 2014.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 3..20
• J. Brillhart, M. Filaseta, and A. Odlyzko, On an irreducibility theorem of A. Cohn, Canad. J. Math. 33 (1981), pp. 1055-1059.
• Morgan Cole, Sharp Bounds Associated With An Irreducibility Theorem For Polynomials Having Non-Negative Coefficients (2013).
• Scott Michael Dunn, Explorations in Elementary and Analytic Number Theory (2014).
• Michael Filaseta, Irreducibility criteria for polynomials with nonnegative coefficients, Canad. J. Math. 40 (1988), pp. 339-351.
• Michael Filaseta and Samuel Gross, 49598666989151226098104244512918, J. Number Theory 137 (2014), pp. 16-49.
• Wikipedia, Cohn's irreducibility criterion