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

3,1

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

Crossrefs

Sequence in context: A232381 A109183 A284872 * A394816 A251518 A031793

Adjacent sequences: A253277 A253278 A253279 * A253281 A253282 A253283

Keyword

nonn,nice

Author

Charles R Greathouse IV, Sep 30 2015

Status

approved