A178868 Numbers n such that the trinomial x^n + A x + B has an irreducible cubic as its lowest-degree factor (for some nonzero integers A,B).

6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 22, 24, 52

Offset: 1

Author

Artur Jasinski, Jun 20 2010,Jun 24 2010

Keywords

• more
• nonn

Comments

a(1)=6 because e.g. x^6+16x+16 have factor x^3-2x^2+4

a(2)=7 because e.g. x^7-183352x+814800 have factor x^3+4x^2+22x-300

a(3)=8 because e.g. x^8+7x-4 have factor x^3-x^2+2x-1

a(4)=9 because e.g. x^9+2187x+2916 have factor x^3+3x^2+9x+9

a(5)=10 because e.g. x^10+297x-243 have factor x^3+3x-3 (*Schinzel*)

a(6)=12 because e.g. x^12+576x+368 have factor x^3+2x^2+4x+2 (*Browkin-Schinzel*)

a(7)=13 because e.g. x^13+768x+1024 have factor x^3+2x^2+4x+4 (*Browkin*)

a(8)=14 because x^14+4x+3 have factor x^3-x^2+1 (*Bremner*)

a(9)=15 because x^15-1059328125x+2378362500 have factor x^3+15x-45 (*Browkin*)

a(10)=16 because x^16+3486328125x+9277343750 have factor x^3+5x^2+25x+50 (*Bremner*)

a(11)=17 because x^17+103x+56 have factor x^3-x^2+x+1 (*Bremner*)

a(12)=22 because x^22+376832x-425984 have factor x^3+ 2 x^2-4 (*Browkin*)

a(13)=24 because x^24+14336x+12032 have factor x^3-2x^2+2 (*Browkin-Schinzel*)

a(14)=52 because x^52+2731599200256x+3539053051904 have factor x^3+2x^2+4x+4 (*Browkin*)

Examples

			a(3)=8 because e.g. x^8+36x-13 has the cubic factor x^3-x^2+3x-1.
		

References

• Schinzel A. 1993. On reducible trinomials. Dissertationes Mathematicae. Warszawa Vol. CCCXXIX, pp.1-83.
• Schinzel A. 2000. On reducible trinomials, II. Publicationes Mathematicae. Debrecen. Tomus 56 Fasc.3-4, pp.575-608.

Extensions

Definition edited by N. J. A. Sloane, Jun 25 2010

11 inserted Artur Jasinski, Jun 25 2010

16 inserted and 11 deleted Artur Jasinski, Jun 29 2010