A192211 Number of zero trace primitive elements in Galois field GF(2^n).

0, 0, 3, 4, 15, 12, 63, 72, 207, 290, 979, 864, 4095, 5250, 13485, 16496, 65535, 69948, 262143, 240000, 888888, 1319758, 4106167, 3318144, 16199225, 22355866, 56730861, 66385676, 266917769, 267331800, 1073741823, 1073809184, 3481794591, 5726404746, 16262257795

Offset: 1

Author

Pasha Zusmanovich, Jun 25 2011

Keywords

• nonn
• hard

References

• R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997. Chapter 2 discusses primitivity in sections 1-2 and trace in section 3.

Links

• W.-S. Chou and S.D. Cohen, Primitive elements with zero traces, Finite Fields and Their Appl. 7 (2001), 125-141; DOI:10.1006/ffta.2000.0284.

Crossrefs

Cf. A192212, A192213, A192214, A192215, A192216 for other primes.

Cf. A152049, A107222.

Programs

GAP

p := 2;
for n in [1..17] do
    F := GF(p^n);
    num := 0;
    for f in F do
        if (f = Zero(F)) then continue; fi;
        if (Trace(f) <> Zero(F)) then continue; fi;
        if (Order(f) <> Size(F) - 1) then continue; fi;
        num := num + 1;
    od;
    Print (num, ",");
od;

Formula

a(n) = n * A152049(n). [Joerg Arndt, Jul 03 2011]

Extensions

Terms 69948, ..., 1073809184 from Joerg Arndt, Jun 26 2011

Terms >1073809184 from Joerg Arndt, Jul 03 2011