A192507
Number of conjugacy classes of primitive elements in GF(3^n) which have trace 0.
Original entry on oeis.org
0, 0, 1, 2, 7, 14, 52, 104, 333, 870, 2571, 4590, 20440, 56736, 133782, 327558, 1265391, 2612694, 10188836, 20769420, 76562106
Offset: 1
Cf.
A027385 (number of primitive polynomials of degree n over GF(3)).
-
p := 3;
a := function(n)
local q, k, cnt, x;
q:=p^n; k:=GF(p, n); cnt:=0;
for x in k do
if Trace(k, GF(p), x)=0*Z(p) and Order(x)=q-1 then
cnt := cnt+1;
fi;
od;
return cnt/n;
end;
for n in [1..16] do Print (a(n), ", "); od;
-
# much more efficient
p=3; # choose characteristic
for n in range(1,66):
F = GF(p^n, 'x')
g = F.multiplicative_generator() # generator
vt = vector(ZZ,p) # stats: trace
m = p^n - 1 # size of multiplicative group
# Compute all irreducible polynomials via Lyndon words:
for w in LyndonWords(p,n): # digits of Lyndon words range form 1,..,p
e = sum( (w[j]-1) * p^j for j in range(0,n) )
if gcd(m, e) == 1: # primitive elements only
f = g^e
t = f.trace().lift(); # trace (over ZZ)
vt[t] += 1
print(vt[0]) # choose index 0,1,..,p-1 for different traces
# Joerg Arndt, Oct 03 2012
A152049
Number of conjugacy classes of primitive elements in GF(2^n) which have trace 0.
Original entry on oeis.org
0, 0, 1, 1, 3, 2, 9, 9, 23, 29, 89, 72, 315, 375, 899, 1031, 3855, 3886, 13797, 12000, 42328, 59989, 178529, 138256, 647969, 859841, 2101143, 2370917, 9204061, 8911060, 34636833, 33556537, 105508927, 168423669, 464635937
Offset: 1
a(3)=1 because of the two primitive degree 3 polynomials over GF(2), namely t^3+t+1 and t^3+t^2+1, only the former has a zero next-to-highest coefficient.
Similarly, a(13)=315, because of half (4096) of the 8192 elements of GF(2^13) have trace 0 and all except 0 (since 1 has trace 1) are primitive, so there are 4095/13=315 conjugacy classes of primitive elements of trace 0.
-
a := function(n)
local q,k,cnt,x; q:=2^n; k:=GF(2,n); cnt:=0;
for x in k do
if Trace(k, GF(2), x)=0*Z(2) and Order(x)=q-1 then
cnt := cnt+1;
fi;
od;
return cnt/n;
end;
for n in [1..32] do Print (a(n), ", "); od;
More terms (13797...8911060) by
Joerg Arndt, Jun 26 2011.
More terms (34636833...464635937) by
Joerg Arndt, Jul 03 2011.
A192215
Number of zero trace primitive elements in Galois field GF(11^n).
Original entry on oeis.org
0, 0, 48, 320, 5925, 33936, 691880, 5107200, 69610716, 628484000
Offset: 1
A192509
Number of conjugacy classes of primitive elements in GF(7^n) which have trace 0.
Original entry on oeis.org
0, 0, 3, 20, 160, 846, 5426, 27360, 196740, 1215548, 8552408, 37330020
Offset: 1
-
p := 7;
a := function(n)
local q, k, cnt, x;
q:=p^n; k:=GF(p, n); cnt:=0;
for x in k do
if Trace(k, GF(p), x)=0*Z(p) and Order(x)=q-1 then
cnt := cnt+1;
fi;
od;
return cnt/n;
end;
for n in [1..16] do Print (a(n), ", "); od;
-
# See A192507 (change first line p=3 to p=7)
A192511
Number of conjugacy classes of primitive elements in GF(13^n) which have trace 0.
Original entry on oeis.org
0, 0, 18, 112, 1904, 17184, 229848, 1686008, 29713758
Offset: 1
-
p := 13;
a := function(n)
local q, k, cnt, x;
q:=p^n; k:=GF(p, n); cnt:=0;
for x in k do
if Trace(k, GF(p), x)=0*Z(p) and Order(x)=q-1 then
cnt := cnt+1;
fi;
od;
return cnt/n;
end;
for n in [1..16] do Print (a(n), ", "); od;
-
# See A192507 (change first line p=3 to p=13)
A192508
Number of conjugacy classes of primitive elements in GF(5^n) which have trace 0.
Original entry on oeis.org
0, 0, 4, 8, 54, 140, 1116, 2976, 19828, 58388, 443892, 1036180, 9390024, 27996724, 175396812
Offset: 1
-
p := 5;
a := function(n)
local q, k, cnt, x;
q:=p^n; k:=GF(p, n); cnt:=0;
for x in k do
if Trace(k, GF(p), x)=0*Z(p) and Order(x)=q-1 then
cnt := cnt+1;
fi;
od;
return cnt/n;
end;
for n in [1..16] do Print (a(n), ", "); od;
-
# See A192507 (change first line p=3 to p=5)
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