Table of trinomials x^r + x^s + 1 over GF(2) with "large" primitive factors.
RPB, 20010830..20031202.

If rm is a Mersenne exponent (so 2^rm - 1 is a Mersenne prime)
and r = rm + rd where 1 < rd < rm (usually rd as small as possible),
we want x^r + x^s + 1 to have an irreducible (necessarily primitive)
factor of degree rm. The other ("small") irreducible factors have total 
degree rd. The period P = f.(2^rm - 1) where f depends on the small 
irreducible (not necessarily primitive) factors. 
Let F be the product of these small factors.

For most of the small examples, see Blake, Gao and Lambert, 
"Construction and Distribution Problems for Irreducible 
Trinomials over Finite Fields", in Applications of Finite Fields 
(D. Gollmann, ed.), Oxford, Clarendon Press, 1996, 19-32
(available from www.math.clemson.edu/~sgao/pub.html).

Larger examples (with rm > 500) are obtained using our highd program
(checked using Magma, NTL and/or an independent C program).

rm      rd      s       f                       F

61      5       17      31              x^5 + x^3 + x^2 + x + 1

107     2       8       3               x^2 + x + 1
                14      3               x^2 + x + 1
                17      3               x^2 + x + 1

2203    3       355     7               x^3 + x^2 + 1

4253    8       1806    255             x^8 + x^7 + x^2 + x + 1
                1960    85              x^8 + x^6 + x^5 + x^4 + x^2 + x + 1

9941    3       1077    7               x^3 + x^2 + 1

11213   6       227     63              x^6 + x^5 + x^3 + x^2 + 1

21701   3       6999    7               x^3 + x^2 + 1
                7587    7               x^3 + x^2 + 1

86243   2       2288    3               x^2 + x + 1

216091  12     42930    3937 = 31.127   x^12 + x^11 + x^5 + x^3 + 1 =
                                        (x^5 + x^4 + x^3 + x + 1).
                                        (x^7 + x^5 + x^4 + x^3 + x^2 + x + 1)

1257787 3      74343    7               x^3 + x^2 + 1 
 
1398269 5     417719    21 = 3.7        x^5 + x^4 + 1 = (x^2+x+1).(x^3+x+1)

2976221 8    1193004    85              x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + 1

13466917                                Not yet attempted 
                                        (none for rd < 3 or rd = 4)

20996011                                Not yet attempted