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