Also "A fast algorithm for testing irreducibility of trinomials mod 2 (preliminary report)", Report PRG TR-13-00, 30 December 2000.
Preprint: dvi (24K), pdf (343K), ps (73K).
Preliminary Report: dvi (27K), pdf (208K), ps (100K).
If 2r - 1 is a Mersenne prime, then an irreducible trinomial of degree r is necessarily primitive. We give primitive trinomials for the Mersenne exponents r = 756839, 859433, and 3021377. The results for r = 859433 extend and correct some computations of Kumada et al [Mathematics of Computation 69 (2000), 811-814]. The two results for r = 3021377 are primitive trinomials of the highest known degree.
On 31 August 2002 we found a larger primitive trinomial, of degree 6972593. For further details see [214, 230].
For an extension to "almost irreducible" and "almost primitive" trinomials, see . A talk on Primitive and Almost Primitive Trinomials over GF(2) is available here.
For the application to random number generators, see [132, 211].
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