Twelve New Primitive Binary Trinomials

We exhibit twelve new primitive trinomials over GF(2) of record degrees 42 643 801, 43 112 609, and 74 207 281. In addition we report the first Mersenne exponent not ruled out by Swan’s theorem [10] — namely 57 885 161 — for which none primitive trinomial exists. This completes the search for the currently known Mersenne prime exponents. Primitive trinomials of degree up to 32 582 657 were reported in [5]. We have completed a search for all new Mersenne prime exponents found by the GIMPS project [7]. Twelve new primitive trinomials were found (see Table 1).