Ten new primitive binary trinomials

We exhibit ten new primitive trinomials over GF(2) of record degrees 24 036 583, 25 964 951, 30 402 457, and 32 582 657. This completes the search for the currently known Mersenne prime exponents. Primitive trinomials of degree up to 6 972 593 were previously known [4]. We have completed a search for all known Mersenne prime exponents [7]. Ten new primitive trinomials were found. Our results are summarized in the following theorem: Theorem 1. For the integers r listed in Table 1, the primitive trinomials x+x+1 of degree r over GF(2) are exactly those given in Table 1, and the corresponding reciprocal trinomials x + xr−s + 1. Proof. From the GIMPS Project [7], the integers r listed in Table 1 are exponents of Mersenne primes 2 − 1. Thus, irreducible trinomials of degree r are necessarily primitive. Irreducibility of the trinomials listed in Table 1 follows from the authors’ computations, using the new algorithm described in [5, 6] (verified using the algorithm of [3] and independently verified by Allan Steel using Magma). Finally, the fact that no irreducible trinomials were missed during the search, for those degrees r, follows from the certificates given on the authors’ web pages [1]. Remarks. The integers r listed in Table 1 are the known Mersenne exponents of the form r = ±1 mod 8 in the interval [100 000, 32 582 657]. For smaller exponents, omitted to save space, see [10] or our web site [1]. According to the GIMPS Project [7], the list is complete for r ≤ 16 300 000. Known Mersenne exponents of the form r = ±3 mod 8 for r > 5 cannot be the degrees of irreducible trinomials due to Swan’s theorem [12]; the possibility x+x+1 permitted by Swan’s theorem is easily ruled out in all known cases with r > 5; see the authors’ web site [1]. Our search used a new algorithm [5, 6] relying on fast arithmetic in GF(2)[x], whose details are given in [2]. Another significant improvement over previous work is that certificates were produced; this enables one to check easily that the claimed nonprimitive trinomials are indeed reducible. A certificate is simply an encoding of a nontrivial factor of smallest degree. A 2.4Ghz Intel Core 2 takes only 15 minutes to check the certificates of all 16 291 325 reducible trinomials (s ≤ r/2) of degree r = 32 582 657 with our check-ntl program based on NTL [11]. Received by the editor April 15, 2008. 2000 Mathematics Subject Classification. Primary 11B83, 11Y16; Secondary 11-04, 11T06, 11Y55, 12-04.