A primitive trinomial of degree 6972593

Brent, Richard P.; Larvala, Samuli; Zimmermann, Paul

doi:10.1090/s0025-5718-04-01673-4

Cited by 4 publications

(2 citation statements)

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“…Since we had caught up with the GIMPS project, we thought (not for the last time) that this game had finished, and published our results in [4,5]. However, GIMPS soon overtook us by finding several larger Mersenne primes with exponents ±1 mod 8: r ′ 41 = 24 036 583, .…”

Section: The Classical Periodmentioning

confidence: 98%

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The great trinomial hunt

Brent¹,

Zimmermann²

2010

Preprint

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Section: The Classical Periodmentioning

confidence: 98%

“…The unique primitive trinomial of degree 6 972 593 is x 6972593 + x 3037958 + 1. It was named Bibury after the village that the three authors of[5] were visiting on the day that it was discovered.…”

mentioning

confidence: 99%

The great trinomial hunt

Brent¹,

Zimmermann²

2010

Preprint

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A multi-level blocking distinct-degree factorization algorithm

Brent¹,

Zimmermann²

2008

Finite Fields and Applications

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Ten new primitive binary trinomials

Brent

Zimmermann

2008

Math. Comp.

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Abstract. 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 r +x s +1 of degree r over GF(2) are exactly those given in Table 1, and the corresponding reciprocal trinomials x r + x r−s + 1. Proof. From the GIMPS Project [7], the integers r listed in Table 1 are exponents of Mersenne primes 2 r − 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]. 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 r +x 2 +1 permitted by Swan's theorem is easily ruled out in all known cases with r > 5; see the authors' web site [1]. Remarks. The integers r listed inOur 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].

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A primitive trinomial of degree 6972593

Abstract: The only primitive trinomials of degree 6972593 over GF(2) are x 6972593 + x 3037958 + 1 and its reciprocal.

Cited by 4 publications

References 4 publications

The great trinomial hunt

The great trinomial hunt

A multi-level blocking distinct-degree factorization algorithm

Ten new primitive binary trinomials

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