The twenty-second Fermat number is composite

Doenias, J.; Norrie, C.; Young, Jeff

doi:10.1090/s0025-5718-1995-1277765-9

Cited by 11 publications

(10 citation statements)

References 9 publications

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“…Between the early 1980s and the present, massive compositeness tests have established that F 20 [31], F 22 [9,29], and as reported herein F 24 are all composite. Thus we now know that every F n with 5 ≤ n ≤ 32 is composite, and the smallest Fermat number of unknown status is, at the time of this writing, the 8-gigabit colossus F 33 .…”

Section: Computational History Of Fermat Numberssupporting

confidence: 54%

“…Incidentally in the case of F 22 , an entirely independent team of J. Carvalho and V. Trevisan [29] surprised the authors of [9] by announcing shortly after the latter group's run a machine proof of the same result, together with identical SelfridgeHurwitz residues. The two proofs used different software and machinery, and so there can be no reasonable doubt that F 22 is composite.…”

Section: Computational History Of Fermat Numbersmentioning

confidence: 99%

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The twenty-fourth Fermat number is composite

Mayer

Papadopoulos

2002

Math. Comp.

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Abstract. We have shown by machine proof that F 24 = 2 2 24 + 1 is composite. The rigorous Pépin primality test was performed using independently developed programs running simultaneously on two different, physically separated processors. Each program employed a floating-point, FFT-based discrete weighted transform (DWT) to effect multiplication modulo F 24 . The final, respective Pépin residues obtained by these two machines were in complete agreement. Using intermediate residues stored periodically during one of the floating-point runs, a separate algorithm for pure-integer negacyclic convolution verified the result in a "wavefront" paradigm, by running simultaneously on numerous additional machines, to effect piecewise verification of a saturating set of deterministic links for the Pépin chain. We deposited a final Pépin residue for possible use by future investigators in the event that a proper factor of F 24 should be discovered; herein we report the more compact, traditional Selfridge-Hurwitz residues. For the sake of completeness, we also generated a Pépin residue for F 23 , and via the Suyama test determined that the known cofactor of this number is composite.

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Section: Computational History Of Fermat Numberssupporting

confidence: 54%

Section: Computational History Of Fermat Numbersmentioning

confidence: 99%

The twenty-fourth Fermat number is composite

Mayer

Papadopoulos

2002

Math. Comp.

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“…In practice, ECM is only feasible on F n for moderate n: the limit is about the same as the limit of feasibility of Pépin's test (currently n ≤ 22, see [27]). …”

Section: Lenstra's Analysis Of Phasementioning

confidence: 99%

“…Because the F n grow rapidly in size, a method which factors F n may be inadequate for F n+1 . Historical details and references can be found in [21,35,36,44,74], and some recent results are given in [17,26,27,34].…”

Section: Introduction and Historical Summarymentioning

confidence: 99%

Factorization of the tenth Fermat number

Brent

1999

Math. Comp.

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Abstract. We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 , . . . , F 11 .

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“…It is known [12] that F n is prime for 0 ≤ n ≤ 4, and composite for 5 ≤ n ≤ 23. For a brief history of attempts to factor Fermat numbers, we refer to [3, §1] and [5].…”

Section: Introductionmentioning

confidence: 99%