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Compositeness.

Prime Fermat numbers

 
F0 = 3,       F1 = 5,       F2 = 17,       F3 = 257,       F = 65537
 

Completely factored Fermat numbers

 

 
FmExtension
5 641 6700417
6274177 67280421310721
759649589127497217 5704689200685129054721
81238926361552897 93461639715357977769163558199606896584051237541638188580280321
92424833 7455602825647884208337395736200454918783366342657 P99
1045592577 6487031809 4659775785220018543264560743076778192897 P252
11 319489 974849 167988556341760475137 3560841906445833920513 P564

 
 

Factorization known to be incomplete

 

 
FmExtension
12114689 26017793 63766529 190274191361 1256132134125569 568630647535356955169033410940867804839360742060818433 C1133
132710954639361 2663848877152141313 3603109844542291969 319546020820551643220672513 C2391
14116928085873074369829035993834596371340386703423373313 C4880
151214251009 2327042503868417 168768817029516972383024127016961 C9808
16825753601 188981757975021318420037633 C19694
1731065037602817 7751061099802522589358967058392886922693580423169 C39395
1813631489 81274690703860512587777 C78884
1970525124609 646730219521 37590055514133754286524446080499713 C157770
214485296422913 C631294
2264658705994591851009055774868504577 C1262577
23167772161 C2525215

 
 

Composite cofactors of Fermat numbers Fm

 

 
mDigitsYearProver
12 1133 2010Different anonymous provers
13 2391 1995 R. P. Brent
14 4880 2010Various different provers
15 9808 1997 R. P. Brent
16 19694 1996 R. P. Brent
17 39395 2011Different anonymous provers
18 78884 1999 R. E. Crandall
19 157770 2009Different anonymous provers
21 631294 1993R. E. Crandall, J. Doenias, C. Norrie & J. Young
22 1262577 2010 An anonymous prover
23 2525215 2000E. W. Mayer, J. S. Papadopoulos & R. E. Crandall

 
 

Composite Fermat numbers Fm without known factor

 

 
mDigitsYearProver
20 315653 1987D. A. Buell & J. Young
24 5050446 1999E. W. Mayer, J. S. Papadopoulos & R. E. Crandall

 
 

Count of factors according to difference n - m

 

 
n - mFrequency
2169
383
444
513
610
75
86
93
103
111
121

 
 
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