172 is a prime number

Fermat primes


These are prime numbers of the form. In order for a number of this kind to be a prime number, it must itself be a power of 2, i.e. of the form. Because if it has an odd factor, i.e. if it is a natural number, then one has the decomposition for with

d. H. is not a prime number. So one calls the -th Fermat number and asks whether it is actually a prime number.

The prime numbers indeed result for

and in 1640 Pierre de Fermat suggested that this was always the case. But already in 1732 Leonhard Euler was able to support the dismantling

demonstrate. Because 641 divides the number
. On the other hand, because of 641 also divides the number, so too. A total of 641 therefore divides the difference.

In 1880, Landry disproved Fermat's conjecture for the case as well, by finding the following decomposition

After some prime factors had already been found for larger Fermat numbers (see the tables below), the complete decomposition of 1970 by Morrison and Brillhart was not successful:

Euler and Lucas had shown that prime factors of a Fermat number always depend on the form

are, for example

,
and
.

To date, no other Fermat prime numbers are known other than those given above, but 213 Fermat numbers are definitely known to be composite. is the greatest of them. Although the Fermat's numbers grow rapidly as the index increases, many more have now been broken down into prime factors. The smallest Fermat number for which a complete prime factorization is not yet known is the 1234-digit number. The first five prime factors of it are known, namely and. We also know that the remaining factor C1187, which at least still consists of 1187 digits, is composed (composite), but no concrete breakdown has yet been made for it.

Up to now we only know of some Fermat's numbers () that they are composed without knowing a single factor. Many Fermat numbers () are so far neither known whether they are prime numbers nor that they are composed.


The following overviews and tables show the current status of the search for prime factors of Fermat's numbers, which has also been carried out on the Internet for several years with the FERMAT program by Leonid Durman. A total of 246 prime factors of Fermat's numbers are known.

The full information on to can already be found in the text above.

F.8

Prent and Pollard found the decomposition in 1980

where is a 62-digit prime number and k itself has 59 digits.

F.9

After Western found the factor as early as 1903, the factorization was complete

not until 1990 Lenstra, Manasse and others. The second factor is the same

and in the third factor the number k already has 96 digits.

F.10

The first factor was found in 1953 by Selfridge, the second in 1962 by Brillhart. In 1995, Brent achieved full factorization by adding the third factor

and determined the fourth factor. The number k already has 248 digits.

F.11

As early as 1899, Cunningham had found the first two factors and. The full factorization was then achieved in 1988 by Brent and Morain, where Brent first found the two other factors and and then, together with Morain, the last factor in which k itself has 560 digits.

F.12

The known prime factors have already been given above. Lucas and Pervushin found the first in 1877, the next two Westerns in 1903. Hallyburton and Brillhart found the fourth factor in 1974 and the last Baillie in 1986, which also proved that the other factor C1187 is composed.

F.13

The first prime factor was found by Hallyburton and Brillhart in 1974, the second and third by Crandall in 1991, and the last Brent to date in 1995, which also demonstrated that the remaining factor C2391 is composite.

F.14

In 1963, Selfridge and Hurwitz proved that the 4933-digit number F14 is composed. More is not known so far.

F.15

Kraitchik found the first prime factor as early as 1925, the second Gostin in 1987, the last for the time being Crandall and van Halewyn in 1997. Also in 1997, Brent demonstrated that the remaining factor is C9808.

F.16

Selfridge found the first prime factor in 1953, the last one so far by Crandall and Dilcher in 1996. In the same year Brent proved that the missing factor C19694 is a composite.

F.17

Gostin found the only prime factor so far in 1978. In 1987, Baillie proved that the remaining factor C39444 is composed.

F.18

Western found the first prime factor as early as 1903, the second Crandall, McIntosh and Tardif in 1999. Crandall proved in 1999 that the remaining factor C78884 is composed.

F.19

Riesel found the first prime factor in 1962, the second Wrathall in 1963. Crandall, Doenias, Norrie and Young proved in 1993 that the remaining factor C157804 is composed.

F.20

In 1987, Buell and Young proved that the 315653 digit number F20 is composed. More is not known so far.

F.21

Wrathall found the only prime factor so far in 1963. Crandall, Doenias, Norrie and Young then showed in 1993 that the remaining factor C631294 is composed.

F.22

In 1993, Crandall, Doenias, Norrie, and Young proved that the 1262,612-digit number F22 is composed. More is not known so far.

F.23

Pervushin found the first prime factor as early as 1878. Mayer, Papadopoulos and Crandall then showed in 2000 that the remaining factor is composed of C2525215.

F.24

In 1999 Mayer, Papadopoulos and Crandall proved that the 5050446 digit number F24 is composed. More is not known so far.

F.25

There are three known prime factors (discovered in 1963 by Wrathall), (discovered in 1985 by Gostin) and (discovered in 1987 by McLaughlin). The further structure is unknown.

F.26

Only one prime factor is known (discovered by Wrathall in 1963). The further structure is unknown.

F.27

Two prime factors are known (discovered in 1963 by Wrathall) and (discovered in 1985 by Gostin). The further structure is unknown.

F.28

Only one prime factor is known (discovered by Taura in 1997). The further structure is unknown.

F.29

Only one prime factor is known (discovered by Gostin and McLaughlin in 1980). The further structure is unknown.

F.30

Two prime factors are known (discovered in 1963 by Wrathall) and (discovered in 1963 by Wrathall). The further structure is unknown.

F.31

Only one prime factor is known (discovered in 2001 by Kruppa and Forbes). The further structure is unknown.

F.32

Only one prime factor is known (discovered by Wrathall in 1963). The further structure is unknown.
As mentioned above, the list of Fermat numbers begins with, the structure of which is still completely unknown. The sporadic knowledge of known prime factors of even larger Fermatian numbers is shown in the following table.
m k n Discovery date Explorer
36 5 39 1886 Seelhoff
36 3759613 38 1981 Gostin, McLaughlin
37 1275438465 39 1991 Gostin
38 3 41 1903 Cullen, Cunningham, Western
38 2653 40 1963 Wrathall
39 21 41 1956 Robinson
42 43485 45 1963 Wrathall
43 212675402445 45 2000 Samidoost, Durman
48 2139543641769 50 2001 Bodschwinna, Durman
52 4119 54 1963 Wrathall
52 21626655 54 1982 basement, cellar
55 29 57 1956 Robinson
58 95 61 1957 Robinson
61 54985063 66 1986 Gostin
62 697 64 1977 Shippee
63 9 67 1956 Robinson
64 17853639 67 1986 Gostin
66 7551 69 1977 Shippee
71 683 73 1977 Shippee
72 76432329 74 1986 Gostin
73 5 75 1906 Morehead
75 3447431 77 1982 Gostin
77 425 79 1957 Robinson, Selfridge
77 5940341195 79 1957 Taura
81 271 84 1957 Robinson, Selfridge
88 119942751127 90 2001 Nohara, Durman
90 198922467387 92 26.03.2001 Grobstich, Durman
91 1421 93 1977 Shippee
93 92341 96 1979 Baillie
94 482524552001 97 18.04.2001 Grobstich, Durman
99 16233 104 1979 Gostin, McLaughlin, Suyama
107 1289179925 111 1992 Gostin
116 3433149787 120 1999 Taura
117 7 120 1956 Robinson
122 5234775 124 1986 Gostin
125 5 127 1956 Robinson
133 88075576149 135 2001 Samidoost, Durman
142 8152599 145 1986 Gostin
144 17 147 1956 Robinson
146 37092477 148 1987 Gostin
147 3125 149 1979 Gostin, McLaughlin
147 124567335 149 1990 Gostin
150 1575 157 1956 Robinson
150 5439 154 1980 Gostin, McLaughlin, Suyama
164 1835601567 167 1993 Gostin
172 20569603303 174 2001 Durman
178 313047661 180 1991 Gostin
184 117012935 187 1990 Gostin
201 4845 204 1980 Gostin, McLaughlin
205 232905 207 1984 basement, cellar
207 3 209 1956 Robinson
215 32111 217 1980 Suyama
226 15 229 1956 Robinson
228 29 231 1956 Robinson
230 372236097 232 2000 Durman
232 70899775 236 1991 Gostin
250 403 252 1957 Robinson, Selfridge
251 85801657 254 1991 Gostin
255 629 257 1979 Baillie
256 36986355 258 1991 Gostin
256 36986355 258 1991 Gostin
259 36654265 262 1991 Gostin
267 177 271 1957 Robinson, Selfridge
268 21 276 1956 Robinson
275 22347 279 1984 basement, cellar
284 7 290 1956 Robinson
284 1061341513 286 2000 Durman
286 78472588395 288 02.11.2002 Vasily Danilov, Durman
287 5915 289 1980 Suyama
297 72677552745 301 13.02.2003 Vasily Danilov, Durman
298 247 302 1979 Baillie
301 7183437 304 1990 Gostin
316 7 320 1956 Robinson
329 1211 333 1981 Suyama
334 27609 341 1984 basement, cellar
338 27654487 342 1990 Gostin
343 4844391185 345 2001 Vasily Danilov, Durman
353 18908555 355 1990 Gostin
370 573230511 373 2000 Durman
375 3733251 377 1986 Gostin
376 810373 378 1986 Gostin
380 321116871 385 2000 Durman
398 120845 401 1984 basement, cellar
416 8619 418 1981 Suyama
416 38039 419 1984 basement, cellar
417 118086729 421 1992 Gostin
431 5769285 434 1990 Gostin
452 27 455 1956 Robinson
468 27114089 471 1992 Gostin
480 5673968845 484 2001 Vasily Danilov, Durman
544 225 547 1979 Baillie
547 77377 550 1986 Gostin
556 127 558 1976 Matthew, Williams
569 6616590375 575 25.02.2003 Sergey Kuzmin, Durman
579 63856313 581 1999 Taura
620 10084141 624 1992 Gostin
635 4258979 645 1991 Gostin
637 11969 643 1984 basement, cellar
642 52943971 644 1999 Taura
667 491628159 669 2001 Taura
692 717 695 1979 Atkin, Rickert
723 554815 730 1991 Gostin
744 17 747 1976 Matthew, Williams
851 497531 859 1988 Gostin
885 16578999 887 1992 Gostin
906 57063 908 1986 Gostin
931 1985 933 1980 basement, cellar
971 541664191 976 2002 Komin, Durman
1069 137883 1073 1992 Gostin
1082 82165 1084 1991 Gostin
1114 11618577 1116 2001 Gostin
1123 25835 1125 1987 Gostin
1225 79707 1231 1991 Gostin
1229 29139 1233 1987 Gostin
1394 62705223 1396 2001 Taura
1451 13143 1454 1986 Gostin
1551 291 1553 1979 Atkin, Rickert
1598 10923781 1600 2000 Taura
1849 98855 1851 1992 Gostin
1945 5 1947 1957 Robinson
1990 150863 1993 1995 Taura
2023 29 2027 1979 Atkin, Rickert, Cormack, Williams
2059 591909 2063 2000 Ballinger, Gallot
2089 431 2099 1983 Suyama
2456 85 2458 1979 Atkin, Rickert
3310 5 3313 1979 Atkin, Rickert, Cormack, Williams
3506 501 3508 1986 Gostin
3723 13308899 3725 2002 Fougeron
4250 173373 4252 1999 Kerchner
4258 1435 4262 1993 Gostin
4332 2466157 4334 2002 Fougeron
4724 29 4727 1979 Cormack, Williams
5320 21341 5323 1998 Taura
5531 1503975 5533 2002 Gostin
5792 8872947 5794 25.02.2003 Fougeron
5957 421435 5960 2001 Gostin
6208 763 6210 1993 Gostin
6355 115185 6358 2000 Kerchner
6390 303 6393 1993 Gostin
6537 17 6539 1979 Cormack, Williams
6835 19 6838 1978 basement, cellar
6909 6021 6912 1993 Gostin
7181 168329 7187 2000 Gostin
7309 145 7312 1992 Dubner
8239 7473 8242 1998 Prethaler, Gallot
8555 645 8557 1993 Dubner
9322 8247 9324 1999 Kerchner
9428 9 9431 1983 basement, cellar
9448 19 9450 1980 basement, cellar
9549 1211 9551 1998 Taura
9691 260435 9693 2001 Gostin
11695 203355 11703 2001 Gostin
12185 81 12189 1993 Dubner
13250 351 13252 1996 Taura
13623 48265 13626 2000 Gostin
14252 1173 14254 1997 Taura
14276 157 14280 1996 Taura
14528 17217 14530 2000 Gostin
15161 55 15164 1993 Dubner
17906 135 17909 1996 Taura
18749 11 18759 1992 Dubner
18757 33 18766 1993 Dubner
19211 13323 19220 2001 Gostin
22296 4777 22298 2001 Gostin
23069 681 23071 1997 Demichel, Gallot, Taura
23288 19 23290 1992 Dubner
23471 5 23473 1984 basement, cellar
24651 99 24653 1996 Taura
25006 57 25010 1993 Young
28281 81 28285 1996 Taura
35563 357 35567 2000 Melo, Gallot
41894 4935 41897 2001 Fougeron
43665 2495 43667 2001 Fougeron
49093 165 49095 1998 Gallot
50078 7619 50081 2002 Axelsson
63679 169 63686 1998 Dubner, Gallot
83861 99 83863 1998 Gusev, Gallot
90057 189 90061 1999 Morenus, Gallot
91213 585 91215 2001 Axelsson
94798 21 94801 1995 Young
95328 7 95330 1994 Young
104448 927 104451 2001 Oleynick
113547 39 113549 1999 Renze, Gallot
114293 13 114296 1995 Young
125410 5 125413 1995 Young
142460 159 142462 2000 Melo, Gallot
146221 57 146223 2000 Lewis, Gallot
157167 3 175169 1995 Young
213319 3 213321 1996 Young
270091 63 270094 2002 Taura, Gallot
282717 51 282719 2002 Odermatt, Gallot
303088 3 303093 1998 Young
382447 3 382449 1999 Cosgrave, Gallot
2145351 3 2145353 21.02.2003 Cosgrave, Jobling, Woltman, Gallot