172 is a prime number
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
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.8Prent and Pollard found the decomposition in 1980
where is a 62-digit prime number and k itself has 59 digits.
F.9After 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.10The 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.11As 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.12The 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.13The 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.14In 1963, Selfridge and Hurwitz proved that the 4933-digit number F14 is composed. More is not known so far.
F.15Kraitchik 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.16Selfridge 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.17Gostin found the only prime factor so far in 1978. In 1987, Baillie proved that the remaining factor C39444 is composed.
F.18Western 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.19Riesel 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.20In 1987, Buell and Young proved that the 315653 digit number F20 is composed. More is not known so far.
F.21Wrathall 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.22In 1993, Crandall, Doenias, Norrie, and Young proved that the 1262,612-digit number F22 is composed. More is not known so far.
F.23Pervushin 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.24In 1999 Mayer, Papadopoulos and Crandall proved that the 5050446 digit number F24 is composed. More is not known so far.
F.25There 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.26Only one prime factor is known (discovered by Wrathall in 1963). The further structure is unknown.
F.27Two prime factors are known (discovered in 1963 by Wrathall) and (discovered in 1985 by Gostin). The further structure is unknown.
F.28Only one prime factor is known (discovered by Taura in 1997). The further structure is unknown.
F.29Only one prime factor is known (discovered by Gostin and McLaughlin in 1980). The further structure is unknown.
F.30Two prime factors are known (discovered in 1963 by Wrathall) and (discovered in 1963 by Wrathall). The further structure is unknown.
F.31Only one prime factor is known (discovered in 2001 by Kruppa and Forbes). The further structure is unknown.
F.32Only 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.
|38||3||41||1903||Cullen, Cunningham, Western|
|99||16233||104||1979||Gostin, McLaughlin, Suyama|
|150||5439||154||1980||Gostin, McLaughlin, Suyama|
|286||78472588395||288||02.11.2002||Vasily Danilov, Durman|
|297||72677552745||301||13.02.2003||Vasily Danilov, Durman|
|343||4844391185||345||2001||Vasily Danilov, Durman|
|480||5673968845||484||2001||Vasily Danilov, Durman|
|569||6616590375||575||25.02.2003||Sergey Kuzmin, Durman|
|2023||29||2027||1979||Atkin, Rickert, Cormack, Williams|
|3310||5||3313||1979||Atkin, Rickert, Cormack, Williams|
|23069||681||23071||1997||Demichel, Gallot, Taura|
|2145351||3||2145353||21.02.2003||Cosgrave, Jobling, Woltman, Gallot|
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