Although P�pin's test and Proth's theorem have been implemented on computers to prove the compositeness of some Fermat numbers, neither test gives a specific nontrivial factor.
22.
In contrast, the equivalently fast P�pin's test for any Fermat number can only be used on a much smaller set of very large numbers before reaching computational limits.
23.
However these arguments give quite different estimates, depending on how much information about Fermat numbers one uses, and some predict no further Fermat primes while others predict infinitely many Fermat primes.
24.
In the 90's, Barry Fagin published different algorithms for long integer multiplication in the sequential and parallel frameworks, the first algorithm being based on Fermat Number Transform ( FNT ).
25.
Any n that gives a prime for n n + 1 must be of the form 2 ^ 2 ^ k, with the result being a ( 2 k + k ) th Fermat number.
26.
From the last equation, we can deduce "'Goldbach's theorem "'( named after Christian Goldbach ) : no two Fermat numbers share a common integer factor greater than 1.
27.
This includes P�pin's test for Fermat numbers ( 1877 ), Proth's theorem ( around 1878 ), the Lucas Lehmer primality test ( originated 1856 ), and the generalized Lucas primality test.
28.
However, the very next Fermat number 2 32 + 1 is composite ( one of its prime factors is 641 ), as Euler discovered later, and in fact no further Fermat numbers are known to be prime.
29.
However, the very next Fermat number 2 32 + 1 is composite ( one of its prime factors is 641 ), as Euler discovered later, and in fact no further Fermat numbers are known to be prime.
30.
For the primes 1093 and 3511, it was shown that neither of them is a divisor of any Mersenne number with prime index nor a divisor of any Fermat number, because 364 and 1755 are neither prime nor powers of 2.
