His exploration of elementary and advanced topics in classical number theory covers a range of numbers, including Fermat numbers
, Mersenne primes, powerful numbers, sublime numbers, Wieferich primes, insolite numbers, Sastry numbers, and voracious numbers.
More recently analysis of these so-called Fermat numbers
have found no other primes above [F.
2]+1, or 5," and later, "the first four Fermat numbers are prime, but [among] the rest, up to and now including the 24th, none are prime.
Almost all number theorists consider the first Fermat prime to be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so that the first five Fermat numbers are prime.
Crandall is now working with Kurowski to set up a system that would allow individuals and teams to join forces to factor large Fermat numbers.
Papadopoulos of the University of Maryland at College Park performed a massive calculation to prove that the 24th Fermat number, which is more than 5 million digits long, is not a prime.
No one has yet found even one prime factor of the 24th Fermat number, for example.
Dubbed "Little Fermat," after the 17th-century French mathematician Pierre de Fermat, it works with instructions and data expressed in 257-bit "words" and uses a special kind of arithmetic based on so-called Fermat numbers.
Number theory offers a way to rid calculations of these intrinsic errors by combining a special procedure called modular arithmetic with a set of numbers known as Fermat numbers.
built a digital signal-processing device based on Fermat arithmetic, demonstrating that the electronic circuitry needed to do modular arithmetic based on Fermat numbers can operate faster than the circuitry used for performing real-number operations.
finished factoring the tenth Fermat number, proving that this 155-digit behemoth is the product of three prime numbers.
To crack the tenth Fermat number (n = 9), Lenstra and Manasse used a recently invented method that significantly speeds the factoring of Fermat-type numbers.