They discuss topics like what prime numbers are, division and multiplication, congruences, Euler's theorem, testing for primality and factorization, Fermat numbers
, perfect numbers, the Newton binomial formula, money and primes, cryptography, new numbers and functions, primes in arithmetic progression, and sequences, with examples, some proofs, and biographical notes about key mathematicians.
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.
Along the way, the lay reader learns to appreciate how mathematicians derive such principles as Fermat numbers
, the Fibonacci Series, and Godel's incompleteness theorem.
2n] + 1, called Fermat numbers, determine when a regular polygon can be inscribed in a circle using ruler and compass.
In a similar effort, Lenstra and Manasse also organized a massive factor-by-mail project using the Number Field Sieve in order to factor the 9th Fermat number, [2.
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
This capacity to exploit Fermat numbers
allows the general-purpose supercomputer to surmount technical problems inherent in other machines.
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
finished factoring the tenth Fermat number
, proving that this 155-digit behemoth is the product of three prime numbers.