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.sub.4].
152), you state, "The first Fermat number is [2.sup.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.
Numbers of the form [2.sup.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.sup.512] + 1, which as the number whose factorization was 'Most Wanted' by mathematicians.
The first four Fermat numbers
are prime, but the fifth is divisible by 641.
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
. These characteristics clearly differentiate the new machine from conventional computers.
have the form 2'"+1, where m = [2.sup.n] and n is zero or a positive whole number.