Smarandache function, the Fermat number
, lower bound estimate, elementary method.
Researchers at Bell Communications Research developed a computer software program that divided the project of factoring the ninth Fermat number
into individual tasks, and then used the Internet to recruit researchers to this task.
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
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.
In 1990, Lenstra and a colleague used the method to factor a 155-digit Fermat number
Two computer scientists factored a record-breaking, 155-digit Fermat number
(137: 389; 138: 90).
Using a Fermat number
as the divisor in modular arithmetic provides a handy way of speeding up certain types of calculations and circumvents the need to deal with real numbers.
finished factoring the tenth Fermat number
, proving that this 155-digit behemoth is the product of three prime numbers.
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.
Almost all number theorists consider the first Fermat prime to be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so that the first five Fermat numbers