The main purpose of this paper is using the elementary method to study the estimate problem of S (Fn), and give a sharper lower bound estimate for it, where Fn = [2.sup.2n] + 1 is called the

Fermat number.

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."

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.

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, [2.sup.m] + 1, where m = [2.sup.9] (SN: 6/23/90, p.389).

* Two computer scientists factored a record-breaking, 155-digit

Fermat number (137: 389; 138: 90).

Systems Research Center in Palo Alto, Calif., 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.sub.4].

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.

This capacity to exploit

Fermat numbers allows the general-purpose supercomputer to surmount technical problems inherent in other machines.