## Primes and Factorization

** News**:

April 2006: report on New Record Elliptic Curve (ECM) prime factor, 66-digits, titled
20 years of ECM, Zimmermann/Dodson accepted for presentation at ANTS VII
(Algorithmic Number Theory
Symposium, 7), Berlin, July 2006, proceedings in Springer Lecture Notes
in CS.
**
Sieving, 1928-2003: **

E. Tromer's presentation [abbreviated version]
**
Method of Trial Division is an exponential calculation **

A favorite method from high school, if n is the product
n = ab, with a smaller than b, then a ^{ 2 } smaller than
ab = n, and n has a prime factor smaller than the
squareroot of n. So check the primes in order to find a factor
of n; or else declare n to be a prime when the divisors reach the
squareroot of n . Prime Count using the Prime Number Theorem.

Trial Division also suggests primality is harder than factoring.
Both numerics and theory support the reverse; emphatically, in an easily
visible way.
Spring 2000 and
Spring 2006.

**
Methods: cryptographic numbers; small and medium prime factors
**

chronology
methods (ECM)

Lehigh ECMNET page ecmnet2004a;
update
ecmnet2004;
update
ecmnet2006.