"The Challenge of Large Numbers," by Richard E. Crandall. Scientific American, February 1997, pages 74-78.
Large numbers have always sparked the human imagination. Today's powerful computers are allowing explorations of properties of dizzingly large numbers. As Crandall puts it, "we can now characterize numbers about which earlier mathematicians could only dream." Looking back at some historical examples, Crandall remarks that Avogadro's number, which is on the order of 10 to the 23rd power, is fairly small among the numbers that scientists and mathematicians deal with today. He discusses efforts to find the prime factorizations of ever larger numbers, including two recently developed methods, the Number Field Sieve and the Elliptic Curve Method. Finding large primes is also an area of intense work, and Crandall describes various algorithms that have been used. The article includes some clever analogies and pictures which provide intriguing ways to envision just how large some numbers are.