Factoring a very large positive integer is a difficult problem.However, Monte Carlo methods, which employ statistical techniques to test for the primality of very large numbers, are beyond the scope of this book.

In 1202, Fibonacci's Book of Calculations contained a list of all the primes and composite natural numbers less than or equal to 100. Pietro Cataldi's Treatise on Perfect Numbers published in Bologna in 1603 contains factors of all positive integers less than 750. In 1657, Frans van Schooten listed all the primes up to 9929. In 1659, the first extensive factor tables were constructed and published by Johann Heinrich Rahn, Latinized Rohnius, in his Algebra. Rahn included all factors of the numbers from 1 to 24 000, omitting from the tables all multiples of 2 and 5. Rahn, who was a student of John Pell's at Zurich, introduced the symbol to denote division. In 1668, Thomas Brancker determined the least factor, greater than 1, of each integer less than 105. Johann Lambert, the FIrst to show that PIE was irrational, published an extensive table of least factors of the integers up to 102 000 in 1770.

Others have not been so fortunate. In 1776, Antonio Felkel, a Viennese schoolteacher, constructed factor tables for the first 408 000 positive integers. The tables were published at the expense of the Austrian Imperial Treasury but, because of the disappointing number of subscribers, the Treasury confiscated all but a few copies and used the paper for cartridges in a war against the Turks, a dubious mathematical application to warfare at best. In 1856, A.L. Crelle, determined the first six million primes and, in 1861, Zacharias Dase extended Crelle's table to include the first nine million primes. Crelle founded and, for many years, edited and published the prestigious Journal fuÈr die reine und angewandte Mathematik.

In 1863, after 22 years of effort to complete the task, J.P. Kulik, a professor at the University of Prague, published factor tables that filled several volumes. His tables included the factors, except for 2, 3, and 5, of the first 100 million positive integers. He donated his work to the library atthe University of Prague, but unfortunately, through someone's negligence, the second volume, the factorizations of the integers from 12 642 600 to 22 852 800, was lost. In 1910, D.H. Lehmer published factor tables for the integers up to 10 million. Lehmer worked on a long table equipped with rollers at each end. For small primes, he made paper stencils with holes through which he recorded multiples.

Fermat devised a number of ingenious methods to factor integers. We know of his work correspondence through his correspondence with Marin Mersenne, a Franciscan friar, number theory enthusiast and philosopher who corresponded with a number of mathematicians and scientists including Galileo and Torricelli. Mersenne was the leader of a group that met regularly in Paris in the 1630s to discuss scientific topics. He once asked Fermat whether he thought that 100 895 598 269 was prime. After a short period, Fermat replied that it was not and, in fact, it was the product of 898 423 and 112 303.

The basis for one of Fermat's factoring methods depends on the ability to write the integer to be factored as the difference of two integral squares. In this case, 100 895 598 269 =505 3632^2 - 393 0602 ^2 = (505 363 + 393 060)(505 363 - 393 060). Fermat assumed that the integer n to be factored was odd, hence its two factors u and v must also be odd.

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