IV. Update to the Introduction for the Second Edition*.

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The tables of the _rst edition contained all factors known to the authors on

October 23, 1982. Since then more than two thousand new factorizations have been

discovered. Appendix C lists the smallest composite cofactors in the tables. In the

_rst edition it contained numbers with 51 to 64 digits. Now it gives numbers of 80 to

100 digits. The lists of \wanted" factorizations in the _rst edition had 25 numbers

with 52 to 71 digits. They have all been factored. Other \wanted" lists have since

been issued and many of their entries have been factored. The \wanted" lists (of the

second edition) now contain numbers with 86 to 291 digits. All remaining numbers

bn _ 1 with exponent n < 100 now appear on the \wanted" lists. All but nine of

the numbers for base b > 2 considered in [11] have been completely factored!

The smallest probable prime (PRP) in Appendix A of the _rst edition had 54

digits. Prime proofs have now been completed for all numbers up to 221 digits.

In this edition we have updated the tables and appendices to June 22, 1987, and

reviewed the developments in technology, factorization and primality testing which

have produced the recent advances. We also include a few references to recent work

which, though it has not contributed to this edition, may produce results in the

future.

Since some of the _rst edition tables had very few composite entries, and since

now most of the Aurifeuillians in those tables with base > 2 have been factored,

we decided to extend the higher base tables in the second edition. The numbers

we have added to these tables have been factored with about the same e_ort that

was applied to numbers in the _rst edition. The factoring for these extensions was

done mostly by Robert Silverman, Peter Montgomery and SSW. Some factors of

10n _ 1 came from Samuel Yates [262] and his updates.

The tables of the _rst edition were found to be nearly free of errors. The most

interesting error was the composite number 1223165341640099735851, which was

listed as a prime factor of 6175 􀀀 1. A. O. L. Atkin found that this number is

34840572551:35107498301. Other errors were the line references in the parentheses

of the _rst lines of 3,399+ and 11,209+. The _rst digit of k for the second factor

of F7 was missing. These errors have been corrected in the present edition.

The format of the tables and appendices has been changed a little in this

edition. In the _rst edition the decimal digits of a prime factor appeared in the

main tables if it had no more than 25 digits; otherwise it was placed in Appendix

A. In the _rst edition no penultimate prime factor had more than 25 digits. In the

present edition many penultimate prime factors have more than 25 digits and they

are given in full in the main tables. Only _nal prime factors are given in Appendix

A. On the other hand, _nal prime factors with 21 to 25 digits have been placed

in Appendix A when a line could be saved in the main tables. At the beginning

of each table _nal factors are given in full as long as they _t, so as to enhance the

pleasing triangular shape of the beginning of each table.

*The text of this part of the Introduction to the second edition is essentially that used in the

second edition. In this section the word \now" means, \on June 22, 1987." A few typographical

errors were corrected and the old status report was deleted. The new text for the third edition

appears in Section V.

lxxix

IV A 1 DEVELOPMENTS IN TECHNOLOGY lxxx

The inclusion of so many large new prime factors forced many more factorizations

to be split into two lines in the main tables. When it was necessary to break

a factorization we aligned the second line as follows, unless it was very long: If the

factorization was incomplete, put the \C" in the column for \C"'s. The last digit

of a broken complete factorization appears two columns to the left of the \C"'s.

(In the _rst edition we tried to align a dot in the second line with a dot in the _rst

line.)

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