VI. How to Use the Main Tables.

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Suppose we wish to _nd the factorization of 2147 􀀀 1 from the _rst main table

(on page 2). On line 147 of that table one _nds:

147 (3,7,21,49) 7_:2741672362528725535068727

The desired factorization is obtained from this line by multiplying together the

unparenthesized factors in the \Prime Factors" column on the lines 3, 7, 21, 49 and

147, i.e.,

3 7 21 49 147

2147 􀀀1 = 7:127:

z }| {

7_:337 :4432676798593:

z }| {

7_:2741672362528725535068727

= 73:127:337:4432676798593:2741672362528725535068727;

which can be checked by comparing it with the factorization in the Short 2􀀀 table.

When an ultimate prime factor in one of the main tables has many digits (prime

factors in the tables with 20 or fewer digits are given in full), it is listed only as

P, followed by the number of its decimal digits. The actual digits are given in the

full Appendix A, where the numbers are listed by: number of digits, base and line

number in the relevant table. For example, the _nal factor at line 71 in the main

10􀀀 table is given as P41, while the factor itself appears in Appendix A as

41 10,71􀀀 4 5994811347 8868463102 2172889522 3034301839

where the digits are in groups of ten. This factorization may be veri_ed by looking

in the Short 10􀀀 table.

In the _rst and second editions, some primality proofs were incomplete for a

few large primes. These were indicated PRP in the tables and Appendix A of those

editions. All large primes with at least 60 digits for which there is no proof given

in Appendix B have been proved prime by ECPP.

In some incomplete factorizations space has been left on the line for the insertion

of new factors when they are found. The cofactor in this case is indicated as

composite by a C followed by the number of its decimal digits and then the line

number repeated. For example, line 1025 of Table 2􀀀 (page 11) has the entry C137,

which, because it has no more than 140 digits, appears explicitly in Appendix C

as:

137 2,1025􀀀 1985892 : : : 4955410801

The composite cofactors in Appendix C are the numbers the authors have not

yet been able to factor. It is expected that interested people who have available

computer time will be able to factor many of these numbers.

For each base, there exists for certain exponents a second, independent factorization

called \Aurifeuillian", which makes the entries in these tables a little

more elaborate. For example, in Table 2LM, which gives the known factors of

24k􀀀2+1 = L.M= (22k􀀀1 􀀀2k +1)(22k􀀀1 +2k +1), each trinomial is given its own

line, denoted at the left by L and M (the line number not being repeated). Thus,

c

ci VI HOW TO USE THE MAIN TABLES

for example, the complete factorization of 2150 + 1 is obtained from the two lines

(on page 28)

150L (2,10L,30M,50M) 63901:13334701

M (6,10M,30L,50L) 1201:1182468601

as

2 10L 30M 50M 150L

5 5_ 1321 5_:268501 63901:13334701

and

6 10M 30L 50L 150M

13 41 61 101:8101 1201:1182468601

so we have the complete factorization

2150 + 1 = 53:13:41:61:101:1201:1321:8101:63901:268501:13334701:1182468601:

Similarly, the factorization of 1155 + 1, given in a three-line format, is obtained

from the lines (on page 146)

55 (1,5) L.M

L (11L) 21537414911:85480219991

M (11M) 4951:411841:131525983711

as

1 5 11L 55L

2:2:3 13421 58367 21537414911:85480219991

and

11M 55M

23:89:199 4951:411841:131525983711

so the complete factorization is 1155 + 1 =

22:3:23:89:199:4951:13421:58367:411841:21537414911:85480219991:131525983711

Throughout the tables when there are more factors than will _t on one line, they

are continued onto a second line, the factorization being broken at a multiplication

dot with the line number repeated at the right of this line. A few primes in the

tables are too long to _t on one line. These are broken with a continuation slash (n)

at the end of the _rst line. It should be noted that although the column heading

in the main tables is Prime Factors, the factors L.M as in line 55 in the example

above are also indicated in this column.

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