since August 04, 1997   2,000 
These are factorization results of various kind of numbers.
I started this project (factorize various numbers under 100 digits)
on 1991 using 80286 PC with 80386 (8 MHz) board.
Used tools were UBASIC's factorization program (RHO, ECMX, MPQSX).
Target numbers are,
primorial±1 (products of prime), factorial±1 (n!±1),
primorial±next prime, Fibonacci and Lucas numbers,
Cullen and Woodall numbers, Wolstenholme numbers 1, 2 (numerator of Σ(1/n^{i})),
An, !n, Kn (sums of factorials), Euler and Bernoulli numbers.
I opened these results on my personal Web site on 1997. Many people joined to this project. By using new tools such that S.Tomabechi's "factor (ECM + PPMPQS)", Paul Zimmermann's ECM, under 100 digits numbers are completely factored on 27 Aug. 1998.
After finished to factor under 100 digits, several members reported further results over 100 digits, so I extended the search range up to 150 or certain digits, and added the new series of numbers, Wolstenholme numbers 3, 4 (numerator of Σ(1/n^{i})), fourier coefficients of j (τ) (elliptic modular function), partition numbers.
Now we are trying up to around 250 digits numbers.
I wish many contributors will join in.
Old log : 2008, 2007, 2006, 2005, 2004, 2003, 2002, 2001, 2000, 1999, until August 04, 1999
0. The Cunningham Project

Related to A3 Mersenne primes. Repunits. Fermat numbers. Primes of shape k.2^n+1 Richard K. Guy, "Unsolved Problems in Number Theory" Third Edition The Cunningham Project (Samuel S. Wagstaff) Factor Tables (Richard Brent) Computational Number Theory (CNT) (Paul Leyland) ECMNET (Paul Zimmerman) NFSNET (Jeff Gilchrist)
Mersenne Prime
Repunit primes and factors (Torbjon Granlund)
aba (Hisanori Mishima)  
1. #Pn (Primorial)±1

Related to A2 Primes connected with factorials "Primorial" is the product of primes from 2 to nth prime, i.e. #Pn = p_{1} * p_{2} * ... * p_{n}named from "prime" and "factorial". The number #Pn + 1 appears in the proof of the theorem that there are infinite amounts of prime numbers. But it is not apparent that these numbers are prime or not. As for primarity testing, the results are maintained at Chris K. Caldwell's site, Primorial and Factorial Primes  
2. n!±1

Related to A2 Primes connected with factorials It is not obvious that these numbers can be factorized or not. (n1)!+1 can be divided by n if and only if n is prime. (Wilson's Theorem). The results up to 400 are maintained at Paul Leyland's site,
N!+1 The current results up to 1000 are maintained at Andrew Walker's site, N!±1 Factoring Status As for primarity testing, the results are maintained at Chris K. Caldwell's site, Primorial and Factorial Primes  
3. Compositorial±1

"Compositorial" is the product of all composite numbers under n. It is a kind of complement of Primorial. Sometimes it is described as,
n! / #Nn
in this notation #Nn means the products of all primes under n. These initial results are owe to Robert G. Wilson, V.  
4. #Pn±NextPrime

#Pn±NextPrime is defined as,
p_{1} * p_{2} * ... * p_{n}±p_{n+1}These numbers does not also have explicit factors by the same reason to "1. #Pn (Primorial)±1". That is, if 2 * 3 * 5 * 7 + 1 (resp.  1) is dividedand if 2 * 3 * 5 * 7 + 11 (resp.  11) is dividedThe other number theoretic properties are unknown.  
5. n!±Next n

According to the same reason with the uncertainty of primarity of n!±1, it is also not obvious that these numbers can be factorized or not.  
6. Compositorial±Next Composite

It is natural to consider these numbers under the following table,
 
7. Fibonacci Numbers

Fibonacci number is defined as follows,F_{0} = 0For more details go here (Japanese version).
Blair Kelly is maintaining the results of from n=0 to 9999 Fn (n = 0 to 1000) 

8. Lucas Numbers

Lucas number is defined as follows,L_{0} = 2Fibonacci numbers and Lucas numbers are simultaneously defined as Lucas sequences. For more details go here (Japanese version).
Blair Kelly is maintaining the results of from n=0 to 9999 Ln (n = 0 to 1000) 

9. Cullen Numbers

Related to B20 Cullen and Woodall numbers Cullen number is defined as
Cn = n.2^{n} + 1 The current results are maintained at Paul Leyland's sites,
Cullen and Woodall numbers
The results of generalized form of Cullen numbers GC(a,n) = n.a^{n} + 1,
Generalized Cullen and Woodall numbers 

10. Woodall (Riesel) Numbers

Related to B20 Cullen and Woodall numbers Woodall (Riesel) number is defined as
Rn = n.2^{n}  1 The current results are maintained at Paul Leyland's sites,
Cullen and Woodall numbers
The results of generalized form of Woodall numbers GW(a,n) = n.a^{n}  1,
Generalized Cullen and Woodall numbers 

11. Wolstenholme Numbers
Wol 1
Wol 2
Wol 3
Wol 4
Numerator of sum of 1/p
Numerator of partial sum of log2
Numerator of Σ (μ(n)/n)
Numerator of Σ(1/i) 1 to m,

Wolstenholme's theorem
According to these theorems, the numerator of Σ(1/p^{i})
Numerator of sum of 1/p
Numerator of partial sum of log2
Numerator of Σ (μ(n)/n)
Numerator of Σ (1/i) 1 to m, where gcd(i,m)=1  
12. An, !n, Kn (Sums of factorials)
An
!n
Kn

Related to B43 Alternating sums of factorials B44 Sums of factorials The definitions are An = n!  (n1)! + (n2)!  ... (1)^{n} 

13. Euler Numbers

Related to B45 Euler numbers The definition is
Richard Guy ,"Unsolved Problems in Number Thoery" B45. How to compute is given here (Japanese version).
Nov. 25, 1998, I got an Email from Dr. Samuel S. Wagstaff, Jr. Related link
Factors of Bernoulli and Euler numbers 

14. Bernoulli Numbers

The definition is
Nov. 25, 1998, I got an Email from Dr. Samuel S. Wagstaff, Jr. Related link
Factors of Bernoulli and Euler numbers 

15. Fourier Coefficients of j (τ)

j (τ) is Elliptic modular function.
where s_{3} (n) is the sum of the cube of the divisors of n, i.e., where q = e ^{2πiz}  
16. Partition Numbers

A representation of n as a sum of positive integers
is called partition. Partition Number is the number of partitions of n.
Some numbers over 100 digits are the target of


17. Cyclotomic Numbers

Let Φ_{n}(X) be cyclotomic polynomials, the value of Φ_{n}(X) at X=x is called as Cyclotomic number.
The project of factoring cyclotomic numbers was started in 1984 How to compute is given here(Japanese version). Related link
Factorization of Cyclotomic Numbers 

18. Decimal expansion of Mathematical constants



19. Smarandache consecutive sequences

"Smarandache consecutive sequences" is the nth member of the consecutive sequence, e. g. Sm(11)=1234567891011, and RSm(11)=1110987654321. Current factorization status is maintained in the following Web site.
SMARANDACHE FACTORS AND REVERSE FACTORS We may consider any other variations of Smrandache notation, such as;
and reverse of all of those. 
Mint (Multiprecision integer library) Mint for Linux (588 KB) (Dec. 27, 2001) Mint for Windows (536 KB) (Dec. 27, 2001) 
"Mint" is a library for computing multiprecision integer. It is consisted of header files for representing multi digits numbers and many number theoretic functions. Especially, it includes the source files of PPSIQS and PPMPQS. I'm convinced that this will help many programmers who try to write effective factoring tools. 
PPSIQS Ver. 1.1 for Linux (188 KB) (Dec. 17, 2001) Ver. 1.1 for Windows (190 KB) (Nov. 09, 2001) 
PPSIQS is the double large primes procedure variation of the selfintializing quadratic sieve. In general, PPSIQS is faster than PPMPQS. Reference Scott Contini,"Factoring integers with the selfinitializing quadratic sieve" ,Performance report of PPSIQS "PPSIQS Version 1.1 of Satoshi Tomabechi: Statistics for 60 Runs", By Tom Hill (January 21, 2003) 
PPMPQS
Ver. 2.8 for Linux (184 KB) (Dec. 17, 2001) Ver. 2.8 for Windows (185 KB) (Jul. 14, 2001) 
If there remain composite numbers up to around 100 digits, and you convince there are no factors which can be found by ECM, then you should try to use this program. The following documents are attached in the zip files. Manual for Ver. 2.71 Average time of factorization using PPMPQS 
factor 
Factorization program for general numbers. It proceeds by rho (x^{2}+1, x^{2}1, x^{2}+3), p1 and PPMPQS. Usage is same as the following programs. 
p1 method  p1 method 
Chapter 12 Factorization Algorithms 
"Mathematician's Secret Room"  Appendix 2 Bibliography 

Chapter 12 (Japanese)  index (Japanese)  Appendix 2 (Japanese) 