Algebraic and Aurifeuillean factorizations of generalized Cullen and Woodall
numbers, n*a^n \pm 1, fall into a few cases, frequently overlapping.  Note
that we only consider 100<n<=1000, a<=12.

This document falls naturally into two parts.  The first part describes each
case in turn.  The second part gives the known factors of each composite
factor where at least two such of a given integer are not yet fully
factored.  These composites, being much smaller than the GC or GW numbers of
which they are factors, should be much easier to factor.

The names of those who first brought them to my attention are given in
braces.

It is possible, indeed probable, that some algebraic factorizations have
been missed and I would appreciate being informed of any further
discoveries.


==================================
PART 1 --- general factorizations.
==================================


Class 1: GW numbers, no restrictions on a or n.

GW(X,144) [12*X^72 - 1, 1; 12*X^72 + 1, 1]
GW(X,196) [14*X^98 - 1, 1; 14*X^98 + 1, 1]
GW(X,216) [6*X^72 - 1, 1; 36*X^144 + 6*X^72 + 1, 1]
GW(X,256) [2*X^32 - 2*X^16 + 1, 1; 2*X^32 - 1, 1; 2*X^32 + 1, 1; 2*X^32 + 2*X^16 + 1, 1; 16*X^128 + 1, 1]
GW(X,324) [18*X^162 - 1, 1; 18*X^162 + 1, 1]
GW(X,400) [20*X^200 - 1, 1; 20*X^200 + 1, 1]
GW(X,484) [22*X^242 - 1, 1; 22*X^242 + 1, 1]
GW(X,576) [24*X^288 - 1, 1; 24*X^288 + 1, 1]
GW(X,676) [26*X^338 - 1, 1; 26*X^338 + 1, 1]
GW(X,729) [9*X^243 - 1, 1; 81*X^486 + 9*X^243 + 1, 1]
GW(X,784) [28*X^392 - 1, 1; 28*X^392 + 1, 1]
GW(X,900) [30*X^450 - 1, 1; 30*X^450 + 1, 1]


Class 2: GC numbers, no restrictions on a or n.

GC(X,216) [6*X^72 + 1, 1; 36*X^144 - 6*X^72 + 1, 1]
GC(X,324) [18*X^162 - 6*X^81 + 1, 1; 18*X^162 + 6*X^81 + 1, 1]
GC(X,729) [9*X^243 + 1, 1; 81*X^486 - 9*X^243 + 1, 1]


Class 3: a is a square (so a=4 or a=9), n is an odd square.

GW(X^2, 121) [11*X^121 - 1, 1; 11*X^121 + 1, 1]
GW(X^2, 169) [13*X^169 - 1, 1; 13*X^169 + 1, 1]
GW(X^2, 225) [15*X^225 - 1, 1; 15*X^225 + 1, 1]
GW(X^2, 289) [17*X^289 - 1, 1; 17*X^289 + 1, 1]
GW(X^2, 361) [19*X^361 - 1, 1; 19*X^361 + 1, 1]
GW(X^2, 441) [21*X^441 - 1, 1; 21*X^441 + 1, 1]
GW(X^2, 529) [23*X^529 - 1, 1; 23*X^529 + 1, 1]
GW(X^2, 625) [25*X^625 - 1, 1; 25*X^625 + 1, 1]
GW(X^2, 729) [3*X^243 - 1, 1; 3*X^243 + 1, 1; 9*X^486 - 3*X^243 + 1, 1; 9*X^486 + 3*X^243 + 1, 1]
GW(X^2, 841) [29*X^841 - 1, 1; 29*X^841 + 1, 1]
GW(X^2, 961) [31*X^961 - 1, 1; 31*X^961 + 1, 1]


Class 4: a is a cube (so a=8) n is a cube != 0 mod 3.

GW(8,125) [5*2^125 - 1, 1; 25*2^250 + 5*2^125 + 1, 1]
GW(8,343) [7*2^343 - 1, 1; 49*2^686 + 7*2^343 + 1, 1]
GW(8,512) included in Class 10 below
GW(8,1000) [10*2^1000 - 1, 1; 100*2^2000 + 10*2^1000 + 1, 1]

GC(8,125) [5*2^125 + 1, 1; 25*2^250 - 5*2^125 + 1, 1]
GC(8,343) [7*2^343 + 1, 1; 49*2^686 - 7*2^343 + 1, 1]
GC(8,512) included in Class 10 below
GC(8,1000) [10*2^1000 + 1, 1; 100*2^2000 - 10*2^1000 + 1, 1]


Class 5: 1+ax^2 is even for some x.

GW(3,147) [7*3^74 - 1, 1; 7*3^74 + 1, 1]
GW(3,243) included in Class 10 below
GW(3,363) [11*3^182 - 1, 1; 11*3^182 + 1, 1]
GW(3,507) [13*3^254 - 1, 1; 13*3^254 + 1, 1]
GW(3,675) [15*3^338 - 1, 1; 15*3^338 + 1, 1]
GW(3,867) [17*3^434 - 1, 1; 17*3^434 + 1, 1]
GW(5,125) included in Class 10 below
GW(5,245) [7*5^123 - 1, 1; 7*5^123 + 1, 1]
GW(5,405) [9*5^203 - 1, 1; 9*5^203 + 1, 1]
GW(5,605) [11*5^303 - 1, 1; 11*5^303 + 1, 1]
GW(5,845) [13*5^423 - 1, 1; 13*5^423 + 1, 1]
GW(7,175) [5*7^88 - 1, 1; 5*7^88 + 1, 1]
GW(7,343) included in Class 10 below
GW(7,567) [3*7^142 - 1, 1; 3*7^142 + 1, 1; 9*7^284 + 1, 1]
GW(7,847) [11*7^424 - 1, 1; 11*7^424 + 1, 1]
GW(9,225) [5*9^113 - 1, 1; 5*9^113 + 1, 1]
GW(9,441) [7*9^365 - 1, 1; 7*9^221 + 1, 1]
GW(9,729) included in Class 10 below
GW(11,275) [5*11^138 - 1, 1; 5*11^138 + 1, 1]
GW(11,539) [7*11^270 - 1, 1; 7*11^270 + 1, 1]
GW(11,891) [3*11^223 - 1, 1; 3*11^223 + 1, 1; 9*11^446 + 1, 1]


Class 6: 1+ax^3 is a multiple of 3 for some x.

GW(4,250) [5*2^167 - 1, 1; 25*2^334 + 5*2^167 + 1, 1]
GW(4,500) [5*4^167 - 1, 1; 25*4^334 + 5*4^167 + 1, 1]
GW(5,320) [4*5^107 - 1, 1; 16*5^214 + 4*5^107 + 1, 1]
GW(7,875) [5*7^292 - 1, 1; 25*7^584 + 5*7^292 + 1, 1]
GW(8,512) included in Class 10 below
GW(11,704) [4*11^235 - 1, 1; 16*11^470 + 4*11^235 + 1, 1]

GC(4,250) [5*2^167 + 1, 1; 25*2^334 - 5*2^167 + 1, 1]
GC(4,500) [5*4^167 + 1, 1; 25*4^334 - 5*4^167 + 1, 1]
GC(5,320) [4*5^107 + 1, 1; 16*5^214 - 4*5^107 + 1, 1]
GC(7,875) [5*7^292 + 1, 1; 25*7^584 - 5*7^292 + 1, 1]
GC(8,512) included in Class 10 below
GC(11,704) [4*11^235 + 1, 1; 16*11^470 - 4*11^235 + 1, 1]


Class 7: 1+ax^5 is a multiple of 5 for some x.

GW(7,224) [N - 1, 1; N^4 + N^3 + N^2 + N + 1, 1] where N=2*7^45
GW(12,384) [N - 1, 1; N^4 + N^3 + N^2 + N + 1, 1] where N=2*12^77

GC(7,224) [N + 1, 1; N^4 - N^3 + N^2 - N + 1, 1] where N=2*7^45
GC(12,384) [N + 1, 1; N^4 - N^3 + N^2 - N + 1, 1] where N=2*12^77


Class 8: 2+a^2x^5  is a multiple of 5 for some x.

GW(3,288) [N - 1, 1; N^4 + N^3 + N^2 + N + 1, 1] where N=2*3^58
GC(3,288) [N + 1, 1; N^4 - N^3 + N^2 - N + 1, 1] where N=2*3^58


Class 9: GW(12,3x^2) for odd x.

Class 10: Cunningham-like numbers.  See standard cyclotomic
          factorizations and the Cunningham tables for further details.

GW(3,243) = 3^248-1
GW(3,729) = 3^735-1
GW(4,128) = 2^263-1
GW(4,256) = 2^520-1
GW(4,512) = 2^1033-1
GW(5,125) = 5^128-1
GW(5,625) = 5^629-1
GW(6,216) = 6^219-1
GW(7,343) = 7^346-1
GW(8,128) = 2^391-1
GW(8,256) = 2^776-1
GW(8,512) = 2^1545-1
GW(9,243) = 3^491-1
GW(9,729) = 3^1464-1
GW(10,1000) = 10^1003-1
GW(11,121) = 11^123-1
GW(12,144) = 12^146-1

GC(3,243) = 3^248+1
GC(3,729) = 3^735+1
GC(4,128) = 2^263+1
GC(4,256) = 2^520+1
GC(4,512) = 2^1033+1
GC(5,125) = 5^128+1
GC(5,625) = 5^629+1
GC(6,216) = 6^219+1
GC(7,343) = 7^346+1
GC(8,128) = 2^391+1
GC(8,256) = 2^776+1
GC(8,512) = 2^1455+1
GC(9,243) = 3^491+1
GC(9,729) = 3^1464+1
GC(10,1000) = 10^1003+1
GC(11,121) = 11^123+1
GC(12,144) = 12^146+1


Class 11: Aurifeuillean factorizations.

GC(4,625) = 4*N^4 + 1 = (2*N^2 - 2*N + 1)*(2*N^2 + 2*N + 1)
  N = 5*4^156)
  
GC(4,729) = N^6 + 1 = (N^2 + 1)*(N^4 - N^2 + 1)

  N = 6m^2 = 6*4^121

  N^4 - N^2 + 1 = 81*4^486 - 9*4^243 + 1
                = (9*4^243 + 18*4^121 + 1)^2 - (18*4^182 + 3*4^61)^2

  L,M = 9*4^243 +- 18*4^182 + 18*4^121 +- 3*4^61 + 1

GC(12,243) = 4*N^4 + 1 = L*M, N = 2^121*3^62 = 18*12^60
             L,M = (54*12^121 +- 3*12^61 + 1)

GC(12,729) = N^3 + 1 = (N + 1)*(N^2 - N + 1), N = 3m^2, m = 6*12^121
             N^2 - N + 1 = 81*12^486 - 9*12^243 + 1 =
            (9*12^243 + 1)^2 - (18*12^121)^2

Class 12: Cyclotomic 7th powers.

N^7 - 1 = (N - 1)*(N^6 + N^5 + N^4 + N^3 + N^2 + N + 1)

GW(3,384) N = 2*3^55

N^7 + 1 = (N+1)*(N^6 - N^5 + N^4 - N^3 + N^2 - N + 1)

GC(3,384) N = 2*3^55

===============================================================
PART 2 --- specific examples, omitting case 10, Cunningham-like
===============================================================

# GW(4,961)		{Jo YU 2003-01-04}
 31*2^961 - 1			61.79.2729.64800187.220797893.643260623.101488022268547533167520973.1098417158770796175489834679.	C206
 31*2^961 + 1			3.3.3.7.7283.1522693087042045046999668577561.	C255

# GW(5,784)		{M Rodenkirch 2002-12-28}
 28*5^392 - 1			3.47.199.283.10829479.165465497.	C254
 28*5^392 + 1			29.599.83181543308053.	C258

# GW(7,676)		{M Rodenkirch 2002-12-28}
 26*7^338 - 1			19.2161.12829.22367083.297659119.1522169087.89566907369.	C243
 26*7^338 + 1			3.5.5.17.2482903.3161728673.40178969489.2334670790225422040412221449.	C231

# GW(7,847)		{M Rodenkirch 2002-12-28}
 11*7^424 - 1			2.5.19.317.19787875578626681.	C339
 11*7^424 + 1			2.2.3.31.71.6043.1003928621.	C343

# GW(7,875)		{M Rodenkirch 2002-12-28}
 5*7^292 - 1			2.2.62311.15095083481.31326323543519.175601409996952669.	C202
 25*7^584 + 5*7^292 + 1		1861.61819.1848165091.	C478

# GW(7,900)		{M Rodenkirch 2002-12-28}
 30*7^450 - 1			113.241.1282956007872338334564421100963295719.	C342
 30*7^450 + 1			23.31.163.117319.662083.124842953.19725863259485618222486123555657.	C327

# GW(8,676)		{M Rodenkirch 2002-12-28}
 26*8^338 - 1			229.37567.3945419988787533211.	C282
 26*8^338 + 1			3.3.5.37.9986627622236941004362233516696174261680007233.4779071546981148366496701348312232311787083439148627.	C204

# GW(8,900)		{M Rodenkirch 2002-12-28}
 30*8^450 - 1			995763833.	C399
 30*8^450 + 1			17.31.113.11083.29429.26464106209.12799590479560739508630235229209.16686128638848771115735799849273.	C322

# GW(9,784)		{M Rodenkirch 2002-12-28}
 28*9^392 - 1			4079.58321.317332557620369.1995597812648217244910906959.159241884437269707946571478977.71641065090904438323791678957685877.	C262
 28*9^392 + 1			29.14627.6321800821248287.74326094040036479.81125636313967662023036485183909725697.	C300

# GW(10,676)		{M Rodenkirch 2002-12-28}
 26*10^338 - 1			113.1723.4389666759037178821.	C316
 26*10^338 + 1			3.3.17.884131.4371041.5349727.14875728974914415302633.	C296

# GW(10,784)		{M Rodenkirch 2002-12-28}
 28*10^392 - 1			3.3.39623.69197.654175495146601.5592249144326200867193.787257133655780092975997149.	C320
 28*10^392 + 1			23.29.1391409432670146751309009.1706891774626364863731413.911785948124781757644909710542637.	C310

# GW(11,784)		{M Rodenkirch 2002-12-28}
 28*11^392 - 1			3.12013750490401638721.8798457188509210971503762543879701.	C357
 28*11^392 + 1			29.20693.461763966059388684645402562109530736676904697.	C360

# GW(11,900)		{M Rodenkirch 2002-12-28}
 30*11^450 - 1			241.347.78889.4805557.47074421.26721480367.406307238859.	C424
 30*11^450 + 1			31.6390836772473.	C456

# GW(12,675)		{Jo YU 2003-01-04}
 90*12^337 - 1			13.1291.28283.	C357
 90*12^337 + 1			8140836150082229073396008979637963282307572009.	C320

# GW(12,676)		{M Rodenkirch 2002-12-28}
 26*12^338 - 1			19.23.2699.43177.300343.25160764349729364104054642730602989086499164505579.	C301
 26*12^338 + 1			5.7.317593.66182613548287.413118327887729.540377684145358571341.	C310

# GW(12,900)		{M Rodenkirch 2002-12-28}
 30*12^450 - 1			70677617.891891733.1006304085833.10171357484773398404441229830553879791503.52724295759850148381281305689860986667668149.	C375
 30*12^450 + 1			29.31.31.167.167.363510259831.1791357382008278467.3742115936291641889493491877375857.	C415