Generalized Cullen and Woodall Numbers  

The original "Cullen numbers", named after Rev. James Cullen who studied them in 1905, are numbers of the form n 2n + 1.

The original "Woodall numbers", named after H. J. Woodall who studied them in 1917, are numbers of the form n 2n - 1. They are sometimes also called Riesel numbers. For every Cullen number you can subtract 2 and get a corresponding Woodall number.

The "generalized" Cullen and Woodall numbers allow the base to be some number other than 2, and thus are of the form a ba ± 1. To avoid having every integer qualify, a and b must both be greater than 1. These "generalized" numbers were named by Paul Leyland, who has extensively studied the factorization of very large numbers of this form.

Because they are so similar, I am going to discuss the Cullen and Woodall/Riesel numbers together by eliminating the ±1 and just considering numbers of the form a×ba. I'll just call these "ABA nunbers".

Here is a table of some of the smaller ABA numbers, showing values less than 10,000:

n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 ---- ---- ---- ---- ---- ---- ---- ---- b=2 8 24 64 160 384 896 2048 4608 b=3 18 81 324 1215 4374 b=4 32 192 1024 5120 b=5 50 375 2500 b=6 72 648 5184 b=7 98 1029 9604 b=8 128 1536 b=9 162 2187 b=10 200 3000 b=11 242 3993 b=12 288 5184 b=13 338 6591 b=14 392 8232 b=15 450 b=16 512 b=17 578 b=18 648 b=19 722 b=20 800 b=21 882 . . . b=69 9522 b=70 9800

For n=2, the b values go up to 70, so there are 69 ABA numbers in the n=2 column and only 27 in all the other columns put together. In general, within a finite interval, the values with n=2 form the majority of ABA numbers.

The first 3 columns of the table are sequences A001105 (2N2), A117642 (3N3), and A141046 (4N4). The rows are A036289 (N2N), A036290 (N3N), A018215 (N4N), A036291 (N5N), and so on.

Notice that the number 648 appears twice (row 6 and row 18). It is the first example of a "double solution" x = a ba = c dc: 648 = 3×63 = 2×182.

Here are the values up to 100000. I show the numbers of the form 2b2 in italics, and numbers with multiple solutions (discussed next) in bold. This is Sloane's sequence A171607:

8, 18, 24, 32, 50, 64, 72, 81, 98, 128, 160, 162, 192, 200, 242, 288, 324, 338, 375, 384, 392, 450, 512, 578, 648, 722, 800, 882, 896, 968, 1024, 1029, 1058, 1152, *1215, 1250, 1352, 1458, 1536, 1568, 1682, 1800, 1922, 2048, 2178, 2187, 2312, 2450, 2500, 2592, 2738, 2888, *3000, 3042, 3200, 3362, 3528, 3698, 3872, 3993, 4050, 4232, 4374, 4418, 4608, 4802, 5000, 5120, 5184, 5202, 5408, 5618, 5832, 6050, 6272, 6498, 6591, 6728, 6962, 7200, 7442, 7688, 7938, 8192, 8232, 8450, 8712, 8978, 9248, 9522, 9604, 9800, 10082, 10125, 10240, 10368, 10658, 10952, 11250, 11552, 11858, 12168, 12288, 12482, 12800, 13122, 13448, 13778, 14112, 14450, 14739, 14792, 15138, *15309, 15488, 15625, 15842, 16200, 16384, 16562, 16928, 17298, 17496, 17672, 18050, 18432, 18818, 19208, 19602, 20000, 20402, 20577, 20808, 21218, 21632, 22050, 22472, *22528, 22898, 23328, 23762, 24000, 24200, 24576, 24642, 25088, 25538, 25992, 26244, 26450, 26912, 27378, 27783, *27848, 28322, 28800, 29282, 29768, 30258, 30752, 31250, *31752, 31944, 32258, 32768, 33282, 33800, 34322, 34848, *35378, 35912, 36450, 36501, 36992, 37538, 38088, 38642, 38880, 39200, 39762, 40000, 40328, 40898, 41472, 42050, *42632, 43218, 43808, 44402, 45000, 45602, 46208, *46818, 46875, 47432, 48050, 48672, 49152, 49298, 49928, *50562*, 51200, 51842, 52488, 52728, 53138, 53792, 54450, 55112, 55778, 56448, 57122, 57800, 58482, 58564, *59049, 59168, 59858, 60552, 61250, 61952, 62658, 63368, 64082, 64800, 65522, 65856, 66248, 66978, 67712, 68450, 69192, 69938, 70688, 71442, 72200, 72962, 73167, 73728, *74498, 75272, 76050, 76832, 77618, 78408, 79202, 80000, *80802, 81000, 81608, 82418, 82944, 83232, 84035, 84050, *84872, 85698, 86528, 87362, 88200, 89042, 89373, 89888, *90738, 91592, 92450, 93312, 93750, 94178, 95048, 95922, *96800*, 97682, 98304, 98568, 99458, 100352, ...

Double and Multiple Solutions

As mentioned above, the number 648 qualifies as an ABA number two different ways. It is the first of many non-trivial cases (Sloane's sequence A171606):

648 = 3×63 = 2×182
2048 = 8×28 = 2×322
4608 = 9×29 = 2×482
5184 = 4×64 = 3×123
41472 = 3×243 = 2×1442
52488 = 8×38 = 2×1622
472392 = 3×543 = 2×4862
500000 = 5×105 = 2×5002
524288 = 8×48 = 2×5122
2654208 = 3×963 = 2×11522
3125000 = 8×58 = 2×12502
4718592 = 18×218 = 2×15362
10125000 = 3×1503 = 2×22502
13436928 = 8×68 = 2×25922
21233664 = 4×484 = 3×1923
30233088 = 3×2163 = 2×38882
46118408 = 8×78 = 2×48022
76236552 = 3×2943 = 2×61742
134217728 = 8×88 = 2×81922
169869312 = 3×3843 = 2×92162
344373768 = 8×98 = 3×4863 = 2×131222
402653184 = 24×224 = 3×5123
512000000 = 5×405 = 2×160002
648000000 = 3×6003 = 2×180002
737894528 = 7×147 = 2×192082
800000000 = 8×108 = 2×200002
838860800 = 25×225 = 2×204802
922640625 = 5×455 = 3×6753
1147971528 = 3×7263 = 2×239582
1207959552 = 9×89 = 2×245762
1714871048 = 8×118 = 2×292822
1934917632 = 3×8643 = 2×311042
2754990144 = 4×1624 = 3×9723
3127772232 = 3×10143 = 2×395462
3439853568 = 8×128 = 2×414722
4879139328 = 3×11763 = 2×493922
6525845768 = 8×138 = 2×571222
6973568802 = 18×318 = 2×590492
7381125000 = 3×13503 = 2×607502 etc...

The first solution that does not involve a 2×b2 form is 5184 = 4×64 = 3×123. The first odd solution is 922640625 = 5×455 = 3×6753.

It appears there are a lot of solutions to the equation a ba = c dc. In fact, there are an infinite number of solutions for any pair of relatively prime numbers a and c.

Here is an example of how to find such a solution. Let a=11 and c=7. We wish to find b and d such that a ba = c dc. Substituting in for a and c, we have 11×b11 = 7×d7. Assume b and d are each a power of 7 times a power of 11: b=7w11x and d=7y11z. Then the whole equation becomes

11×(7w11x)11 = 7×(7y11z)7

Therefore we have

7(11w) 11(11x+1) = 7(7y+1) 11(7z)

and therefore, 11w=7y+1 and 7z=11x+1. It is easy to find that the smallest solutions are w=2, y=3, z=8, x=5. So our soluton is:

11×(72115)11 = 7×(73118)7

or

11×789149911 = 7×735250961837

Additional solutions for any n>0 can be found by adding 7n to w and x, and 11n to y and z.

A similar procedure generates solutions for any two relatively prime exponents. Using variables for all quantities now, and repeating the method above, we get:

a ba = c dc

b=cwax   and   d=cyaz

a (cwax)a = c (cyaz)c

caw a(ax+1) = c(cy+1) acz

so w and y are constrained by

aw = cy+1

while x and z are constrained by

cz = ax+1

Suppose a=2 and c=3; the smallest solution results from w=2, y=1, x=1, z=1 which gives b=32×21=18 and d=31×21=6, yielding the solution 2×182 = 648 = 3×33. The next-higher w and y are (w=5, y=3), and the next-higher x and z are (x=4, z=3). These can be combined in either combination with the smaller (w,y) and the smaller (x,z) giving 3 more solutions:

(w=2,y=1), (x=4,z=3) give b=3224=144 and d=3123=24
  for the solution 2×1442 = 41472 = 3×243

(w=5,y=3), (x=1,z=1) give b=3521=486 and d=3321=54
  for the solution 2×4862 = 472392 = 3×543

(w=5,y=3), (x=4,z=3) give b=3524=3888 and d=3323=216
  for the solution 2×38882 = 30233088 = 3×2163

Continuing, we find (w=8, y=5) and (x=7, z=5) which can be mixed and matched with each other or with the smaller (w,y) and (x,z) pairs, yielding another 5 solutions, and so on.


If you like this sort of thing, you might also be interested in the solution to xy = yx, which has an infinite number of solutions with rational values of x and y; it is discussed here.

The equation xy=xy also has an infinte number of rational solutions, see here.


I have also written a lot about other integer sequences; start with this page.



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