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 b^a ± 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×b^a. I'll just call these "AB^A nunbers".

Here is a table of some of the smaller AB^A 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 AB^A 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 AB^A numbers.

The first 3 columns of the table are sequences A001105 (2N²), A117642 (3N³), and A141046 (4N⁴). The rows are A036289 (N2^N), A036290 (N3^N), A018215 (N4^N), A036291 (N5^N), 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 b^a = c d^c: 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 AB^A 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 b^a = c d^c. 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 b^a = c d^c. 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=7^w11^x and d=7^y11^z. Then the whole equation becomes

11×(7^w11^x)¹¹ = 7×(7^y11^z)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

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 b^a = c d^c

b=c^wa^x and d=c^ya^z

a (c^wa^x)^a = c (c^ya^z)^c

c^aw a^(ax+1) = c^(cy+1) a^cz

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 x^y = y^x, which has an infinite number of solutions with rational values of x and y; it is discussed here.

The equation x^y=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.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2018 Feb 04.

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