# 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 2^{n} + 1.

The original "Woodall numbers", named after H. J. Woodall who studied
them in 1917, are numbers of the form n 2^{n} - 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:

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
(2N^{2}), A117642 (3N^{3}), and A141046 (4N^{4}).
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 = 3×6^{3} = 2×18^{2}.

Here are the values up to 100000. I show the numbers of the form
2b^{2} 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 = 3×6^{3} = 2×18^{2}

2048 = 8×2^{8} = 2×32^{2}

4608 = 9×2^{9} = 2×48^{2}

5184 = 4×6^{4} = 3×12^{3}

41472 = 3×24^{3} = 2×144^{2}

52488 = 8×3^{8} = 2×162^{2}

472392 = 3×54^{3} = 2×486^{2}

500000 = 5×10^{5} = 2×500^{2}

524288 = 8×4^{8} = 2×512^{2}

2654208 = 3×96^{3} = 2×1152^{2}

3125000 = 8×5^{8} = 2×1250^{2}

4718592 = 18×2^{18} = 2×1536^{2}

10125000 = 3×150^{3} = 2×2250^{2}

13436928 = 8×6^{8} = 2×2592^{2}

21233664 = 4×48^{4} = 3×192^{3}

30233088 = 3×216^{3} = 2×3888^{2}

46118408 = 8×7^{8} = 2×4802^{2}

76236552 = 3×294^{3} = 2×6174^{2}

134217728 = 8×8^{8} = 2×8192^{2}

169869312 = 3×384^{3} = 2×9216^{2}

344373768 = 8×9^{8} = 3×486^{3} = 2×13122^{2}

402653184 = 24×2^{24} = 3×512^{3}

512000000 = 5×40^{5} = 2×16000^{2}

648000000 = 3×600^{3} = 2×18000^{2}

737894528 = 7×14^{7} = 2×19208^{2}

800000000 = 8×10^{8} = 2×20000^{2}

838860800 = 25×2^{25} = 2×20480^{2}

922640625 = 5×45^{5} = 3×675^{3}

1147971528 = 3×726^{3} = 2×23958^{2}

1207959552 = 9×8^{9} = 2×24576^{2}

1714871048 = 8×11^{8} = 2×29282^{2}

1934917632 = 3×864^{3} = 2×31104^{2}

2754990144 = 4×162^{4} = 3×972^{3}

3127772232 = 3×1014^{3} = 2×39546^{2}

3439853568 = 8×12^{8} = 2×41472^{2}

4879139328 = 3×1176^{3} = 2×49392^{2}

6525845768 = 8×13^{8} = 2×57122^{2}

6973568802 = 18×3^{18} = 2×59049^{2}

7381125000 = 3×1350^{3} = 2×60750^{2}
etc...

The first solution that does not involve a 2×b^{2} form is 5184
= 4×6^{4} = 3×12^{3}. The first odd solution is 922640625 =
5×45^{5} = 3×675^{3}.

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×b^{11} =
7×d^{7}. Assume b and d are each a power of 7 times a power of
11: b=7^{w}11^{x} and d=7^{y}11^{z}.
Then the whole equation becomes

11×(7^{w}11^{x})^{11} = 7×(7^{y}11^{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×(7^{2}11^{5})^{11} = 7×(7^{3}11^{8})^{7}

or

11×7891499^{11} = 7×73525096183^{7}

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^{w}a^{x} and d=c^{y}a^{z}

a (c^{w}a^{x})^{a} = c (c^{y}a^{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=3^{2}×2^{1}=18 and
d=3^{1}×2^{1}=6, yielding the solution 2×18^{2} = 648 =
3×3^{3}. 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=3^{2}2^{4}=144 and d=3^{1}2^{3}=24

for the solution 2×144^{2} = 41472 = 3×24^{3}

(w=5,y=3), (x=1,z=1) give b=3^{5}2^{1}=486 and d=3^{3}2^{1}=54

for the solution 2×486^{2} = 472392 = 3×54^{3}

(w=5,y=3), (x=4,z=3) give b=3^{5}2^{4}=3888 and d=3^{3}2^{3}=216

for the solution 2×3888^{2} = 30233088 = 3×216^{3}

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 2019 Jan 05. s.11