I came across automorphic numbers in a tweet by Cliff Pickover. He pointed out that:
☛
I thought I'd investigate how many of these numbers there up to one million. It turns out that there aren't many. They are:
- 0 0
- 1 1
- 5 25
- 6 36
- 25 625
- 76 5776
- 376 141376
- 625 390625
- 9376 87909376
- 90625 8212890625
- 109376 11963109376
- 890625 793212890625
Surprisingly when we consider the cubes of numbers, the count increases substantially but the same numbers as for the squares reappear:
- 0 0 square also
- 1 1 square also
- 5 125 square also
- 6 216 square also
- 25 15625 square also
- 76 438976 square also
- 376 53157376 square also
- 625 244140625 square also
- 9376 824238309376 square also
- 90625 744293212890625 square also
- 109376 1308477051109376 square also
- 890625 706455230712890625 square also
A033819 | Trimorphic numbers: |
0, 1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999, 1249, 3751, 4375, 4999, 5001, 5625, 6249, 8751, 9375, 9376, 9999, 18751, 31249, 40625, 49999, 50001, 59375, 68751, 81249, 90624, 90625, ...
With fourth powers however, the count again is more modest and the same numbers reappear:
- 0 0 square and cube also
- 1 1 square and cube also
- 5 625 square and cube also
- 6 1296 square and cube also
- 25 390625 square and cube also
- 76 33362176 square and cube also
- 376 19987173376 square and cube also
- 625 152587890625 square and cube also
- 9376 7728058388709376 square and cube also
- 90625 67451572418212890625 square and cube also
- 109376 143115985942139109376 square and cube also
- 890625 629186689853668212890625 square and cube also
This property of these numbers continues indefinitely and as Wikipedia states:
There are four 10-adic fixed points of
, the last 10 digits of which are one of these:
Thus we see why all the automorphic number appear as they do, forming OEIS A003226. Apparently such numbers can also be called curious numbers or circular numbers.
A003226 | Automorphic numbers: |
0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376, 59918212890625, 259918212890625, 740081787109376, ...
Of course, automorphic numbers can exist in any base. For a given base
Applied to base 30 (that is comprised of three prime factors) it can be seen that there are
- 0 0
- 1 1
- 6 16
- a 3a
- f 7f
- g 8g
- l el
- p kp
- 3a b3a
- 7f 1q7f
- ap 3rap
- j6 c8j6
- mg grmg
- ql nmql
- 13a 1713a
- 2j6 6t2j6
- 3mg e23mg
- q7f mt1q7f
- rap osirap
- sql roisql
- 1q7f 3fe1q7f
- b2j6 42s9b2j6
- csql 5i15csql
- h13a 9k7oh13a
- irap brjsirap
- s3mg qb0fs3mg
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