Automorphic Numbers

I came across automorphic numbers in a tweet by Cliff Pickover. He pointed out that:

☛ \(376\) is an automorphic number, meaning a number whose square "ends" in the same digits as the number itself. \(376\) has the property that its cube and fourth power also end in the same digits.

• \(376^2=141376\)
• \(376^3=53157376\)
• \(376^4=19987173376\)

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

However, it should be emphasised that there are many other trimorphic numbers apart from these e.g. \(24^2=13824\). The trimorphic numbers are listed in OEIS A033819.

A033819

Trimorphic numbers: \(n^3\) ends with \(n\).

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 \( f(x)=x^{2}\), the last 10 digits of which are one of these:

• \( \ldots 0000000000 \)
• \(  \ldots 0000000001 \)
• \(  \ldots 8212890625 \)
  (sequence A018247 in the OEIS)
• \(  \ldots 1787109376 \)
  (sequence A018248 in the OEIS)

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: \(m^2\) ends with \(m\).

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 \(b\), the number of \(b\)-adic fixed points is determined by 2^(number of distinct prime factors). Because 10 is the product of two distinct prime factors, it has \(2^2=4\) fixed points. Likewise with 6 and 12 (even though \(12=2^2 \times 3\), it has only two distinct prime factors). Of course, for prime numbered bases such as 2, 3, 5 etc. and perfect powers such as 4, 8, 9, 16 etc., there are only 2 fixed points and these are the trivial 0 and 1. Here is a permalink that will generate automorphic numbers in any base (up to 36) and for any power.

Applied to base 30 (that is comprised of three prime factors) it can be seen that there are \(2^3=8\) distinct 30-adic fixed points. Here are the 30-morphic numbers up to one million:

• 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

There's a lot more to this topic of course but at least this post serves as an introduction to the topic and perhaps I can pursue other aspects later.