Automorphic number

In mathematics an automorphic number (sometimes referred to as a circular number) is a number whose square "ends" in the same digits as the number itself. For example, 5² = 25, 6² = 36, 76² = 5776, and 890625² = 793212890625, so 5, 6, 76 and 890625 are all automorphic numbers. The only automorphic Kaprekar number is 1, because the square of a Kaprekar number cannot start with zero.

The sequence of automorphic numbers begins 1, 5, 6, 25, 76, 376, 625, 9376, ... (sequence A003226 in the OEIS).

Given a k-digit automorphic number n > 1, an automorphic number n′ with at most 2k-digits can be found with the formula:

$n'=(3\cdot n^2 - 2\cdot n^3) \ \bmod{10^{2k}}\, .$

For k greater than 1, there are at most two automorphic numbers with k digits, one ending in 5 and one ending in 6. One of them has the form:

$n\equiv 0 \pmod{2^{k}}, \quad n\equiv 1 \pmod{5^{k}}\, ,$

and the other has the form:

$n\equiv 1 \pmod{2^{k}}, \quad n\equiv 0 \pmod{5^{k}}\, .$

The sum of the two numbers is 10^k + 1. The smaller of these two numbers may be less than 10^{k − 1}; for example with k = 4 the two numbers are 9376 and 625. In this case there is only one k digit automorphic number; the smaller number could only form a k-digit automorphic number if a leading 0 were added to its digits.

The following digit sequence can be used to find the two k-digit automorphic numbers, where k ≤ 1000.

12781 25400 13369 00860 34889 08436 40238 75765 93682 19796 
26181 91783 35204 92704 19932 48752 37825 86714 82789 05344 
89744 01426 12317 03569 95484 19499 44461 06081 46207 25403 
65599 98271 58835 60350 49327 79554 07419 61849 28095 20937 
53026 85239 09375 62839 14857 16123 67351 97060 92242 42398 
77700 75749 55787 27155 97674 13458 99753 76955 15862 71888 
79415 16307 56966 88163 52155 04889 82717 04378 50802 84340 
84412 64412 68218 48514 15772 99160 34497 01789 23357 96684 
99144 73895 66001 93254 58276 78000 61832 98544 26232 82725 
75561 10733 16069 70158 64984 22229 12554 85729 87933 71478 
66323 17240 55157 56102 35254 39949 99345 60808 38011 90741 
53006 00560 55744 81870 96927 85099 77591 80500 75416 42852 
77081 62011 35024 68060 58163 27617 16767 65260 93752 80568 
44214 48619 39604 99834 47280 67219 06670 41724 00942 34466 
19781 24266 90787 53594 46166 98508 06463 61371 66384 04902 
92193 41881 90958 16595 24477 86184 61409 12878 29843 84317 
03248 17342 88865 72737 66314 65191 04988 02944 79608 14673 
76050 39571 96893 71467 18013 75619 05546 29968 14764 26390 
39530 07319 10816 98029 38509 89006 21665 09580 86381 10005 
57423 42323 08961 09004 10661 99773 92256 25991 82128 90625 (sequence A018247 in the OEIS)

One automorphic number is found by taking the last k digits of this sequence; the second is found by subtracting the first number from 10^k + 1.

n-automorphic number

An n-automorphic number is a number k such that nk² has its last digit(s) equal to k. For example, since 3*92² = 25,392 and 25,392 ends with 92, so 92 is 3-automorphic.

n	last 10 digits of the n-automorphic numbers	OEIS sequence of the n-automorphic numbers
0	0000000000	A000004
1	0000000000, 0000000001, 8212890625, 1787109376	A000004, A000012, A007185, A016090
2	0000000000, 0893554688	A000004, A030984
3	0000000000, 7262369792, 9404296875, 6666666667	A000004, A030985, A030986, A067275
4	0000000000, 0446777344	A000004, A030987
5	0000000000, 3642578125	A000004, A030988
6	0000000000, 3631184896	A000004, A030989
7	0000000000, 7142857143, 4548984375, 1683872768	A000004, A030990, A030991, A030992
8	0000000000, 0223388672	A000004, A030993
9	0000000000, 5754123264, 3134765625, 8888888889	A000004, A030994, A030995, A059482

Other radixes

Automorphic numbers are radix dependent, and the description above applies to automorphic numbers in base 10. Using other radixes there are different automorphic numbers. 0 and 1 are automorphic numbers in every radix; automorphic numbers other than 0 and 1 only exist when the radix has at least two distinct prime factors.

A single digit number x is automorphic in radix b > x when b divides x² − x. So 6 is automorphic in a radix which is a divisor of 6² − 6 = 30 that is greater than 6; these divisors are 10, 15 and 30.

In any given radix there are 2^p sequences of automorphic numbers where p is the number of distinct prime factors in the radix. For base 10 this gives 2² = 4 sequences, which are 0,1,5 and 6 for 1 digit or 00, 01, 25, 76 for two digits and so on. A prime power radix (such as 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, etc) can only have 0 and 1 (prepended by one or more zeroes) as automorphic numbers. Base 6 is the first radix with non-trivial automorphic numbers and base 15 the first such odd radix. Base 30 is the first radix with three distinct prime factors and has 8 sequences of automorphic numbers. Here some examples of non-trivial 1, 2 and 4 digit automorphic numbers in some non-primepower radixes (using A−Z to represent digits 10 to 35):

These numbers are the "infinite" solution of x² − x = 0 in base b. In fact, they are the solution of this equation in the ring of b-adic numbers. If b is a prime power, then the ring is also a field. If it is a field, then the equation can be written as b(b − 1) = 0, so the only solutions are 0 and 1.

number of digits (n)	radix	numbers formed by last n digits (excluding 0 and 1)	expressed in decimal
1	6	3, 4	3, 4
2	6	13, 44	9, 28
4	6	0213, 5344	81, 1216
1	10	5, 6	5, 6
2	10	25, 76	25, 76
4	10	0625, 9376	625, 9376
1	12	4, 9	4, 9
2	12	54, 69	64, 81
4	12	3854, 8369	6400, 14337
1	14	7, 8	7, 8
2	14	37, A8	49, 148
4	14	0C37, D1A8	2401, 344
1	15	6, A	6, 10
2	15	86, 6A	126, 100
4	15	DA86, 146A	46251, 4375
1	18	9, A	9, 10
2	18	49, DA	81, 244
4	18	1249, GFDA	6561, 98416
1	20	5, G	5, 16
2	20	B5, 8G	225, 176
4	20	B6B5, 8D8G	90625, 69376
1	21	7, F	7, 15
2	21	G7, 4F	343, 99
4	21	H7G7, 3D4F	160867, 33615
1	22	B, C	11, 12
2	22	5B, GC	121, 364
4	22	185B, KDGC	14641, 219616
1	24	9, G	9, 16
2	24	L9, 2G	513, 64
4	24	D0L9, AN2G	180225, 151552
1	30	6, A, F, G, L, P	6, 10, 15, 16, 21, 25
2	30	J6, 3A, 7F, MG, QL, AP	576, 100, 225, 676, 801, 325
4	30	B2J6, H13A, 1Q7F, S3MG, CSQL, IRAP	299376, 460000, 50625, 759376, 350001, 510625
1	36	9, S	9, 28
2	36	29, XS	81, 1216
4	36	DN29, MCXS	636417, 1043200

Note that:

• the 4-digit numbers that are automorphic in bases 20 and 30, when expressed in decimal, are also automorphic in the last 4 digits.
• the 2-digit numbers that are automorphic in bases 20 and 30, when expressed in decimal, are also automorphic in the last 2 digits.

External links

• Weisstein, Eric W. "Automorphic number". MathWorld.

References

http://planetmath.org/examplesof1automorphicnumbers