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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, 42 = 14, 692 = 3969, 38542 = 11873854, X083692 = 851X4EX08369, so 4, 69, 3854, and X08369 are all automorphic numbers.

The first few automorphic numbers are

0, 1, 4, 9, 54, 69, 369, 854, 3854, 8369, E3854, 1E3854, X08369, 5X08369, 61E3854, E61E3854, 1E61E3854, X05X08369, 21E61E3854, 9X05X08369, E21E61E3854, 2E21E61E3854, 909X05X08369, ...

Since all square numbers end with square digits (i.e. 0, 1, 4 or 9), automorphic numbers can only end with square digits.

These numbers are the "infinite" solution of x2 − x = 0. In fact, they are the solution of this equation in the ring of dozadic (10-adic) numbers.

The automorphic numbers with 1 digit are exactly the 1-digit square numbers (0, 1, 4, and 9) (the only two 2-digit automorphic numbers are also squares (54=82 and 69=92), however, not all automorphic numbers are square numbers, the first counterexample is 369), and for k greater than 1, there are at most two automorphic numbers with k digits, one ending in 4 and one ending in 9. One of them has the form:

${\displaystyle n\equiv 0 \pmod{4^{k}}, \quad n\equiv 1 \pmod{3^{k}}\, ,}$

and the other has the form:

${\displaystyle n\equiv 1 \pmod{4^{k}}, \quad n\equiv 0 \pmod{3^{k}}\, .}$

The sum of the two numbers is 10k + 1. The smaller of these two numbers may be less than 10k − 1; for example with k = 5 the two numbers are E3854 and 8369. 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, with k ≤ 1000:

888E09 64E177 240540 35EE96 9X9942 844662 X5E70X 809669 6X1163 298X6E 416587 X25850
3E5799 61499X 706E12 760298 23X082 178607 231189 7651X3 X39498 814974 985556 2E9EE2
454X00 849841 909447 462582 366397 1X246E X46418 992279 069693 153960 2408EE X22346
770803 38E9E5 95650E 875498 480X55 XX1252 6X7098 06EXEX 084597 7E6145 257504 566920
515EE4 5E406X 321295 06E578 048X48 E34886 998677 623638 3185E0 258197 560766 X18064
08X254 X99E21 X96414 61X713 608X48 337X37 87E135 890904 X40812 2X56X3 46187E 90E7E4
373845 384E8E E0X168 X05195 5X5X62 140496 36234E 2EE708 45E091 87X658 6E4045 E4235E
E98026 659357 085000 70E57X 4XX59E 45E813 E78239 48X9X3 E53205 9224X0 925182 38E220
E60300 701955 4E3E01 83E963 317603 91545E 888E63 67736X 73181E 717540 888E4X 2E9EX8
439637 199198 657X38 191493 0671E4 710X86 818E05 877269 11XE47 381927 36261E X99813
303382 280263 840220 689X92 12E331 24484E 1E1586 673751 E4214E 0337X5 E4X672 99X240
957340 135782 3EX816 28E293 044791 757518 03E809 154X19 303247 181091 68829E 4067X3
204322 1E5E02 942508 6X5721 58E821 7E6155 475E8E 80EE82 EEX2E0 674192 662332 05651E
632756 X65428 673X08 0213E3 EX189E 760X13 46EXX7 6842E5 893E5X 2X7447 201341 5EE842
X41389 314X24 113641 567300 X59E5X X31X57 9041EE 6X8407 589387 452544 412503 49528E
825102 5X33E5 6X1538 0X9339 827029 23903E 69390X 16EE15 0E7090 120863 08X3X1 59X874
4E6014 9EXE75 662875 E2EE55 64X014 946929 61364X 347469 919557 78E764 26X509 897612
7955XX 3E6826 39E99X 640525 261693 E1XX6X 6559X0 2909X7 8X7X51 E64752 2X9787 775262
137EXX 491655 E83831 2E16X3 111562 0E8425 X746E0 E0607E 021E17 E39E26 43102X 841559
0E84X6 4EXE72 943763 E43175 713386 241467 870230 478624 786326 813990 E6X844 889523
778008 619453 299042 123155 5X0899 X24503 1870X4 50X253 0769X0 868262 E14001 677844
230904 79E6E2 100137 36X3E4 E9843E 242538 67X44X 8X0338 504325 479764 7E6E71 E00244
855X59 60E398 54XX76 313EX1 X51637 901420 4335X1 XX78E1 13X4E2 X1E2E3 706278 7E5027
456672 0E2X15 25678E 790135 998304 02E646 8X133X 81X3X1 6986E2 67E345 2E21E6 1E3854


## 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 3×2,9232 = 1X,E42,923 and 1X,E42,923 ends with 2,923, so 2,923 is 3-automorphic.

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