# Automorphic number

In mathematics an automorphic number is a number whose square "ends" in the number itself. For example, 52 = 25, 762 = 5776, and 8906252 = 793212890625.

The automorphic numbers begin 1, 5, 6, 25, 76, 376, 625, 9376, ... Template:OEIS

Given a k-digit automorphic number [itex]n>1[itex], an at-most 2k-digit automorphic number [itex]n'[itex] can be found by the formula [itex]n'=3\cdot n^2 - 2\cdot n^3\bmod{10^{2k}}[itex].

There are at most two automorphic numbers with k digits, one ending in 5 and one ending in 6 (unless [itex]k=1[itex], when there are three). One of them has the form [itex]n\equiv 0\pmod{2^{k}}, n\equiv 1\pmod{5^{k}}[itex] and the other has the form [itex]n\equiv 1\pmod{2^{k}}, n\equiv 0\pmod{5^{k}}[itex]. The sum of the two is 10k + 1.

The following sequence allows one to find a k-digit automorphic number, where [itex]k\leq1000[itex].

12781254001336900860348890843640238757659368219796\ 26181917833520492704199324875237825867148278905344\ 89744014261231703569954841949944461060814620725403\ 65599982715883560350493277955407419618492809520937\ 53026852390937562839148571612367351970609224242398\ 77700757495578727155976741345899753769551586271888\ 79415163075696688163521550488982717043785080284340\ 84412644126821848514157729916034497017892335796684\ 99144738956600193254582767800061832985442623282725\ 75561107331606970158649842222912554857298793371478\ 66323172405515756102352543994999345608083801190741\ 53006005605574481870969278509977591805007541642852\ 77081620113502468060581632761716767652609375280568\ 44214486193960499834472806721906670417240094234466\ 19781242669078753594461669850806463613716638404902\ 92193418819095816595244778618461409128782984384317\ 03248173428886572737663146519104988029447960814673\ 76050395719689371467180137561905546299681476426390\ 39530073191081698029385098900621665095808638110005\ 57423423230896109004106619977392256259918212890625 Template:OEIS

Just take the last k digits. Remember that the backslash means that the number continues in the next line. The other automorphic number is found by subtracting the number from [itex]10^k + 1[itex].

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