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, ...
Given a k-digit automorphic number n > 1, an at-most 2k-digit automorphic number n' can be found by the formula 
.
There are at most two automorphic numbers with k digits, one ending in 5 and one ending in 6 (unless k = 1, when there are three). One of them has the form 
and the other has the form 
. The sum of the two is 10k + 1.
The following sequence allows one to find a k-digit automorphic number, where 
.
12781254001336900860348890843640238757659368219796\
26181917833520492704199324875237825867148278905344\
89744014261231703569954841949944461060814620725403\
65599982715883560350493277955407419618492809520937\
53026852390937562839148571612367351970609224242398\
77700757495578727155976741345899753769551586271888\
79415163075696688163521550488982717043785080284340\
84412644126821848514157729916034497017892335796684\
99144738956600193254582767800061832985442623282725\
75561107331606970158649842222912554857298793371478\
66323172405515756102352543994999345608083801190741\
53006005605574481870969278509977591805007541642852\
77081620113502468060581632761716767652609375280568\
44214486193960499834472806721906670417240094234466\
19781242669078753594461669850806463613716638404902\
92193418819095816595244778618461409128782984384317\
03248173428886572737663146519104988029447960814673\
76050395719689371467180137561905546299681476426390\
39530073191081698029385098900621665095808638110005\
57423423230896109004106619977392256259918212890625 (sequence A018247 in 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 10k + 1.
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