Automorphic number

In mathematics an automorphic number is a number whose square ends in the same digit or digits as that or those of the number itself. For example, 5² = 25, 76² = 5776, and 890625² = 793212890625.

Given a k-digit automorphic n, a 2k-digit automorphic can be found by the formula: $n'\equiv 3\cdot n^{2}-2\cdot n^{3}{\pmod {10^{2k}}}$

There are two automorphic numbers of k digits. One of them has the form $n\equiv 0{\pmod {2^{k}}},n\equiv 1{\pmod {5^{k}}}$ and the other has the form $n\equiv 1{\pmod {2^{k}}},n\equiv 0{\pmod {5^{k}}}$ . The sum of both automorphic numbers is 10^k + 1.

The following sequence allows us to find a k-digit automorphic number, where k <= 1000.

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

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 10^k + 1.

Tables of automorphic numbers

n n² 5 25 25 625 625 390625 90625 8212890625 890625 793212890625 2890625 8355712890625 12890625 166168212890625 212890625 45322418212890625 8212890625 67451572418212890625 18212890625 331709384918212890625 918212890625 843114912509918212890625 9918212890625 98370946943759918212890625 59918212890625 3590192236006259918212890625 259918212890625 67557477392256259918212890625 6259918212890625 39186576032079756259918212890625 56259918212890625 3165178397321142256259918212890625 256259918212890625 65669145682477392256259918212890625 2256259918212890625 5090708818534039892256259918212890625 92256259918212890625 8511217494096854352392256259918212890625 392256259918212890625 153864973445024588727392256259918212890625 7392256259918212890625 54645452612300005057477392256259918212890625 77392256259918212890625 5989561329000849809744977392256259918212890625 977392256259918212890625 955295622596853633012869977392256259918212890625 9977392256259918212890625 99548356235275381465044119977392256259918212890625 19977392256259918212890625 399096201360473745722856619977392256259918212890625 619977392256259918212890625 384371966908872375601191606619977392256259918212890625 6619977392256259918212890625 43824100673983991394155879106619977392256259918212890625 106619977392256259918212890625 11367819579125235975036734004106619977392256259918212890625 4106619977392256259918212890625 16864327638717175315320739859004106619977392256259918212890625 9004106619977392256259918212890625 81073936023920699329853843152771109004106619977392256259918212890625n n² 6 36 76 5776 376 141376 9376 87909376 109376 11963109376 7109376 50543227109376 87109376 7588043387109376 787109376 619541169787109376 1787109376 3193759921787109376 81787109376 6689131260081787109376 40081787109376 1606549657881340081787109376 740081787109376 547721051611007740081787109376 3740081787109376 13988211774267263740081787109376 43740081787109376 1913194754743017343740081787109376 743740081787109376 553149309256696143743740081787109376 7743740081787109376 59965510454276227407743740081787109376 607743740081787109376 369352453608598807478607743740081787109376 2607743740081787109376 6800327413935747244982607743740081787109376 22607743740081787109376 511110077017207231620022607743740081787109376 80022607743740081787109376 6403617750108490103144731780022607743740081787109376 380022607743740081787109376 144417182396352539175410357380022607743740081787109376 3380022607743740081787109376 11424552828858793029898066613380022607743740081787109376 893380022607743740081787109376 798127864794612716138610952755893380022607743740081787109376 5893380022607743740081787109376 34731928090872050116956482046515893380022607743740081787109376 995893380022607743740081787109376 991803624372854204655478894958610995893380022607743740081787109376