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: \nThere are two automorphic numbers of k digits. One of them has the form and the other has the form . 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\\\n26181917833520492704199324875237825867148278905344\\\n89744014261231703569954841949944461060814620725403\\\n65599982715883560350493277955407419618492809520937\\\n53026852390937562839148571612367351970609224242398\\\n77700757495578727155976741345899753769551586271888\\\n79415163075696688163521550488982717043785080284340\\\n84412644126821848514157729916034497017892335796684\\\n99144738956600193254582767800061832985442623282725\\\n75561107331606970158649842222912554857298793371478\\\n66323172405515756102352543994999345608083801190741\\\n53006005605574481870969278509977591805007541642852\\\n77081620113502468060581632761716767652609375280568\\\n44214486193960499834472806721906670417240094234466\\\n19781242669078753594461669850806463613716638404902\\\n92193418819095816595244778618461409128782984384317\\\n03248173428886572737663146519104988029447960814673\\\n76050395719689371467180137561905546299681476426390\\\n39530073191081698029385098900621665095808638110005\\\n57423423230896109004106619977392256259918212890625 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\nn n²\n 5 25\n 25 625\n 625 390625\n 90625 8212890625\n 890625 793212890625\n 2890625 8355712890625\n 12890625 166168212890625\n 212890625 45322418212890625\n 8212890625 67451572418212890625\n 18212890625 331709384918212890625\n 918212890625 843114912509918212890625\n 9918212890625 98370946943759918212890625\n 59918212890625 3590192236006259918212890625\n 259918212890625 67557477392256259918212890625\n 6259918212890625 39186576032079756259918212890625\n 56259918212890625 3165178397321142256259918212890625\n 256259918212890625\n 65669145682477392256259918212890625\n 2256259918212890625\n 5090708818534039892256259918212890625\n 92256259918212890625\n 8511217494096854352392256259918212890625\n 392256259918212890625\n 153864973445024588727392256259918212890625\n 7392256259918212890625\n 54645452612300005057477392256259918212890625\n 77392256259918212890625\n 5989561329000849809744977392256259918212890625\n 977392256259918212890625\n 955295622596853633012869977392256259918212890625\n 9977392256259918212890625\n 99548356235275381465044119977392256259918212890625\n 19977392256259918212890625\n 399096201360473745722856619977392256259918212890625\n 619977392256259918212890625\n 384371966908872375601191606619977392256259918212890625\n 6619977392256259918212890625\n 43824100673983991394155879106619977392256259918212890625\n 106619977392256259918212890625\n 11367819579125235975036734004106619977392256259918212890625\n 4106619977392256259918212890625\n 16864327638717175315320739859004106619977392256259918212890625\n 9004106619977392256259918212890625\n 81073936023920699329853843152771109004106619977392256259918212890625\n \nn n²\n 6 36\n 76 5776\n 376 141376\n 9376 87909376\n 109376 11963109376\n 7109376 50543227109376\n 87109376 7588043387109376\n 787109376 619541169787109376\n 1787109376 3193759921787109376\n 81787109376 6689131260081787109376\n 40081787109376 1606549657881340081787109376\n 740081787109376 547721051611007740081787109376\n 3740081787109376\n 13988211774267263740081787109376\n 43740081787109376\n 1913194754743017343740081787109376\n 743740081787109376\n 553149309256696143743740081787109376\n 7743740081787109376\n 59965510454276227407743740081787109376\n 607743740081787109376\n 369352453608598807478607743740081787109376\n 2607743740081787109376\n 6800327413935747244982607743740081787109376\n 22607743740081787109376\n 511110077017207231620022607743740081787109376\n 80022607743740081787109376\n 6403617750108490103144731780022607743740081787109376\n 380022607743740081787109376\n 144417182396352539175410357380022607743740081787109376\n 3380022607743740081787109376\n 11424552828858793029898066613380022607743740081787109376\n 893380022607743740081787109376\n 798127864794612716138610952755893380022607743740081787109376\n 5893380022607743740081787109376\n 34731928090872050116956482046515893380022607743740081787109376\n 995893380022607743740081787109376\n 991803624372854204655478894958610995893380022607743740081787109376\n \n

"The concept is interesting and well-formed, but in order to earn better than a 'C', the idea must be feasible." - A Yale University management professor in response to student Fred Smith's paper proposing reliable overnight delivery service (Smith went on to found Federal Express Corp.)