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Reappearing Integers

While working on my Bachelor Degree at U of M, I noticed an interesting behavior in some integers when they were squared;  they "reappeared."  For example, 76² = 5776 and the "76" reappears in "5776."  If you were to keep squaring the squares, the "76" would never go away.  I wrote a generator for all integers that behave this way and its proof of correctness.  This paper was never submitted for publication.

One very strange thing happens as you generate these integers:  when you sum them, you get a very special palindrome!  As you'll see in my paper, this result is not that strange after all.

As the paper shows, there are really only two solutions per number of digits.  For example, here are the two 231-digit integers that reappear in their squares:  ( You can try out other N-digit integers that reappear in their squares with the Java Applet below. )

Solution 1
5244778618461409128782984384317032481734288865727376631465191049880294479608146737605039571968937146
7180137561905546299681476426390395300731910816980293850989006216650958086381100055742342323089610900
4106619977392256259918212890625

Solution 2
4755221381538590871217015615682967518265711134272623368534808950119705520391853262394960428031062853
2819862438094453700318523573609604699268089183019706149010993783349041913618899944257657676910389099
5893380022607743740081787109376

Solution 1 squared ( Solution 1 has reappeared in its square )
2750770275666996728979490810631200565544668981277854667529169366797961832472999984219686696897412952
4637963339589421590700752500019984869164309000507253142938002398329221902836316069611800362054233828
9939801814303863239892163487909524477861846140912878298438431703248173428886572737663146519104988029
4479608146737605039571968937146718013756190554629968147642639039530073191081698029385098900621665095
80863811000557423423230896109004106619977392256259918212890625

Solution 1 + Solution 2 ( A very special palindrome )
1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000001

Try out my Reappear formula with this Java Applet

View java souce
Download class file

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Integers That Reappear in Their Squares

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