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We build a special number from ##5##, by squaring it, appending the next digit of the square to it and repeating the steps.
##5^2=25.## The next digit is 2, which is added to 5 to give 25.
##25^2=625.## The next digit is 6, which is added to 25 to give 625.
##625^2=390625.## The next digit is 0, which is added to 625 to give 0625.
##0625^2=390625.## The next digit is 9, which is added to 0625 to give 90625.
##90625^2=8212890625.## The next digit is 8, which is added to 90625 to give 890625.
We end up with the special number ##F=\,...\,4106619977392256259918212890625.##
Why is it that the number in every step shares the same digits as its square? In other words, prove the existence of this number.
Source: https://divisbyzero.com/2008/12/29/a-10-adic-number-that-is-a-zero-divisor/
(The article mentioned that the digit 6 can be used. But I don't think so because ##36\times36=1296##, which doesn't end with 36.)
##5^2=25.## The next digit is 2, which is added to 5 to give 25.
##25^2=625.## The next digit is 6, which is added to 25 to give 625.
##625^2=390625.## The next digit is 0, which is added to 625 to give 0625.
##0625^2=390625.## The next digit is 9, which is added to 0625 to give 90625.
##90625^2=8212890625.## The next digit is 8, which is added to 90625 to give 890625.
We end up with the special number ##F=\,...\,4106619977392256259918212890625.##
Why is it that the number in every step shares the same digits as its square? In other words, prove the existence of this number.
Source: https://divisbyzero.com/2008/12/29/a-10-adic-number-that-is-a-zero-divisor/
(The article mentioned that the digit 6 can be used. But I don't think so because ##36\times36=1296##, which doesn't end with 36.)
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