TITLE: Given any finite string of number, is it true there exists a perfect square whose leading numbers are the string
QUESTION [5 upvotes]: Given any finite string of number, is it true there exists a perfect square whose leading numbers are the string? For example, given the string 123456, can I find a perfect square with leading digits 123456? 
I think the answer is a positive one because number of integers with the string are infinite so one can keep increasing the number to obtain a perfect square?  Appreciate all advice, thank you

REPLY [2 votes]: There is a well-known algorithm for extracting the square root of a number.
Treating your desired string of digits as an integer, add $1$,
then use the standard square-root algorithm to compute the square root to
a sufficient number of digits.
The result (an approximate square root)
is a decimal number whose square is the desired
string of digits followed by some other digits.
Multiply this approximate square root by a suitable power of $10$ so that the
result is an integer.
Now you have an integer whose square starts with the desired string of digits.
Full details and justification of the algorithm can be found in an earlier answer to a related question.