Of the 22 two-digit endings of any square, only these four (29, 41, 61, 89) are prime. This speeds up our initial search considerably and the six-digit number 136,161 (3692) soon appears.
Richard England has also discovered that all squares ending in 41 or 89 have the previous digit even, so these need not be considered. The smallest start for our eight-digit square is 11, so its root must be greater than 3,316. One does not then have to go far to reach the answer 11,377,129 = 3,3732.
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