Consider two 5-digit perfect squares, the first two digits of each of which form a 2-digit prime number, and the last three digits form a 3-digit prime number.

For sake of discussion, let the digits be called ABCDE and VWXYZ. The two squares I'm thinking of can form, from those digits, another 5-digit square: ABXYZ. It is of the same type as the other two as AB is prime as is XYZ.

The use of different letters does not imply that all the letters represent different digits; any two may be the same or different, but the combined square does share its first two digits with those of one of the two original squares and its last three with the last three of the other.

What are the three squares?

There are eight 5-digit squares where both the first two digits of each form a prime and last three digits of each form a prime: 11449, 11881, 19881, 23409, 29241, 29929, 83521, and 89401. Of these, the only pair that shares a common final 3-digits are: 11881 and 19881. Of these two, the only square that shares a common beginning 2-digits of the other six 5-digit squares is: 11881 (with that of the square 11449.)

Thus, the three squares are:
11449 (= 107²), 11881 (= 109²), and 19881 (= 141²). 

Edited on April 10, 2009, 8:55 am

Posted by Dej Mar on 2009-04-09 16:48:55