West London reader Richard England has been looking at the arcane subject of “squares consisting entirely of two-digit primes”. There are very few of these numbers, the lowest being 4,761, which is 692 (primes with a leading zero such as 03 not being taken into account). Richard’s first question is: What is the lowest six-digit square of this type? Then his main problem is: What is the lowest eight-digit square that consists of four two-digit primes?
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 possible 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.
Please post all Puzzler enquiries to Jim Howson, 5 Hilltop Gardens, Dartford, Kent DA1 5JF
This was first published in March 2008