We show that all perfect odd integer squares not divisible by 3, can be
usefully written as sqrt(N) = a + 18p, where the constant a is determined by
the basic properties of N. The equation can be solved deterministically by an
efficient four step algorithm that is solely based on integer arithmetic. There
is no required multiplication or division by multiple digit integers, nor does
the algorithm need a seed value. It finds the integer p when N is a perfect
square, and certifies N as a non-square when the algorithm terminates without a
solution. The number of iterations scales approximately as log(sqrt(N)/2) for
square roots. The paper also outlines how one of the methods discussed for
squares can be extended to finding an arbitrary root of N. Finally, we present
a rule that distinguishes products of twin primes from squares.Comment: 12 pages, 8 figure