mute@tiny:~$ time gawk -f ./square1 <<< $'90000000000\n4\n1'
90000000000 is the perfect square of 300000
4 is the perfect square of 2
1 is the perfect square of 1

real    0m0.081s
user    0m0.063s
sys     0m0.016s

{
  number=$1
  series=1
  square=1
  root=1
  while (square <= number) {
    if (square == number) {
      printf("%d is the perfect square of %d\n", number, root)
      next
    }
    root++
    series+=2
    square+=series
  }
  print "Not a perfect square"
  next
}

mute@tiny:~$ time bawk -f ./square2 90000000000
90000000000 is a perfect square of 300000

real    0m0.002s
user    0m0.001s
sys     0m0.001s

#!/usr/bin/awk -f
BEGIN {
  num = ARGV[1]
  i = 10
  while (i * i < num)
    i *= 10

  lower = i/10
  upper = i

  while ( (avg = (upper + lower) / 2) != lower )
    if (avg * avg > num)
      upper = avg
    else
      lower = avg

  if (avg * avg == num)
    print num" is a perfect square of "avg
  else
    print num" is not a perfect square"

  exit
}