OS X (Apple)

48,49c48,49
< 	    [[ num -lt 1 ]]
< 	then	printf "Out of range: 0 < x <= $LONG_MAX: x=$num\n"
---
> 	    [[ num -lt 0 ]]
> 	then	printf "Out of range: 0 <= x <= $LONG_MAX: x=$num\n"
58c58
< 		low=$((10 ** ((${#num} - 1) / 2)))
---
> 		low=$((10 ** ((${#num} - 1) / 2) - 1))

           2    2          2
  ( a + b )  = a  + 2ab + b

           2    2          
  ( n + 1 )  = n  + 2n + 1

           2    2
  ( n + 1 )  = n  + n + ( n + 1 ))

   n
  ---             2
  \   2i + 1   = n
  /__ 
  i=0

  Pass  x y   sum
  1     0 1     1
  2     1 2     4
  3     2 3     9
  4     3 4    16

#! /bin/bash

typeset -i iInput=$1
typeset -i iCnt1=0
typeset -i iCnt2=1

while [ $iInput -gt 0 ] ; do
     (( iInput = iInput - iCnt1 ))
     if [ $iInput -lt $iCnt2 ] ; then
          echo "Not a perfect square."
          exit 0
     fi
     (( iInput = iInput - iCnt2 ))
     (( iCnt1++ ))
     (( iCnt2++ ))
done

echo Solution: $iCnt1

exit 0

#!/bin/bash

XX=$1
YY=$((${#XX}&1 ? 1 : 2))
ZZ=${XX:0:$YY}
AA=$(((${#XX}-YY)/2))
VV=${XX:$YY:$((AA*2))}
BB=0
DD=1
EE=1
while (($EE))
do
	ZZ=$((ZZ-DD))
	BB=$((ZZ>=0 ? BB+1 : BB))
	DD=$((ZZ>=0 ? DD+2 : DD-2))
	EE=$((ZZ>=0 ? 1 : 0))
	ZZ=$((ZZ>=0 ? ZZ : ZZ+DD+2))
done
rac=$rac$BB
II=0
while (($AA))
do
	BB=0
	EE=1
	DD=$((((DD+1)*10)+1))
	ZZ=$((ZZ*100))
	UU=${VV:$II:1}
	UU=$((UU*10))
	UU=$((UU+${VV:II+1:1}))
	ZZ=$((ZZ+UU))
	while (($EE))
	do
		ZZ=$((ZZ-DD))
		BB=$((ZZ>=0 ? BB+1 : BB))
		DD=$((ZZ>=0 ? DD+2 : DD-2))
		EE=$((ZZ>=0 ? 1 : 0))
		ZZ=$((ZZ>=0 ? ZZ : ZZ+DD+2))
	done
	rac=$rac$BB	
	II=$((II+2))
	AA=$((AA-1))
done
(( ! ZZ )) && echo $XX is perfect square of $rac || echo $XX is not a perfect square

$ ./p_square.sh 391362529049165025
391362529049165025 is perfect square of 625589745
$ ./p_square.sh 98743978937000250000
98743978937000250000 is perfect square of 9937000500
$ ./p_square.sh 98743978937000250011
98743978937000250011 is not a perfect square