'''---------------------------------------------------------------

 This program prints all excellent numbers of a given length ...

 ... is "excellent": when split on half m=ab, b*b – a*a = m.  For
 example, 48 is excellent, because 8**2 - 4**2 == 48.
   [ link ]
 I take advantage of the fact that, with n = length/2, we can
 rewrite m to be a*10**n + b, set the two m's equal to another,
 simplify to b * b – b == a * (a + 10**n), and solve for b:

           1 + sqrt[4a^2 + 4(10^n)a + 1]
       b = -----------------------------
                        2

 So we loop through all values for a with n digits, find its b,
 and see if b is an integer.  In Python:

---------------------------------------------------------------'''

from is_excellent import is_excellent
import sys

def EXP(base, exp):
  return base ** exp

def SQRT(n):
  return n ** 0.5

def EVEN(n):
  #return (n == int(n)) and ((int(n) % 2) == 0)
  return n == 2 * (int(n) // 2)

def IS_ROOT(r, n):
  return n == r*r

'''
for a in range(lower, 10*lower):
    b = ((4*a**2+400000*a+1)**0.5+1) / 2.0
    if b == int(b):
        result = int(str(a)+str(int(b)))
        print(result)
THEN
for a in range(lower, 10*lower):
    b1 = SQRT(4*EXP(a, 2)+4*EXP(10, n)*a+1) + 1
    print('candidate', b1)
    if EVEN(b1):
        b = b1 / 2.0
        result = int(a * EXP(10, n) + b)   # int(str(a)+str(int(b)))
        print(result)
'''

def print_candidate_and_continue(a, n, upper, candidate):
  print(candidate)
  return a_loop(a+1, n, upper)

def a_loop_3(a, n, upper, det, root, candidate):
    if IS_ROOT(root, det) and EVEN(root + 1) and is_excellent(candidate):
        print_candidate_and_continue(a, n, upper, candidate)
    else:
        return a_loop(a+1, n, upper)

def a_loop_2(a, n, upper, det, root):
    a_loop_3(a, n, upper, det, root, int(a * EXP(10, n) + ((root + 1) / 2.0)))

def a_loop_1(a, n, upper, det):
    a_loop_2(a, n, upper, det, SQRT(det))

def a_loop(a, n, upper):
  if a < upper:
    a_loop_1(a, n, upper, 4*EXP(a, 2) + 4*EXP(10, n)*a + 1)

def create_loop(a, n):
  a_loop(a, n, 10*a)

# ----------------------------------------------------------------

sys.setrecursionlimit(50000)  # for 8-digit numbers, still crashes Python

length = int(sys.argv[1]) # 6      # int(sys.argv[1])
n      = length // 2
lower  = EXP(10, n-1)

create_loop(lower, n)

# ----------------------------------------------------------------