$N$ is a 6 digit Natural number such that its sum of the digits is $43$.
Find $N$ if Exactly One of the statements below is False:
$(1)$ $N$ is a Perfect Square
$(2)$ $N$ is a Perfect Cube
$(3)$ N $\lt 500000$
My Try: The least 6 digit number with Sum of the digits $43$ is $169999$ and the Highest number is $999970$.
Case(i):-> Let Statement 2 is False, That is $N$ is not Perfect Cube.
Now since $N$ $\lt 500000$ and $N$ Last digit can only be either of 0,1,4,5,6 and 9.
Since Least number with sum of digits $43$ is $169999$ and Highest number $\lt 500000$ with sum of digits $43$ is $499993$. Now we need to check a Perfect Square in the Interval $[169999\: 499993]$
I tried to list out numbers in the above interval , but its becoming very Lengthy...Help Required. Thanks