When is $999\cdots$ a perfect square?

I'm interesting to look more about property of this number $999\cdots$ , for even digits which form that number it's clear that's not a perfect square for example :$99=10^2-1,9999=10^4-1,\cdots$ , Now my question is: what about if the numbers of digits of $999\cdots$ is odd ? how i proof or disproof that is a perfect square or nor ?

Note: The trivial case is for $n=1$ which is $9$

• An odd perfect square is always 1 mod 4. Aug 31 '18 at 13:01
• @JohnBrevik Why are you answering in a comment? Aug 31 '18 at 13:04
• @Arthur I thought it would be helpful. Aug 31 '18 at 13:05
• @JohnBrevik Sure. It is, however, more helpful if you put it in an answer post. That's what they're there for, after all. Aug 31 '18 at 13:27
• We know now that no perfect square can end in two or more nines. But, using a slight modification, $...99969$ will occur at the end of perfect squares no matter how many nines are placed before the $69$. Aug 31 '18 at 21:12

Write: $$10^{2n+1}-1 = a^2$$

so $a=2k+1$ and now we have $$2^{2n+1}5^{2n+1}= 2\underbrace{(2k^2+ 2k+1)}_{\rm odd}$$

so $2^{2n+1}=2$ and thus $n=0$. Finally we have $9 = a^2$ so $a=3$.

The only number consisting entirely of nines that's a perfect square is $9$ itself.

Suppose $999$ were a perfect square. It would have the form $(2n+1)^2$ with$n$ a whole number. Therefore

$(2n+1)^2=4n^2+4n+1=999$

$4n(n+1)=998$

So $998$ has to be a multiple of $4$, but that doesn't work because the last two digits $98$ would have to be a multiple of $4$ and they aren't really so. So, the second equation above has no whole number solutions, the first equation can't have any such solutions either and $999$ fails to be a perfect square.

The same thing happens with any number ending with $99$, since the last two digits are what ultimately decides what can be a multiple of $4$. Once we have two nines at the end, therefore, there can never be a perfect square.

Any odd perfect square is of form $4k+1$ for some $k$. Now let $$a^2=10^{n}-1$$if $n\ge 2$ we have $$a^2=10^2\cdot10^{n-2}-1=4k-1$$which can't be a perfect square. So the only perfect square may be among those numbers with $$0\le n<2$$or $$0,9$$which are the only solutions.

• Odd squares are of the form $4k+1$, so you just need to consider $n\ge2$ or $n<2$. Aug 31 '18 at 14:40

To make $\underbrace{999\cdots 9}_{n\text{ many}}$ a perfect square, we need $9\cdot\underbrace{111\cdots 1}_{n\text{ many}}$ a perfect square. As, $9$ is already a perfect square, we need $\underbrace{111\cdots 1}_{n\text{ many}}$ to be a perfect square. But, observe that, $$11\equiv 3\pmod{4}\\ 111\equiv 3\pmod{4} \\ 1111\equiv 3\pmod{4}$$ in general, $$\underbrace{111\cdots 1}_{n\text{ many}}=2\underbrace{77\cdots 7}_{n-2\text{ many}}×4+3\equiv 3\pmod{4}$$

But, any odd perfect square can be of the form $4k+1$. Hence, none of the numbers is a perfect square.