Prove that none of $\{11, 111, 1111,\dots \}$ is the perfect square of an integer

Please help me with solving this :
prove that none of $\{11, 111, 1111 \ldots \}$ is the square of any $x\in\mathbb{Z}$ (that is, there is no $x\in\mathbb{Z}$ such that $x^2\in\{11, 111, 1111, \ldots\}$).

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Hint: Perfect squares are not of the form $4k+3$, where $k$ is an integer.


Hint: For an even integer, $n=2j$, then $n^2 = (2j)^2 = ??$ For an odd integer, $n=2j+1$, then $n^2 = (2j+1)^2 = ??$.

Let suppose that there exists a number that squared gives $11 \cdots111$. Let $ba$ be its last two digits. Then either $a=1$ or $a=9$.

But if $a=1$, then the tens digit is $b + b \pmod{10}$ , which is even.

If $a=9$, , then the tens digit is $9 b + 8 + 9 b \pmod{10} = 18 b + 8 \pmod{10}$; which is also even.

Then, the tens digit cannot be $1$.

By the same reasoning, you can get the stronger result (see Philip Gibbs’ answer) that a square cannot end with two odd digits.

A square number can never end with two odd digits

If it did it would have to be the square of an odd number $x = 10a+b$ where $b$ is odd.

$x^2 = 100a^2 + 20ab + b^2$ so you just have to check for $x = 1,3,5,7$ or $9$ that the 10s digit is even and the rest follow.

Every integer is of one of the forms $\color{green}4k$, $\color{green}4k+\color{red}1$, $\color{green}4k+\color{red}2$ or $\color{green}4k+\color{red}3$ with $k$ integer. The square of an integer has therefore one of the forms

  • $(\color{green}4k)^2 = 16k^2 = \color{green}4(4k^2) = \color{green}4k_0$,
  • $(\color{green}4k+\color{red}1)^2 = 16k^2+8k+1 = \color{green}4(4k^2+2k)+\color{red}1 = \color{green}4k_1+\color{red}{1}$,
  • $(\color{green}4k+\color{red}2)^2 = 16k^2+16k+4 = \color{green}4(4k^2+4k+1)=\color{green}4k_2$ or
  • $(\color{green}4k+\color{red}3)^2 = 16k^2+24k+9 = \color{green}4(4k^2+6k+2)+\color{red}1 = \color{green}4k_3+\color{red}1$.

That is, a perfect square is equivalent either to $\color{red}0$ or to $\color{red}1$ modulo $\color{green}4$.

On the other hand, $R_n=\underbrace{1\ldots1}_n=(10^n-1)/9$ for $n>1$ has the form

  • $R_n=100R_{n-2}+11 = \color{green}4(25R_{n-2}+2)+\color{red}3 = \color{green}4k_r+\color{red}3$.

That is, for $n>1$, $R_n$ is equivalent to $\color{red}3$ modulo $\color{green}4$, which as shown above does not happen for perfect squares.

If $n\equiv11\pmod{100}$, then
$$
\begin{align}
n
&=100k+11\\
&=4(25k+2)+3\\
&\equiv3\pmod{4}
\end{align}
$$
If $n$ is even, $n=2k$ and $n^2=4k^2$. Thus, $n^2\equiv0\pmod{4}$.
If $n$ is odd, $n=2k+1$ and $n^2=4(k^2+k)+1$. Thus, $n^2\equiv1\pmod{4}$

Therefore, whether $n$ is even or odd, $n^2\not\equiv3\pmod{4}$, and consequently,
$$
n^2\not\equiv11\pmod{100}
$$

Any odd square is congruent to 1 modulo 4
$ (2n+1)^2 = 4n^2+4n+1 = 4(n^2+n)+1 \cong 1 \bmod{4} $

But 11 is congruent to 3 modulo 4.

Just as well any positive integer ending with 11 is not a perfect square.

If you examine the sequence formed by the last 2 digits of a square of an integer, or equivalently, look at the sequence $n^2 \pmod{100}$, it is easy to show that a square of an integer can only end with one of the following pairs of digits: 00, 01, 21, 41, 61, 81, 04, 24, 44, 64, 84, 06, 16, 36, 56, 76, 96, 09, 29, 49, 69, 89 or 25.

Each number in your sequence ends with $..11$, hence none of them can be squares.

From the above observation, it is also clear that many other sequences can also never contain a square, for example:

22, 222, 2222, …

14, 414, 1414, 41414, …

One could spend many happy hours creating such sequences.

Let’s first reformulate the claim:
$$ \nexists n \in \mathbb{Z}, k \in \{11, \space 111, \space \ldots \} : n^2 = k $$

Proof:
First, consider all non-negative integers $n$ such that $n$ is even. That is,
$$ n = 2j \space \space \forall j \in \mathbb{Z} $$
For these $n$ we know that $n^2$ is of the form:
$$ n^2 = (2j)^2 = 2^2 \cdot j^2 = 4j^2 $$
This means $n^2$ can be divided by 2 at least twice. Another way of saying this is that half of $n^2$ is even. However, $k$ is of the following form:
$$ k = 10a + 1 \space \space \forall a \in \{1, \space 11, \space \ldots \} $$
Since $10a$ is even, $10a + 1$ is not. However, since $n^2$ is divisible by 4, it is also even. Hence we know that:
$$ \nexists n \in \mathbb{Z}, j \in \mathbb{Z}, k \in \{11, \space 111, \space \ldots \} : n = 2j \space \land \space n^2 = k $$
That leaves us with all non-negative integers $n$ such that $n$ is uneven. Those can be written as:
$$ n = 2j + 1 \space \space \forall j \in \mathbb{Z} $$
In this case $n^2$ will have the form:
$$ n^2 = (2j + 1)^2 = 4j^2 + 4j + 1 $$
We can see right away that this number is odd. But $k$ is odd too, so that is of no help. However, the other terms of $n^2$ all contain a factor of 4. That means $n^2 – 1$ is divisible by 4. Since that is even stronger than knowing a number is even, we should compare $n^2 – 1$ to $k – 1$ and see if $k – 1$ fits the same bill. Let’s re-write this argument in Mathese and see if $k – 1$ is divisible by 4:
$$ n^2 = k \iff n^2 – 1 = k – 1 $$
$$ n^2 – 1 = 4j^2 + 4j = 4(j + 1) \implies 4 \space | \space n^2 – 1 $$
Hence $n^2 = k \iff 4 \space | \space k – 1$.
Let’s try to reformulate $k$ in such a way that shows whether $k$ is divisible by 4:

$$ k \in \{11, \space 111, \space \ldots \} \iff k – 1 \in \{ 11 – 1, \space 111 – 1, \space \ldots \} $$
$$ \kern 47pt \iff k – 1 \in \{ 10, \space 110, \space \ldots \} $$
$$ \kern 73pt \iff k – 1 \in \{ 10 \cdot 1, \space 10 \cdot 11, \space \ldots \} $$
Since $10$ is even but not divisible by 4, the other factor that composes $k – 1$ must be even for $k – 1$ to be divisible by 4 as well.
We can write an equation that separates the factor $10$ from the factor that is an element of $\{1, \space 11, \space \ldots \}$.
$$ k – 1 = 10 \cdot j \space \forall j \in \{1, \space 11, \space \ldots \} $$
Since we know already that $\{11, \space 111, \space \ldots \}$ contains only uneven numbers and $1$ is uneven, we know the set $\{1\} \cup \{11, \space 111, \space \ldots \} \equiv \{1, \space 11, \space \ldots \}$ contains only uneven numbers as well.

Hence $4 \nmid k – 1 \space \forall k \in \{11, \space 111, \space \ldots \}$.

Hence $ \nexists n \in \mathbb{Z}, k \in \{11, \space 111, \space \ldots \} : n^2 = k $

Which is that what was to be proven.

Well, consider an odd number $n = 2k + 1$. Then, $n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1$.

So, $n^2 \equiv 1 \pmod{4}$, or, in other words, $n^2$ will always give you a remainder of 1 on division by 4 if n is odd.

On the other hand, if n is even, then, say, $n = 2k \Longrightarrow n^2 = 4k^2 \Longrightarrow n^2 \equiv 0 \pmod{4}$. Or, $n^2$ gives you a remainder of 0 on division by 4.

So for any integer $n$, we know that $n^2 \equiv 0 \pmod{4}$ if n is even and $n^2 \equiv 1 \pmod{4}$ if n is odd.

Now, let’s look at the numbers in the given sequence. Firstly, note that 100 is divisible by 4. And, say, split each number into a multiple of 100 added to 11.

For example, $1111 = 1100 + 11, 11111 = 11100 + 11, \cdots$.

Now, each number in the sequence is equivalent to $11 \pmod{4}$. But $11 \equiv 3 \pmod{4}$, so each number in the sequence gives a remainder of 3 on division by 4.

But this is impossible for a square number, as we’ve shown above.

Therefore, no number in this sequence is a square. Q.E.D.