Why is $\underbrace{444\dots44}_{2n} + \underbrace{888\dots88}_{n} + 4$ never a perfect square?

For all naturals $n$ we have $$f(n+1)-f(n)=44\cdot10^{2n}+8\cdot{10^n}.$$

Hence, $$\frac{f(n+1)-f(n)}{16}=11\cdot25\cdot10^{2(n-1)}+5\cdot10^{n-1}\in\mathbb N.$$

It follows that $$f(n)\equiv f(1)\equiv8\pmod{16}$$ for all $n\in\mathbb N$.

Because every square is equivalent to $0,1,4$ or $9$ modulo $16$, it follows that $f(n)$ is never a square.


Hint:

Can you show that $\sqrt{f(n)}$ is between consecutive integers $\dfrac2310^n+\dfrac13$ and $\dfrac2310^n+\dfrac43$?

Tags:

Elementary Number Theory

Square Numbers

Related

What's the generalization for $\zeta(2n)$? Without calculating the square roots, determine which of the numbers:$a=\sqrt{7}+\sqrt{10}\;\;,\;\; b=\sqrt{3}+\sqrt{19}$ is greater. Evaluate in closed form: $ \sum_{m=0}^\infty \sum_{n=0}^\infty \sum_{p=0}^\infty\frac{m!n!p!}{(m+n+p+2)!}$ Applications of uncountability of the real numbers Can someone help explain the behaviour of the implicit equation $x^n+y^n-n^x-n^y=0$, specifically the significance of the number 1.9667894071088...? Why is $\frac{1}{\sqrt 5}\left[\left(\frac{1+\sqrt 5}{2}\right)^n-\left(\frac{1-\sqrt 5}{2}\right)^n\right]$ an integer? Sphere inside of a Sphere Behavior of a gradient system Solving a 2nd order DE (non-constant coefficients) If $f(17) = 17$, calculate $A(x) = \int_{1}^{x} f(t)dt$ for $x>0$. How the recursive structure of Apollonian gaskets can be described in order to be able to reproduce them? Prove that a tree with a vertex $v$ of degree $k > 1$ has at least $k$ leaves

Recent Posts