A positive integer $a$ is called a double number if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.
Problem
Source: Indian IMOTC 2013, Team Selection Test 4, Problem 1
Tags: modular arithmetic, Diophantine equation, number theory proposed, number theory
03.08.2013 17:37
Basically,we have to prove whether the equation $n(10^k+1)=s^2-1$ have infinitely many positive integer solutions with $10^{k-1}\le n \le 10^k$ except the trivial solution $n=10^k-1,s=10^k$.
04.08.2013 18:59
And a pell's equation $s^2-(10^k+1)x^2=1$ where the conventional $d=10^k+1$ and $n=x^2$.Now the problem looks something tackalable with $10^{k-1}\le n<10^k.$
05.08.2013 08:37
How would you know that the Pell equation has a solution satisfying the given constraint? Also, I could not find any example for small k (< 8) where a perfect square solution $n$ exists. In general, it seems that for most $n$ there are several solutions that satisfy $s-n = $ constant, but I could not come up with a general form ...
23.03.2014 17:39
Once we know that there are infinitely many solutions (I was told so by the attendees of the camp), its not very difficult to construct the solutions.
P.S. USA people would probably hate me for creating a solution with an infinitude of 9/11's
03.04.2014 14:30
Maybe I don't understand. Why we can't take only the numbers in the form $a=10^{10k-2}-1$?
03.04.2014 16:58
If you do that, then $a+1$ is a power of $10$.
03.04.2014 18:16
I thought that $a+1$ couldn't be in form $k^{10}$, not $10^k$. Sorry -.-
03.04.2014 18:26
This problem was pretty simple but almost everyone lost 1 mark because we didnt prove the fact that a+1 is not a power of 10, even if it was pretty obvious in most constructions, silly mistake indeed