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One of my computer science teachers came up with this neat little problem that he and I have both brute-forced and done some analytical work, but we haven't gotten far.

There exist chains of squares such that each square is the previous one but with a digit slapped on the end. For example...

$1\rightarrow 16 \rightarrow 169$
$1^2\rightarrow 4^2 \rightarrow 13^2$

$4 \rightarrow 49$
$2^2 \rightarrow 7^2$

$25 \rightarrow 256$
$5^2 \rightarrow 16^2$

There are an infinite* number of 2-square chains, but beyond the first one, we haven't found any 3-square chains and I have brute-forced it up to 20k (if I recall correctly) to look for more 3-square chains.

I have done some algebraic stuff, but I haven't come up with anything incredibly useful, unfortunately. Here's some of my work...

$n^2 = 10m^2 + k$
$n = m\sqrt{10} + \epsilon$
$n^2 = 10m^2 + 2\epsilon m\sqrt{10} + \epsilon^2$
$k = 2\epsilon m\sqrt{10} + \epsilon^2$
$0 = \epsilon^2 + 2\epsilon m\sqrt{10} + k$
Application of quadratic formula, simplifying, and noting that only the more positive answer is useful...
$\epsilon = -m\sqrt{10}+\sqrt{10m^2-k}$
While neat, this equation hasn't been of much use to me thus far... :P

*The continued fraction for $\sqrt{10}$ gives 2-square chains composed of the squares of the numerator and denominator.

So yeah...anyone have a way to construct 3-square chains? Even a proof that more than two exist would be very helpful.

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See oeis.org/A031150 and oeis.org/A023110, although there doesn't seem to be a huge amount to work with there. –  Michael Lugo May 6 '11 at 3:29
    
Interesting: With respect to 2-square chains: I've been playing around a bit: $4^2 \rightarrow13^2$, $6^2 \rightarrow 19^2$, ...and more. What I've noticed in every pair I've encountered so far: exactly one of the two is prime. Hmmm... –  amWhy May 6 '11 at 4:14
    
Oooo...interesting! Something new to investigate! :P –  El'endia Starman May 6 '11 at 4:23
    
Running my brute-force program again, there are two counter-examples right after the ones you put. 12^2 -> 38^2 and 18^2 -> 57^2. –  El'endia Starman May 6 '11 at 4:43
    
Thanks for getting back...oh, well...Nice question, though! –  amWhy May 6 '11 at 4:51

1 Answer 1

up vote 7 down vote accepted

I believe your search is sufficient to prove there are no others. If you let $m$ be the square root of the first in the chain and $n$ be the square root of the third, you have $100m^2+a=n^2$ with $9 \lt a \lt 100$. This becomes $(n-10m)(n+10m)=a$, and $n \gt 10m$, so once $10m \gt 50$ there are no more solutions. Another way to see it is that the smallest $n^2$ can be is $(10m+1)^2=100m^2+2m+1$ and with $m \gt 50$ this will not start with $m$ because of the carry.

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