When preparing the challenge this month, I made a small mistake that made the question much easier than intended.
Here is the solution, as I derived it:
Let y^2 be a k-digits-long number, giving us 10^k x^2 + y^2 = z^2.
Write z=10^k a + b and we get
10^k x^2 + y^2 = (10^k a + b)^2 = 10^k a(10^k a+ 2b) + b^2.
So, b^2=y^2 (mod 10^k). Here I made the mistake of concluding that b=y.
Those who want to try to solve a more difficult version of the question, can stop here and ponder a little on solving this version.
My mistake was, of course, that b can be -y mod(10^k), which lends itself to several trivial solutions like
x = 5 * 10^(k-1) - 1; y = 10^k -1; and z = 5 * 10^(2 * k-1) - 10^k + 1.
If we continue with the assumption that b=y, we get
x^2 = a^2 10^k + 2ya
so y = (x^2-a^2 10^k) / 2a.
For every k and a, we can compute x as ceil(a*sqrt(10^k)) and check whether the resulting y is divisible by 2a.
The fact that sqrt(10^37) is surprisingly close to an integer (~3162277660168379331.99889) gives us, for k=37, many possible solutions. Choosing a=463 gives us the intended solution of
János Kramár and Andreas Stiller sent us a solution with y=1, where one can solve Pell's equation: 10*x^2 + 1 = z^2 and get:
Jan Fricke also used Pell's equation to get to a solution with 60,503 digits.
If you have any problems you think we might enjoy, please send them in. All replies should be sent to: firstname.lastname@example.org