# Classifying mathematical “coincidences”

Doing homework a few years ago, I noticed that the sum of the squares of $88$ and $33$ is $8833$. What would this kind of mathematical "curiosity" be called? Does this or any other similar coincidence have any deeper meaning or structure?

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Goldblatt in Topoi mentions $2+2 = 2 \times 2$ – alancalvitti Jan 30 '13 at 1:35
Did he leave out $2^2$? – Todd Wilcox Jan 30 '13 at 20:59
And $2 \mathbin{\uparrow\uparrow} 2$? – Trevor Wilson Jan 31 '13 at 0:23

This particular "curiosity" says that one solution of the Diophantine equation $a^2 + b^2 = 100 a + b$ is $a=88, b=33$. The other solutions in positive integers, by the way, are $a=12, b=33$ and $a=100,b=1$.
Alf van der Poorten wrote a paper, The Hermite-Serret algorithm and $12^2+33^2$. He notes a number of similar curiosities, such as $$25840^2+43776^2=2584043776$$ and $$1675455088^2+3734621953^2=16754550883734621953$$ and develops the theory behind these things, including a discussion of the way to go from a solution of $z^2\equiv-1\pmod n$ to a solution of $x^2+y^2=n$.