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?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
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 $x^2+y^2=n$.
The paper also appears in Cryptography and Computational Number Theory (Singapore, 1999), 129–136, Progr. Comput. Sci. Appl. Logic, 20, Birkhäuser, Basel, 2001. The Math Reviews data would be MR1944724 (2003i:11047)
See also http://oeis.org/A064942