Modern Cryptography and Elliptic Curves: Integer Solutions to $987654319 = x^2 + y^2 + z^2$
by Hidenori
Exercise from P.62 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide.
Are there integer solutions to $987654319 = x^2 + y^2 + z^2$
No.
By checking $(4n + k)^2 \pmod 8$ for each $k = 0, \cdots, 3$, it becomes obvious that $m^2 \equiv 0, 1, 4 \pmod 8$ for any $m \in \mathbb{Z}$. However, $987654319 \equiv 319 \equiv 7 \pmod 8$.
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