I am asked to find all bases for which 561 is an Euler pseudoprime. Unfortunately, the lecturer did not tell us how to do this in a systematic way.
Any help?
2026-03-26 10:57:24.1774522644
561 Euler pseudoprime bases
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You want to find $a$ such that $a^{280} \equiv \pm 1 (\text{mod } 561)$. Since $561 = 3 \cdot 11 \cdot 17$, the congruence must be true mod $3$, mod $11$ and mod $17$. For example, $a^{280} \equiv 1 (\text{mod } 11)$ for all $a$ coprime to $11$, because $280$ is divisible by $10$. Now, what about mod $3$ and mod $17$?