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I came to know that, pseudoprime with $base-2$ called as poulet numbers. In this context, poulet numbers are infinite and starts from $341$ or $341$ is the first poulet number.

Now my question is how to disprove $323$ is not poulet number and $121$ is not poluet but Fermat Pseudo prime.

I know that by definition, $a^{p-1} \equiv 1 \pmod p$ if $a = 2$ then we call it as poluet number.

But, how to diprove $2^{322} \not\equiv 1 \pmod {323}$

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The first step is to factor $323=17\times 19$. Now using Fermat's Little Theorem you should be able to work out $2^{322}$ mod $17$ and mod $19$. In order for it to be a Poulet number, both would have to be $1$. This is not the case, because

$2^{322}=(2^{16})^{20}\times 2^2\equiv 1\times 4$ (mod $17$).

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    how 1 X 4 (mod 17) you got? could you explain please – 2017-02-28
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    $2^{16}\equiv 1$ (Fermat's Little Theorem), so $(2^{16})^{20}\equiv 1^{20}\equiv 1$, and $2^2=4$, so $(2^{16})^{20}\times 2^2\equiv 4$. – 2017-02-28