-gaussian integer
<h3 class=dred>Gaussian integers</h3>
<p>\(\mathbb{Z}(i) = \{a+bi : a,b \in \mathbb{Z} \} \)</p>
-eisenstein integer


<h3 class=dred>Carmichael's totient function</h3>
<p>Function \(\lambda (n)\) denoting the smallest \(m :  \gcd (a,n) \implies a^m \equiv 1 \mod n\)</p>
<p>\(\lambda (p) = \varphi (p)\)</p>


<h3 class=dred>Pseudoprime</h3>
<p>Composite number that follows a set of properties that prime numbers exhibit.</p>

<h3 class=dred>Carmichael number</h3>
<p>Pseudoprime that passes the Fermat primality test for all coprime \(a\)</p>
<p>\(n \text{ is a Carmichael number } \iff n \text{ is composite } \land \forall a \in \mathbb{N} : \gcd(a,n) = 1, a^{n-1} \equiv 1 \mod n\)</p>


<h3 class=dred>Korselt's theorem</h3>
<p>\(n \text{ is a Carmichael number } \iff n=\prod_{i=1} p_i : \text{each prime is distinct } \land p_i | n-1 \)</p>
<h4 class=dred>Proof</h4>
<p>let \gcd(a,n)=1, then \(\forall p_i, a^{p_i -1} \equiv 1 \mod p_i \) and since \(p_i -1 | n-1\) (by the assumption), then \forall p_i, a^{n-1} \equiv 1 \mod p_i\) and therefore since \(a^{n-1}-1\) divides all the prime factors, \( a^{n-1} \equiv 1 \mod n\)</p>


<p>\(q \text{ is a quadratic residue modulo } n \iff \exists x\in \mathbb{Z}_{n} : x^2 \equiv q \mod n\)</p>