Euclid There are infinitely many primes.\ithm{infinitely many primes}


Euler's Theorem \ithm{Euler's} If $\gcd(x,n)=1$, then $$    x^{\vphi(n)} \con 1\pmod{n}. $$


Chinese Remainder Theorem \ithm{Chinese remainder} Let $a, b\in\Z$ and $n,m\in\N$ such that $\gcd(n,m)=1$.  Then there exists $x\in\Z$ such that \begin{align*} x&\con a\pmod{m},\\ x&\con b\pmod{n}. \end{align*} Moreover~$x$ is unique modulo~$mn$.


Pseudoprimality An integer~$p>1$ is prime if and only if for {\em every} $a\not\con 0\pmod{p}$, $$       a^{p-1}\con 1\pmod{p}. $$


Primitive Roots There is a primitive root modulo any prime~$p$.  In particular, the group $(\zmod{p})^*$ is cyclic.


Gauss's Quadratic Reciprocity Law \ithm{quadratic reciprocity} Suppose~$p$ and~$q$ are distinct odd primes.  Then $$ \kr{p}{q} = (-1)^{\frac{p-1}{2}\cdot \frac{q-1}{2}}\kr{q}{p}. $$ Also $$   \kr{-1}{p} = (-1)^{(p-1)/2}\qquad\text{\rm and}\qquad   \kr{2}{p}  = \begin{cases} \hfill1 & \text{\rm if } p\con \pm 1\pmod{8}\\     -1 & \text{\rm if } p \con \pm 3\pmod{8}.             \end{cases} $$


Continued Fraction Limit Let $a_0, a_1, \ldots $ be a sequence of integers such that $a_n > 0$ for all $n\geq 1$, and for each $n\geq 0$, set $c_n = [a_0, a_1, \ldots a_n].$ Then $\ds\lim_{n\ra \infty} c_n$ exists.


Let $a_0, a_1, a_2, \ldots $ be a sequence of real numbers such that $a_n > 0$ for all $n\geq 1$, and for each $n\geq 0$, set $c_n = [a_0, a_1, \ldots a_n].$ Then $\ds\lim_{n\ra \infty} c_n$ exists if and only if the sum $\sum_{n=0}^{\infty} a_n$ diverges.


\ithm{continued fraction existence} Let $x\in\R$ be a real number.  Then $x$ is the value of the (possibly infinite) simple continued fraction $   [a_0, a_1, a_2, \ldots] $ produced by the continued fraction procedure.


Periodic Characterization \ithm{period continued fraction} An infinite simple continued fraction is periodic if and only if it represents a quadratic irrational.


A positive integer~$n$ is a sum of two squares if and only if all   prime factors of~$p\mid n$ such that $p\con 3\pmod{4}$ have even   exponent in the prime factorization of~$n$.


The binary operation $+$ defined in Algorithm~\ref{alg:grouplaw}   endows the set $E(K)$ with an abelian group structure, with identity $\O$.


Suppose $a,b,n\in\Z$.  Then $\gcd(a,b) = \gcd(a,b-an)$.


For any integers $a,b,n$, we have $$\gcd(an,bn) = \gcd(a,b)\cdot |n|.$$


Suppose $a,b,n\in\Z$ are such that $n\mid a$ and $n\mid b$.  Then   $n\mid \gcd(a,b)$.


If~$R$ is a complete set of residues modulo~$n$ and $a\in\Z$ with $\gcd(a,n)=1$, then $aR = \{ax : x \in R\}$ is also a complete set of residues modulo~$n$.


Suppose that $m,n\in\N$ and $\gcd(m,n)=1$. Then the map \begin{equation}\label{eqn:crtprod} \psi: (\zmod{mn})^* \ra (\zmod{m})^* \cross (\zmod{n})^*. \end{equation} defined by $$  \psi(c) = (c\text{ mod } m, \,\,c \text{ mod }n) $$ is a bijection.


The map $\psi:(\zmod{p})^*\to \{\pm 1\}$ given by $\psi(a) = \kr{a}{p}$ is a surjective group homomorphism.


Gauss's Lemma Let~$p$ be an odd prime and let~$a$ be an integer $\not\con 0\pmod{p}$.  Form the numbers $$ a,\, 2a,\, 3a,\, \ldots,\, \frac{p-1}{2} a $$ and reduce them modulo~$p$ to lie in the interval $(-\frac{p}{2},\,\, \frac{p}{2})$, i.e., for each of the above products $k\cdot a$ find a number in the interval $(-\frac{p}{2},\,\, \frac{p}{2})$ that is congruent to $k\cdot a$ modulo $p$.  Let $\nu$ be the number of negative numbers in the resulting set.  Then $$ \kr{a}{p} = (-1)^{\nu}. $$


If $x$ and $y$ are arbitrary integers, then $$ \sum_{n=0}^{p-1} \zeta^{(x-y)n} = \begin{cases} p & \text{\rm if $x\con y\pmod{p}$},\\ 0 & \text{\rm otherwise}. \end{cases} $$


We have $g_0=0$.


For any integer $a$, $$  g_a = \kr{a}{p}g_1. $$


If~$n$ is divisible by a prime~$p\con 3\pmod{4}$, then~$n$ has no primitive representations.


If $x\in\R$ and $n\in\N$, then there is a fraction $\ds\frac{a}{b}$ in lowest terms such that $0<b\leq n$ and $$\left| x - \frac{a}{b} \right| \leq \frac{1}{b(n+1)}.$$


The equation $x^2\con a\pmod{p}$ has no solution if and only if $a^{(p-1)/2}\con -1\pmod{p}$. Thus $\kr{a}{p} \con a^{(p-1)/2}\pmod{p}$.


Cancellation If $\gcd(c,n)=1$ and $$    ac\con bc\pmod{n}, $$ then $a \con b\pmod{n}$.


Units If $\gcd(a,n)=1$, then the equation $  ax\con b\pmod{n} $ has a solution, and that solution is unique modulo~$n$.


Solvability The equation $ax\con b\pmod{n}$ has a solution if and only if $\gcd(a,n)$ divides~$b$.


Multiplicativity of $\vphi The function $\vphi$ is multiplicative. \index{phi function!is multiplicative}\index{Euler!phi function!is multiplicative}


Let~$p$ be a prime number and let~$d$ be a divisor of $p-1$.  Then $f = x^d-1\in(\zmod{p})[x]$ has exactly~$d$ roots in $\zmod{p}$.


Suppose~$p$ is a prime such that $(p-1)/2$ is also prime. Then each element of $(\zmod{p})^*$ has order one of~$1$,~$2$, $(p-1)/2$, or $p-1$.


Decryption Key Let~$n$ be an integer that is a product of distinct primes and let $d,e\in\N$ be such that $p-1\mid de-1$ for each prime $p\mid n$.  Then $a^{de} \con a\pmod{n}$ for all $a\in\Z$.


Euler's Criterion \iprop{Euler's criterion} We have $\kr{a}{p}=1$ if and only if $$ a^{(p-1)/2}\con 1\pmod{p}. $$


Euler Let~$p$ be an odd prime and let~$a$ be a positive integer with $p\nmid a$. If $q$ is a prime with $q\con \pm p\pmod{4a}$, then $\kr{a}{p} = \kr{a}{q}$.


Legendre Symbol of $2 Let~$p$ be an odd prime.  Then $$ \kr{2}{p} = \begin{cases} \hfill1 & \text{ if } p\con \pm 1\pmod{8}\\                          -1 & \text{ if } p\con \pm 3\pmod{8}.  \end{cases} $$


How Convergents Converge \iprop{how convergents converge}% The even indexed convergents $c_{2n}$ increase strictly with~$n$, and the odd indexed convergents $c_{2n+1}$ decrease strictly with~$n$.  Also, the odd indexed convergents $c_{2n+1}$ are greater than all of the even indexed convergents $c_{2m}$.


Geometric Group Law \iprop{geometric group law} Suppose $P_i=(x_i,y_i)$, $i=1,2$ are distinct points on an elliptic curve $y^2=x^3+ax+b$, and that $x_1\neq x_2$.  Let $L$ be the unique line through $P_1$ and $P_2$.   Then $L$ intersects the graph of~$E$ at exactly one other point $$   Q = \left(\lambda^2 -x_1 - x_2,\quad \lambda x_3 + \nu\right), $$ where $\lambda = (y_1-y_2)/(x_1-x_2)$ and $\nu = y_1 - \lambda x_1$.


Congruent Number Criterion % \iprop{congruent number criterion}%   The rational number~$n$ is a congruent number if and only if there is a point $P=(x,y)\in E_n(\Q)$ with $y\neq 0$.