corpus.jsonl

{"_id": "0", "title": "Fundamental Theorem of Arithmetic", "text": "\\index{fundamental theorem of arithmetic}% \\index{integers!factor uniquely}% \\ithm{unique factorization}% Every natural number can be written as a product of primes uniquely up to order."}
{"_id": "1", "title": "Euclid", "text": "\\index{Euclid's theorem!on divisibility}% \\ithm{Euclid} Let~$p$ be a prime and $a, b\\in \\N$. If $p\\mid ab$ then $p\\mid a$ or $p\\mid b$."}
{"_id": "2", "title": "Euclid", "text": "There are infinitely many primes.\\ithm{infinitely many primes}"}
{"_id": "3", "title": "Dirichlet", "text": "\\ithm{Dirichlet} Let~$a$ and~$b$ be integers with $\\gcd(a,b)=1$. Then there are infinitely many primes of the form $ax+b$."}
{"_id": "4", "title": "Prime Number Theorem", "text": "\\ithm{prime number} The function $\\pi(x)$ is asymptotic to $x/\\log(x)$, in the sense that    $$\\lim_{x\\ra \\infty}  \\frac{\\pi(x)}{ x/\\log(x)} = 1.$$"}
{"_id": "5", "title": "Euler's Theorem", "text": "\\ithm{Euler's} If $\\gcd(x,n)=1$, then $$    x^{\\vphi(n)} \\con 1\\pmod{n}. $$"}
{"_id": "6", "title": "Chinese Remainder Theorem", "text": "\\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~$mn$."}
{"_id": "7", "title": "Pseudoprimality", "text": "An integer~$p>1$ is prime if and only if for {\\em every} $a\\not\\con 0\\pmod{p}$, $$       a^{p-1}\\con 1\\pmod{p}. $$"}
{"_id": "8", "title": "Primitive Roots", "text": "There is a primitive root modulo any prime~$p$.  In particular, the group $(\\zmod{p})^*$ is cyclic."}
{"_id": "9", "title": "Primitive Roots mod $p^n", "text": "\\ithm{primitive root mod prime powers} Let~$p^n$ be a power of an odd prime.  Then there is a primitive root modulo~$p^n$."}
{"_id": "10", "title": "Gauss's Quadratic Reciprocity Law", "text": "\\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 $$   \\kr{-1}{p} = (-1)^{(p-1)/2}\\qquad\\text{\\rm and}\\qquad   \\kr{2}{p}  = \\begin{cases} \\hfill1 & \\text{\\rm if } p\\con \\pm 1\\pmod{8}\\\\     -1 & \\text{\\rm if } p \\con \\pm 3\\pmod{8}.             \\end{cases} $$"}
{"_id": "11", "title": "Continued Fraction Limit", "text": "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."}
{"_id": "12", "title": "", "text": "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."}
{"_id": "13", "title": "", "text": "\\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."}
{"_id": "14", "title": "Periodic Characterization", "text": "\\ithm{period continued fraction} An infinite simple continued fraction is periodic if and only if it represents a quadratic irrational."}
{"_id": "15", "title": "", "text": "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~$n$."}
{"_id": "16", "title": "", "text": "The binary operation $+$ defined in Algorithm~\\ref{alg:grouplaw}   endows the set $E(K)$ with an abelian group structure, with identity $\\O$."}
{"_id": "17", "title": "Mordell", "text": "The group $E(\\Q)$ is finitely generated. That is, there are points $P_1,\\ldots, P_s \\in E(\\Q)$ such that every element of $E(\\Q)$ is of the form $n_1 P_1 + \\cdots + n_s P_s$ for integers $n_1, \\ldots n_s\\in\\Z$."}
{"_id": "18", "title": "Mazur, 1976", "text": "\\ithm{Mazur} Let~$E$ be an elliptic curve over~$\\Q$.  Then $E(\\Q)_{\\tor}$ is isomorphic to one of the following 15 groups: \\begin{align*} \\zmod{n} & \\qquad\\text{ for } n\\leq 10 \\text{ or } n=12,\\\\ \\Z/2\\Z\\cross \\Z/2n &\\qquad \\text{ for } n \\leq 4. \\end{align*}"}
{"_id": "19", "title": "Infinitely Many Triangles", "text": "If $n$ is a congruent number, then there are infinitely many distinct right triangles with rational side lengths and area~$n$."}
{"_id": "20", "title": "", "text": "For any integers $a$ and $b$, we have $$    \\gcd(a,b)= \\gcd(b,a) = \\gcd(\\pm a, \\pm b) = \\gcd(a,b-a) = \\gcd(a,b+a). $$"}
{"_id": "21", "title": "", "text": "Suppose $a,b,n\\in\\Z$.  Then $\\gcd(a,b) = \\gcd(a,b-an)$."}
{"_id": "22", "title": "", "text": "For any integers $a,b,n$, we have $$\\gcd(an,bn) = \\gcd(a,b)\\cdot |n|.$$"}
{"_id": "23", "title": "", "text": "Suppose $a,b,n\\in\\Z$ are such that $n\\mid a$ and $n\\mid b$.  Then   $n\\mid \\gcd(a,b)$."}
{"_id": "24", "title": "", "text": "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~$n$."}
{"_id": "25", "title": "", "text": "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."}
{"_id": "26", "title": "", "text": "Suppose $a,b\\in(\\zmod{n})^*$ have orders~$r$ and~$s$, respectively,   and that $\\gcd(r,s)=1$.  Then $ab$ has order $rs$."}
{"_id": "27", "title": "", "text": "The map $\\psi:(\\zmod{p})^*\\to \\{\\pm 1\\}$ given by $\\psi(a) = \\kr{a}{p}$ is a surjective group homomorphism."}
{"_id": "28", "title": "Gauss's Lemma", "text": "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 $$ \\kr{a}{p} = (-1)^{\\nu}. $$"}
{"_id": "29", "title": "", "text": "Let $a, b\\in\\Q$.  Then for any integer~$n$, $$\\#\\left((a,b)\\intersect \\Z\\right) \\con \\#\\left((a,b+2n)\\intersect \\Z\\right) \\pmod{2}$$ and $$ \\#\\left((a,b)\\intersect \\Z\\right) \\con \\#\\left((a-2n,b)\\intersect \\Z\\right) \\pmod{2}, $$ provided that each interval involved in the congruence is nonempty."}
{"_id": "30", "title": "", "text": "For any integer~$a$, $$ \\sum_{n=0}^{p-1} \\zeta^{an} = \\begin{cases}         p & \\text{\\rm if $a \\con 0\\pmod{p}$,}\\\\         0 & \\text{\\rm otherwise.}  \\end{cases} $$"}
{"_id": "31", "title": "", "text": "If $x$ and $y$ are arbitrary integers, then $$ \\sum_{n=0}^{p-1} \\zeta^{(x-y)n} = \\begin{cases} p & \\text{\\rm if $x\\con y\\pmod{p}$},\\\\ 0 & \\text{\\rm otherwise}. \\end{cases} $$"}
{"_id": "32", "title": "", "text": "We have $g_0=0$."}
{"_id": "33", "title": "", "text": "For any integer $a$, $$  g_a = \\kr{a}{p}g_1. $$"}
{"_id": "34", "title": "", "text": "For every $n$ such that $a_n$ is defined, we have $$x = [a_0, a_1, \\ldots, a_{n}+t_n],$$ and if $t_{n}\\neq 0$, then $   x = [a_0, a_1, \\ldots, a_{n}, \\frac{1}{t_n}]. $"}
{"_id": "35", "title": "", "text": "If~$n$ is divisible by a prime~$p\\con 3\\pmod{4}$, then~$n$ has no primitive representations."}
{"_id": "36", "title": "", "text": "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 $$\\left| x - \\frac{a}{b} \\right| \\leq \\frac{1}{b(n+1)}.$$"}
{"_id": "37", "title": "", "text": "The equation $x^2\\con a\\pmod{p}$ has no solution if and only if $a^{(p-1)/2}\\con -1\\pmod{p}$. Thus $\\kr{a}{p} \\con a^{(p-1)/2}\\pmod{p}$."}
{"_id": "38", "title": "Convergents in lowest terms", "text": "If $[a_0,a_1,\\ldots,a_m]$ is a simple continued fraction, so each $a_i$ is an integer, then the $p_n$ and $q_n$ are integers and the fraction $p_n/q_n$ is in lowest terms."}
{"_id": "39", "title": "Convergence of continued fraction", "text": "\\iprop{convergence of continued fraction}% Let $a_0,a_1,\\ldots$ define a simple continued fraction, and let $x=[a_0,a_1,\\ldots]\\in\\R$ be its value. Then for all~$m$, $$   \\left| x - \\frac{p_m}{q_m}\\right|   < \\frac{1}{q_m \\cdot q_{m+1}}. $$"}
{"_id": "40", "title": "", "text": "Suppose that~$a$ and~$b$ are integers with $b\\neq 0$.  Then there   exists unique integers~$q$ and~$r$ such that $0\\leq r< |b|$ and $a =   bq + r$."}
{"_id": "41", "title": "", "text": "Every natural number is a product of primes."}
{"_id": "42", "title": "", "text": "\\iprop{infinitely many primes} There are infinitely many primes of the form $4x-1$."}
{"_id": "43", "title": "", "text": "A number $n\\in\\Z$ is divisible by~$3$ if and only if the sum of the digits of~$n$ is divisible by~$3$."}
{"_id": "44", "title": "Cancellation", "text": "If $\\gcd(c,n)=1$ and $$    ac\\con bc\\pmod{n}, $$ then $a \\con b\\pmod{n}$."}
{"_id": "45", "title": "Units", "text": "If $\\gcd(a,n)=1$, then the equation $  ax\\con b\\pmod{n} $ has a solution, and that solution is unique modulo~$n$."}
{"_id": "46", "title": "Solvability", "text": "The equation $ax\\con b\\pmod{n}$ has a solution if and only if $\\gcd(a,n)$ divides~$b$."}
{"_id": "47", "title": "Wilson's Theorem", "text": "An integer $p>1$ is prime if and only if $(p-1)! \\con -1 \\pmod{p}.$"}
{"_id": "48", "title": "Multiplicativity of $\\vphi", "text": "The function $\\vphi$ is multiplicative. \\index{phi function!is multiplicative}\\index{Euler!phi function!is multiplicative}"}
{"_id": "49", "title": "Extended Euclidean Representation", "text": "Suppose $a,b\\in\\Z$ and let $g=\\gcd(a,b)$. Then there exists $x,y\\in\\Z$ such that $$    ax + by = g. $$"}
{"_id": "50", "title": "Root Bound", "text": "\\iprop{root bound} Let $f\\in k[x]$ be a nonzero polynomial over a field $k$.  Then there are at most $\\deg(f)$ elements $\\alpha\\in k$ such that $f(\\alpha)=0$."}
{"_id": "51", "title": "", "text": "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 $\\zmod{p}$."}
{"_id": "52", "title": "Number of Primitive Roots", "text": "\\iprop{number of primitive roots} If there is a primitive root modulo~$n$, then there are exactly $\\vphi(\\vphi(n))$ primitive roots modulo~$n$."}
{"_id": "53", "title": "", "text": "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 $p-1$."}
{"_id": "54", "title": "Decryption Key", "text": "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 $a\\in\\Z$."}
{"_id": "55", "title": "Euler's Criterion", "text": "\\iprop{Euler's criterion} We have $\\kr{a}{p}=1$ if and only if $$ a^{(p-1)/2}\\con 1\\pmod{p}. $$"}
{"_id": "56", "title": "Euler", "text": "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 $\\kr{a}{p} = \\kr{a}{q}$."}
{"_id": "57", "title": "Legendre Symbol of $2", "text": "Let~$p$ be an odd prime.  Then $$ \\kr{2}{p} = \\begin{cases} \\hfill1 & \\text{ if } p\\con \\pm 1\\pmod{8}\\\\                          -1 & \\text{ if } p\\con \\pm 3\\pmod{8}.  \\end{cases} $$"}
{"_id": "58", "title": "Gauss Sum", "text": "For any~$a$ not divisible by~$p$, $$ \\ds g_a^2 = (-1)^{(p-1)/2}p. $$"}
{"_id": "59", "title": "Partial Convergents", "text": "\\iprop{partial convergents} For $n\\geq 0$ with $n\\leq m$ we have $$ [a_0, \\ldots, a_n] = \\frac{p_n}{q_n}.$$"}
{"_id": "60", "title": "", "text": "For $n\\geq 0$ with $n\\leq m$ we have \\begin{equation}\\label{eqn:detsign} p_n q_{n-1} - q_n p_{n-1} = (-1)^{n-1} \\end{equation} and \\begin{equation}\\label{eqn:detsignan} p_nq_{n-2} - q_n p_{n-2} = (-1)^n a_n. \\end{equation} Equivalently, $$\\frac{p_n}{q_n} - \\frac{p_{n-1}}{q_{n-1}} =                   (-1)^{n-1}\\cdot\\frac{1}{q_n q_{n-1}}$$ and $$\\frac{p_n}{q_n} - \\frac{p_{n-2}}{q_{n-2}} =                   (-1)^{n}\\cdot\\frac{a_n}{q_n q_{n-2}}.$$"}
{"_id": "61", "title": "How Convergents Converge", "text": "\\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 $c_{2m}$."}
{"_id": "62", "title": "Rational Continued Fractions", "text": "Every nonzero rational number can be represented by a simple   continued fraction."}
{"_id": "63", "title": "", "text": "If~$x$ is a rational number, then the sequence $a_0, a_1, \\ldots $ produced by the continued fraction procedure\\index{continued fraction procedure} terminates."}
{"_id": "64", "title": "Geometric Group Law", "text": "\\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 $\\nu = y_1 - \\lambda x_1$."}
{"_id": "65", "title": "", "text": "Suppose~$n$ is the area of a right triangle with rational side lengths $a, b, c$, with $a\\leq b<c$. Let $A=(c/2)^2$.  Then $$A-n, \\quad A,\\, \\text{ and } A+n$$ are all perfect squares of rational numbers."}
{"_id": "66", "title": "Congruent Numbers and Elliptic Curves", "text": "\\iprop{congruent numbers and elliptic curves} Let~$n$ be a rational number.   There is a bijection between $$  A = \\left\\{(a,b,c) \\in \\Q^3 \\,:\\, \\frac{ab}{2} = n,\\, a^2 + b^2 = c^2\\right\\} $$ and $$  B = \\left\\{(x,y) \\in \\Q^2 \\,:\\, y^2 = x^3 - n^2 x, \\,\\,\\text{\\rm with } y \\neq 0\\right\\} $$ given explicitly by the maps $$   f(a,b,c) = \\left(-\\frac{nb}{a+c},\\,\\, \\frac{2n^2}{a+c}\\right) $$ and $$   g(x,y) = \\left(\\frac{n^2-x^2}{y},\\,\\,                          -\\frac{2xn}{y},\\,\\, \\frac{n^2+x^2}{y}\\right). $$"}
{"_id": "67", "title": "Congruent Number Criterion", "text": "% \\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 $y\\neq 0$."}
{"_id": "68", "title": "Commutative Ring", "text": "% A \\defn{commutative ring} is a set~$R$ equipped with binary operations % $+:R\\times R \\to R$ and $\\times:R\\times R \\to R$ such that % $(R,+)$ is an abelian group, $\\times$ is associative and commutative, % and for any $x,y,z\\in R$ we have $x\\times (y+z)=x\\times y+x \\times z$. % We assume that $R$ contains an element $1$ such that $1x=x$ for % all $x\\in R$. %"}
{"_id": "69", "title": "Divides", "text": "If $a, b\\in \\Z$ we say that~$a$ \\defn{divides}~$b$, written $a\\mid b$, if $ac=b$ for some $c\\in \\Z$.  In this case, we say~$a$ is a \\defn{divisor} of~$b$.  We say that~$a$ \\defn{does not divide}~$b$, written $a\\nmid b$, if there is no $c\\in \\Z$ such that $ac=b$."}
{"_id": "70", "title": "Unit", "text": "% A \\defn{unit} is$ % that has a multiplicative inverse, i.e., for which there exists % $y\\in R$ such that $xy=1$. %"}
{"_id": "71", "title": "Irreducible", "text": "%   Let $R$ be a ring and suppose $x\\in R$ is not a unit.  Then~$x$ is %   \\defn{irreducible} if whenever $x=yz$ with $y,z\\in R$, then~$y$ %   or~$z$ is a unit. %"}
{"_id": "72", "title": "Ideal", "text": "% A subset $I$ of a ring~$R$ is an \\defn{ideal} % if $I$ is closed under addition and if whenever % $x\\in R$ and $y\\in I$, then $xy\\in I$. %"}
{"_id": "73", "title": "Prime Ideal", "text": "% An ideal~$I\\neq R$ of a ring $R$ is a \\defn{prime ideal} if % whenever $xy\\in I$ then either $x\\in I$ or $y\\in I$. %"}
{"_id": "74", "title": "Prime Element", "text": "% An element~$x$ of a ring~$R$ is \\defn{prime} if the ideal $xR$ % generated by~$x$ is prime. %"}
{"_id": "75", "title": "Prime and Composite", "text": "An integer $n>1$ is \\defn{prime} if the only positive divisors of $n$ are $1$ and~$n$. We call~$n$ \\defn{composite} if~$n$ is not prime."}
{"_id": "76", "title": "Greatest Common Divisor", "text": "Let $$   \\gcd(a,b)=\\max\\left\\{d\\in \\Z : d \\mid a\\text{ and } d\\mid b\\right\\}, $$ unless both~$a$ and~$b$ are~$0$ in which case $\\gcd(0,0)=0$."}
{"_id": "77", "title": "Group", "text": "A {\\em group} is a set $G$ equipped with a binary operation $G \\times G \\to G$ (denoted by multiplication below) and an identity element $1\\in G$ such that: \\begin{enumerate} \\item For all $a,b,c\\in G$, we have $(ab)c = a(bc)$. \\item For each $a\\in G$, we have $1a=a1=a$, and there exists $b\\in G$ such that $ab = 1$. \\end{enumerate}"}
{"_id": "78", "title": "Abelian Group", "text": "An \\defn{abelian group} is a group $G$ such that $ab=ba$ for every $a,b\\in G$."}
{"_id": "79", "title": "Ring", "text": "A \\defn{ring} $R$ is a set equipped with binary operations $+$ and $\\times$ and elements $0,1\\in R$ such that $R$ is an abelian group under $+$, and for all $a,b,c \\in R$ we have \\begin{itemize} \\item $1a = a1 = a$ \\item $(ab)c = a(bc)$ \\item $a(b+c) = ab + ac$. \\end{itemize} If, in addition, $ab=ba$ for all $a,b\\in R$, then we call $R$ a \\defn{commutative ring}."}
{"_id": "80", "title": "Integers Modulo $n", "text": "The ring $\\zmod{n}$ of \\defn{integers modulo~$n$} is the set of   equivalence classes of integers modulo~$n$.  It is equipped with its   natural ring structure: $$   (a + n\\Z) + (b + n\\Z) = (a+b) + n\\Z $$ $$   (a + n\\Z) \\cdot (b + n\\Z) = (a\\cdot b) + n\\Z. $$"}
{"_id": "81", "title": "Field", "text": "A \\defn{field} $K$ is a ring such that for every nonzero element $a\\in K$ there is an element $b\\in K$ such that $ab=1$."}
{"_id": "82", "title": "Reduction Map and Lift", "text": "We call the natural reduction map $\\Z \\to \\zmod{n}$, which sends $a$   to $a+n\\Z$, \\defn{reduction modulo~$n$}.  We also say that $a$ is a   \\defn{lift} of $a+n\\Z$.  Thus, e.g., $7$ is a lift of $1$ mod $3$,   since $7+3\\Z = 1+3\\Z$."}
{"_id": "83", "title": "Complete Set of Residues", "text": "We call a subset $R\\subset\\Z$ of size~$n$ whose reductions modulo~$n$ are pairwise distinct a \\defn{complete set of residues} modulo~$n$.  In other words, a complete set of residues is a choice of representative for each equivalence class in $\\zmod{n}$."}
{"_id": "84", "title": "Order of an Element", "text": "Let $n\\in\\N$ and $x\\in\\Z$ and suppose that $\\gcd(x,n)=1$. The \\defn{order} of $x$ modulo~$n$ is the smallest $m\\in\\N$ such that $$   x^m \\con 1\\pmod{n}. $$"}
{"_id": "85", "title": "Euler's $\\vphi$-function", "text": "For $n\\in\\N$, let $$  \\vphi(n) = \\#\\{a \\in \\N : a \\leq n \\text{ and } \\gcd(a,n)=1\\}. $$"}
{"_id": "86", "title": "Multiplicative Function", "text": "A function $f:\\N\\ra \\C$ is \\defn{multiplicative} if, whenever $m, n\\in\\N$ and $\\gcd(m,n)=1$, we have $$    f(mn) = f(m)\\cdot f(n). $$"}
{"_id": "87", "title": "Primitive Root", "text": "A \\defn{primitive root} modulo an integer~$n$ is an element of $(\\zmod{n})^*$ of order $\\vphi(n)$."}
{"_id": "88", "title": "Group Homomorphism", "text": "Let $G$ and $H$ be groups.  A map $\\vphi:G \\to H$ is a \\defn{group homomorphism} if for all $a,b\\in G$ we have $\\vphi(ab) = \\vphi(a) \\vphi(b)$. A group homomorphism is called \\defn{surjective} if for every $c\\in H$ there is $a\\in G$ such that $\\vphi(a) = c$. The \\defn{kernel} of a group homomorphism $\\vphi:G\\to H$ is the set $\\ker(\\vphi)$ of elements $a\\in G$ such that $\\vphi(a) = 1$. A group homomorphism is \\defn{injective} if $\\ker(\\vphi) = \\{1\\}$."}
{"_id": "89", "title": "Subgroup", "text": "If $G$ is a group and $H$ is a subset of $G$, then $H$ is a   \\defn{subgroup} if $H$ is a group under the group operation on $G$."}
{"_id": "90", "title": "Quadratic Residue", "text": "\\index{quadratic residue|nn} Fix a prime $p$.  An integer~$a$ not divisible by~$p$ is a \\defn{quadratic residue} modulo~$p$ if~$a$ is a square modulo~$p$; otherwise,~$a$ is a \\defn{quadratic nonresidue}."}
{"_id": "91", "title": "Legendre Symbol", "text": "Let~$p$ be an odd prime and let~$a$ be an integer. Set\\index{a@$\\kr{a}{p}$|nn} $$ \\kr{a}{p} = \\begin{cases}          0 & \\text{if $\\gcd(a,p) \\neq 1$},\\\\         +1 & \\text{if $a$ is a quadratic residue, and}\\\\         -1 & \\text{if $a$ is a quadratic nonresidue}. \\end{cases} $$ We call this symbol the \\defn{Legendre Symbol}."}
{"_id": "92", "title": "Root of Unity", "text": "\\index{root of unity|nn}   \\index{units!roots of unity|nn} An $n$th \\defn{root of unity} is a   complex number~$\\zeta$ such that $\\zeta^n=1$.  A root of   unity~$\\zeta$ is a \\defn{primitive} $n$th\\index{primitive root of     unity|nn}\\index{root of unity!primitive|nn} root of unity if~$n$   is the smallest positive integer such that $\\zeta^n=1$."}
{"_id": "93", "title": "Gauss Sum", "text": "Fix an odd prime $p$.  The \\defn{Gauss sum} associated to an integer~$a$ is $$ g_a = \\sum_{n=1}^{p-1} \\kr{n}{p} \\zeta^{an}, $$ where $\\zeta=\\zeta_p = \\cos(2\\pi{} /p) + i \\sin(2\\pi{}/p) = e^{2\\pi i/p}$."}
{"_id": "94", "title": "Homomorphism of Rings", "text": "Let $R$ and $S$ be rings. A \\defn{homomorphism of rings} $\\vphi:R\\to S$ is a map such that for all $a,b\\in R$, we have \\begin{itemize} \\item $\\vphi(ab) =\\vphi(a)\\vphi(b)$, \\item $\\vphi(a+b) = \\vphi(a) + \\vphi(b)$, and \\item $\\vphi(1) = 1$. \\end{itemize} An \\defn{isomorphism} $\\vphi:R\\to S$ of rings is a ring homomorphism that is bijective."}
{"_id": "95", "title": "Finite Continued Fraction", "text": "\\index{continued fraction!of finite length|nn}\\index{finite continued fraction|nn} A \\defn{finite continued fraction} is an expression $$     a_0 + \\frac{1}{\\ds a_1+\\frac{1}{\\ds a_2 +          \\ds \\frac{1}{ \\cdots +\\frac{1}{a_n}}}} $$ where each $a_m$ is a real number and $a_m>0$ for all $m\\geq 1$."}
{"_id": "96", "title": "Simple Continued Fraction", "text": "A \\defn{simple continued fraction} is a finite or infinite continued fraction in which the $a_i$ are all integers."}
{"_id": "97", "title": "Partial Convergents", "text": "For $0\\leq n\\leq m$, the $n$th \\defn{convergent} of the continued fraction $[a_0,\\ldots,a_m]$ is $[a_0,\\ldots, a_n]$.  These convergents for $n<m$ are also called \\defn{partial convergents}."}
{"_id": "98", "title": "Quadratic Irrational", "text": "A \\defn{quadratic irrational} is a real number $\\alpha\\in\\R$ that is irrational and satisfies a quadratic polynomial with coefficients in~$\\Q$."}
{"_id": "99", "title": "Periodic Continued Fraction", "text": "A \\defn{periodic continued fraction} is a continued fraction $[a_0, a_1, \\ldots, a_n, \\ldots]$ such that $$   a_n = a_{n+h} $$ for some fixed positive integer~$h$ and all sufficiently large~$n$. We call the minimal such~$h$ the \\defn{period of the continued fraction}."}
{"_id": "100", "title": "Algebraic Number", "text": "An \\defn{algebraic number} is a root of a polynomial $f\\in\\Q[x]$."}
{"_id": "101", "title": "Primitive", "text": "A representation $n=x^2 + y^2$ is \\defn{primitive}\\index{primitive!representation} if~$x$ and~$y$ are coprime."}
{"_id": "102", "title": "Elliptic Curve", "text": "An \\defn{elliptic curve} over a field~$K$ is a curve defined by an equation of the form $$  y^2 = x^3 + ax+b, $$ where~$a, b\\in{}K$ and $-16(4a^3+27b^2)\\neq 0$."}
{"_id": "103", "title": "Power Smooth", "text": "%   \\index{power smooth|nn}\\index{smooth|nn} Let~$B$ be a positive   integer. If~$n$ is a positive integer with prime factorization   $n=\\prod p_i^{e_i}$, then $n$ is \\defn{$B$-power smooth} if   $p_i^{e_i}\\leq B$ for all $i$."}
{"_id": "104", "title": "Congruent Number", "text": "We call a nonzero rational number~$n$ a \\defn{congruent number} if $\\pm n$ is the area of a right triangle with rational side lengths.  Equivalently,~$n$ is a \\defn{congruent number} if the system of two equations \\begin{eqnarray*} a^2+b^2&=&c^2\\\\ \\frac{1}{2}ab&=&n \\end{eqnarray*} has a solution with $a,b,c\\in\\Q$."}