queries.jsonl

{"_id": "2", "text": "Euclid There are infinitely many primes.\\ithm{infinitely many primes}"}
{"_id": "5", "text": "Euler's Theorem \\ithm{Euler's} If $\\gcd(x,n)=1$, then $$    x^{\\vphi(n)} \\con 1\\pmod{n}. $$"}
{"_id": "6", "text": "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~$mn$."}
{"_id": "7", "text": "Pseudoprimality An integer~$p>1$ is prime if and only if for {\\em every} $a\\not\\con 0\\pmod{p}$, $$       a^{p-1}\\con 1\\pmod{p}. $$"}
{"_id": "8", "text": "Primitive Roots There is a primitive root modulo any prime~$p$.  In particular, the group $(\\zmod{p})^*$ is cyclic."}
{"_id": "10", "text": "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 $$   \\kr{-1}{p} = (-1)^{(p-1)/2}\\qquad\\text{\\rm and}\\qquad   \\kr{2}{p}  = \\begin{cases} \\hfill1 & \\text{\\rm if } p\\con \\pm 1\\pmod{8}\\\\     -1 & \\text{\\rm if } p \\con \\pm 3\\pmod{8}.             \\end{cases} $$"}
{"_id": "11", "text": "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."}
{"_id": "12", "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", "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", "text": "Periodic Characterization \\ithm{period continued fraction} An infinite simple continued fraction is periodic if and only if it represents a quadratic irrational."}
{"_id": "15", "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", "text": "The binary operation $+$ defined in Algorithm~\\ref{alg:grouplaw}   endows the set $E(K)$ with an abelian group structure, with identity $\\O$."}
{"_id": "21", "text": "Suppose $a,b,n\\in\\Z$.  Then $\\gcd(a,b) = \\gcd(a,b-an)$."}
{"_id": "22", "text": "For any integers $a,b,n$, we have $$\\gcd(an,bn) = \\gcd(a,b)\\cdot |n|.$$"}
{"_id": "23", "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", "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", "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": "27", "text": "The map $\\psi:(\\zmod{p})^*\\to \\{\\pm 1\\}$ given by $\\psi(a) = \\kr{a}{p}$ is a surjective group homomorphism."}
{"_id": "28", "text": "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 $$ \\kr{a}{p} = (-1)^{\\nu}. $$"}
{"_id": "31", "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", "text": "We have $g_0=0$."}
{"_id": "33", "text": "For any integer $a$, $$  g_a = \\kr{a}{p}g_1. $$"}
{"_id": "35", "text": "If~$n$ is divisible by a prime~$p\\con 3\\pmod{4}$, then~$n$ has no primitive representations."}
{"_id": "36", "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", "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": "44", "text": "Cancellation If $\\gcd(c,n)=1$ and $$    ac\\con bc\\pmod{n}, $$ then $a \\con b\\pmod{n}$."}
{"_id": "45", "text": "Units 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", "text": "Solvability The equation $ax\\con b\\pmod{n}$ has a solution if and only if $\\gcd(a,n)$ divides~$b$."}
{"_id": "48", "text": "Multiplicativity of $\\vphi The function $\\vphi$ is multiplicative. \\index{phi function!is multiplicative}\\index{Euler!phi function!is multiplicative}"}
{"_id": "51", "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": "53", "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", "text": "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 $a\\in\\Z$."}
{"_id": "55", "text": "Euler's Criterion \\iprop{Euler's criterion} We have $\\kr{a}{p}=1$ if and only if $$ a^{(p-1)/2}\\con 1\\pmod{p}. $$"}
{"_id": "56", "text": "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 $\\kr{a}{p} = \\kr{a}{q}$."}
{"_id": "57", "text": "Legendre Symbol of $2 Let~$p$ be an odd prime.  Then $$ \\kr{2}{p} = \\begin{cases} \\hfill1 & \\text{ if } p\\con \\pm 1\\pmod{8}\\\\                          -1 & \\text{ if } p\\con \\pm 3\\pmod{8}.  \\end{cases} $$"}
{"_id": "61", "text": "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 $c_{2m}$."}
{"_id": "64", "text": "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 $\\nu = y_1 - \\lambda x_1$."}
{"_id": "67", "text": "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 $y\\neq 0$."}