The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other...

Carl Gustav Jacob Jacobi (/dʒəˈkoʊbi/; German: [jaˈkoːbi]; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions...

Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order. The notational convenience of the Legendre symbol inspired introduction...

symbol, written as ( a n ) {\displaystyle \left({\frac {a}{n}}\right)} or ( a | n ) {\displaystyle (a|n)} , is a generalization of the Jacobi symbol to...

doubly-periodic functions Jacobi polynomials, a class of orthogonal polynomials Jacobi symbol, a generalization of the Legendre symbol Jacobi coordinates, a simplification...

and p. This interpretation of the Legendre symbol as the sign of a permutation can be extended to the Jacobi symbol ( a n ) , {\displaystyle \left({\frac {a}{n}}\right)...

it is 0 if p divides a. The same notation is used for the Jacobi symbol and Kronecker symbol, which are generalizations where p is respectively any odd...

function and a certain Dirichlet L-function The Jacobi symbol is a generalization of the Legendre symbol; the main difference is that the bottom number...

and let ( D n ) {\displaystyle \left({\tfrac {D}{n}}\right)} be the Jacobi symbol. We define δ ( n ) = n − ( D n ) . {\displaystyle \delta (n)=n-\left({\tfrac...

{a}{n}}\right)} is the Jacobi symbol. If n is an odd composite integer that satisfies the above congruence, then n is called an Euler–Jacobi pseudoprime (or...

\left({\tfrac {a}{p}}\right)} is the Legendre symbol. The Jacobi symbol is a generalisation of the Legendre symbol to ( a n ) {\displaystyle \left({\tfrac {a}{n}}\right)}...

Legendre symbol ( a p ) {\displaystyle \left({\frac {a}{p}}\right)} defined for p a prime, a an integer, and takes values 0, 1, or −1. Jacobi symbol ( a b...

quadratic residue modulo N (i.e., x = y2 mod N for some y), when the Jacobi symbol for x is +1. The quadratic residue problem is easily solved given the...

the Jacobi symbol ( a n ) {\displaystyle \left({\frac {a}{n}}\right)} . If n {\displaystyle n} is an odd prime, this is equal to the Legendre symbol, and...

q-Jacobi polynomials Pseudo Jacobi polynomials Sieved Jacobi polynomials Jacobi preconditioner Jacobi rotation Jacobi set Jacobi sum Jacobi symbol Jacobi...

F_{n\;-\,\left({\frac {5}{n}}\right)},} where the Legendre symbol has been replaced by the Jacobi symbol, then this is evidence that n is a prime, and if it...

{\mathfrak {p}}}.} The n-th power symbol has properties completely analogous to those of the classical (quadratic) Jacobi symbol ( ζ {\displaystyle \zeta } is...

≡ 2 or 3 (mod 4). Extend the Jacobi symbol to accept even numbers in the "denominator" by defining the Kronecker symbol: ( a 2 ) = { 0  if  a  is even...

( M ( n ) log ⁡ n ) {\displaystyle O(M(n)\log n)} algorithm for the Jacobi symbol". International Algorithmic Number Theory Symposium. Springer. pp. 83–95...

number p divides C(p + 1)/2 when the Jacobi symbol (2 | p) is −1, and that p divides C(3p − 1)/2 when the Jacobi symbol (2 | p) is + 1. It is unknown whether...

Lou Jacobi (born Louis Harold Jacobovitch; December 28, 1913 – October 23, 2009) was a Canadian character actor. Jacobi came to prominence for his role...

W(p + 1) / 2 if the Jacobi symbol ( 2 p ) {\displaystyle \left({\frac {2}{p}}\right)} is +1 and W(3p − 1) / 2 if the Jacobi symbol ( 2 p ) {\displaystyle...

{b}{F_{n}}}\right)} is the Jacobi symbol. In fact, Pépin's test is the same as the Euler-Jacobi test for Fermat numbers, since the Jacobi symbol ( b F n ) {\displaystyle...

the symbol was discontinued by Legendre, but it was taken up again by Carl Gustav Jacob Jacobi in 1841, whose usage became widely adopted. The symbol is...

"denominator" in the same way the Legendre symbol is generalized into the Jacobi symbol. As with the Jacobi symbol, this extension sacrifices the "numerator...

(Colossus computer); Carl Gustav Jacob Jacobi (Jacobi elliptic functions, Jacobian matrix and determinant, Jacobi symbol). Sidney Altman (Molecular biology...

\left({\frac {a}{N}}\right)=-1} (See Jacobi symbol) then N {\displaystyle N} is composite. If N = Fn > 3, then the above Jacobi symbol is always equal to −1 for...

multiplicative function as are Dirichlet characters, the Jacobi symbol and the Legendre symbol. A completely multiplicative function is completely determined...

{c}{a}}\right)&{\text{if}}\ c\equiv 0{\pmod {4}}.\end{cases}}} Here (⁠a/c⁠) is the Jacobi symbol. This is the famous formula of Carl Friedrich Gauss. For b > 0 the Gauss...

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are...