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...

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

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

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...

( a n ) {\displaystyle \left({\frac {a}{n}}\right)} is the Jacobi symbol. The Jacobi symbol evaluates to 0 if a and n are not coprime, so the test can...

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...

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...

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...

polynomials Jacobi symbol, a generalization of the Legendre symbol Jacobi coordinates, a simplification of coordinates for an n-body system Jacobi identity...

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)...

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F_{n\;-\,\left({\frac {5}{n}}\right)},} where the Legendre symbol has been replaced by the Jacobi symbol, then this is evidence that n is a prime, and if it...

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...

\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)}...

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

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...

slightly stronger test uses the Jacobi symbol to predict which of the two results will be found. The resultant Euler-Jacobi probable prime test verifies...

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...

} where ( D N ) {\displaystyle \left({\frac {D}{N}}\right)} is the Jacobi symbol. Unlike the standard Lucas pseudoprimes, there is no known efficient...

≡ 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...

{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...

\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...

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...

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...

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

{\displaystyle p} ⁠, the ± 1 {\displaystyle \pm 1} term is the (negated) Jacobi symbol, which can be calculated using quadratic reciprocity. Indeed, much of...

( D n ) {\displaystyle \delta =\left({\tfrac {D}{n}}\right)} is the Jacobi symbol. When condition (2) is satisfied, condition (3) becomes equivalent to...

Legendre symbol for composite values of p, the Jacobi symbol, but its properties are not as simple: if m is composite and the Jacobi symbol ( a m ) =...

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...

For instance, while RSA is conjectured to be a one-way function, the Jacobi symbol of the preimage can be easily computed from that of the image.: 121 ...