Search results
Results From The WOW.Com Content Network
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of an odd prime number p: its value at a (nonzero) quadratic residue mod p is 1 and at a non-quadratic residue (non-residue) is −1. Its value at zero is 0. The Legendre symbol was introduced by Adrien-Marie Legendre ...
1. If n is (an odd) prime, then the Jacobi symbol (a / n) is equal to (and written the same as) the corresponding Legendre symbol. 2. If a ≡ b (mod n), then. 3. If either the top or bottom argument is fixed, the Jacobi symbol is a completely multiplicative function in the remaining argument: 4. 5.
The Jacobi symbol is a generalization of the Legendre symbol; the main difference is that the bottom number has to be positive and odd, but does not have to be prime. If it is prime, the two symbols agree. It obeys the same rules of manipulation as the Legendre symbol. In particular
The general Legendre equation reads ″ ′ + [(+)] =, where the numbers λ and μ may be complex, and are called the degree and order of the relevant function, respectively. . The polynomial solutions when λ is an integer (denoted n), and μ = 0 are the Legendre polynomials P n; and when λ is an integer (denoted n), and μ = m is also an integer with | m | < n are the associated Legendre ...
Symbol (number theory) In number theory, a symbol is any of many different generalizations of the Legendre symbol. This article describes the relations between these various generalizations. The symbols below are arranged roughly in order of the date they were introduced, which is usually (but not always) in order of increasing generality.
The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre [3] as the coefficients in the expansion of the Newtonian potential where r and r′ are the lengths of the vectors x and x′ respectively and γ is the angle between those two vectors. The series converges when r > r′.
Legendre's formula can be used to prove Kummer's theorem. As one special case, it can be used to prove that if n is a positive integer then 4 divides ( 2 n n ) {\displaystyle {\binom {2n}{n}}} if and only if n is not a power of 2.
In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then [1][2][3] Euler's criterion can be concisely reformulated using the Legendre symbol: [4] The criterion dates from a 1748 paper by Leonhard Euler. [5][6]