In number theory, Proth's theorem is a theorem which forms the basis of a primality test for Proth numbers (sometimes called Proth Numbers of the First...

larger than 1 would be Proth numbers. The primality of a Proth number can be tested with Proth's theorem, which states that a Proth number p {\displaystyle...

specific number forms include Pépin's test for Fermat numbers (1877), Proth's theorem (c. 1878), the Lucas–Lehmer primality test (originated 1856), and the...

Principal ideal theorem (algebraic number theory) Proth's theorem (number theory) Quadratic reciprocity theorem Ramanujan–Skolem's theorem (Diophantine equations)...

stated four primality-related theorems. The most famous of these, Proth's theorem, can be used to test whether a Proth number (a number of the form k2n + 1...

a = 3, and this special case of Proth's theorem is known as Pépin's test. Although Pépin's test and Proth's theorem have been implemented on computers...

)+cn+d} for some constants c and d. For this recurrence relation, the master theorem for divide-and-conquer recurrences gives the asymptotic bound T ( n ) =...

it can be used as a basic tool for proving theorems in number theory such as Lagrange's four-square theorem and the uniqueness of prime factorizations...

needed] For numbers of the form N = k · 2n + 1 (Proth numbers), either application of Proth's theorem (a Las Vegas algorithm) or one of the deterministic...

Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius Solovay–Strassen Miller–Rabin Prime-generating...

Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius Solovay–Strassen Miller–Rabin Prime-generating...

exist for testing whether a number is prime, such as the Lucas test and Proth's test. These tests typically require factorization of n + 1, n − 1, or a...

Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius Solovay–Strassen Miller–Rabin Prime-generating...

This theorem is a generalization to polynomials of Fermat's little theorem. In one direction it can easily be proven using the binomial theorem together...

always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer n using mental or pen-and-paper arithmetic...

Compute γ := g p e − 1 {\displaystyle \gamma :=g^{p^{e-1}}} . By Lagrange's theorem, this element has order p {\displaystyle p} . For all k ∈ { 0 , … , e −...

Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius Solovay–Strassen Miller–Rabin Prime-generating...

which is more accurate in the polynomial case, leading to the following theorem. If a and b are two nonzero polynomials, then the extended Euclidean algorithm...

following description is based on (Hoffstein, Pipher & Silverman 2008, Theorem 6.68), with the corrections from the errata. INPUT a lattice basis b1,...

Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius Solovay–Strassen Miller–Rabin Prime-generating...

Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius Solovay–Strassen Miller–Rabin Prime-generating...

Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius Solovay–Strassen Miller–Rabin Prime-generating...

The idea behind the test was discovered by M. M. Artjuhov in 1967 (see Theorem E in the paper). This test has been largely superseded by the Baillie–PSW...

Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius Solovay–Strassen Miller–Rabin Prime-generating...

{\displaystyle 3^{16}\equiv 1{\pmod {17}}} —as follows from Fermat's little theorem— it also follows that if n {\displaystyle n} is an integer then 3 4 + 16...

m_{p}} as the order of the group E p {\displaystyle E_{p}} . By Hasse's theorem on elliptic curves we know m p ≤ p + 1 + 2 p = ( p + 1 ) 2 ≤ ( N 4 + 1...

number of solutions to 4x2 + y2 = n is odd and the number is squarefree (theorem 6.1 of ). All numbers n with modulo-sixty remainder 7, 19, 31, or 43 have...

Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius Solovay–Strassen Miller–Rabin Prime-generating...

postulate is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem. Chebyshev's theorem can also be stated as a relationship with π ( x )...

theorem guarantees that the continued fractions algorithm will recover j / r {\displaystyle j/r} from k / 2 2 n {\displaystyle k/2^{2{n}}} : Theorem—If...