Proth's theorem

In number theory, Proth's theorem is a primality test for Proth numbers.

It states that if p is a Proth number, of the form k2ⁿ + 1 with k odd and k < 2ⁿ, and if there exists an integer a for which

a^{{(p-1)/2}}\equiv -1\mod {p},

then p is prime. In this case p is called a Proth prime. This is a practical test because if p is prime, any chosen a has about a 50 percent chance of working.

If a is a quadratic nonresidue modulo p then the converse is also true, and the test is conclusive. Such an a may be found by iterating a over small primes and computing the Jacobi symbol until:

\left({\frac {a}{p}}\right)=-1.

Thus, in contrast to many Monte Carlo primality tests (randomized algorithms that can return a false positive), the primality testing algorithm based on Proth's theorem is a Las Vegas algorithm, always returning the correct answer but with a running time that varies randomly.

Numerical examples

Examples of the theorem include:

for p = 3 = 1(2¹) + 1, we have that 2^(3-1)/2 + 1 = 3 is divisible by 3, so 3 is prime.
for p = 5 = 1(2²) + 1, we have that 3^(5-1)/2 + 1 = 10 is divisible by 5, so 5 is prime.
for p = 13 = 3(2²) + 1, we have that 5^(13-1)/2 + 1 = 15626 is divisible by 13, so 13 is prime.
for p = 9, which is not prime, there is no a such that a^(9-1)/2 + 1 is divisible by 9.

The first Proth primes are (sequence A080076 in the OEIS):

3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153 ….

The largest known Proth prime as of 2016 is $10223\cdot 2^{31172165}+1$ , and is 9,383,761 digits long.^[1] It was found by Szabolcs Peter in the PrimeGrid distributed computing project which announced it on 6 November 2016.^[2] It is also the largest known non-Mersenne prime.^[3] The second largest known Proth prime is 19249 · 2^13018586 + 1, found by Seventeen or Bust.^[4]

Proof

The proof for this theorem uses the Pocklington-Lehmer Primality Test, and closely resembles the proof of Pépin's test. Using the Pocklington Test as a backbone, "n-1=2

History

François Proth (1852–1879) published the theorem around 1878.

References

↑ Chris Caldwell, The Top Twenty: Proth, from The Prime Pages.
↑ "World Record Colbert Number discovered!".
↑ Chris Caldwell, The Top Twenty: Largest Known Primes, from The Prime Pages.
↑ Caldwell, Chris K. "The Top Twenty: Largest Known Primes".

External links

Weisstein, Eric W. "Proth's Theorem". MathWorld.

Number-theoretic algorithms
Primality tests	AKS test APR test Baillie–PSW Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius test Solovay–Strassen Miller–Rabin
Prime-generating	Sieve of Atkin Sieve of Eratosthenes Sieve of Sundaram Wheel factorization
Integer factorization	Continued fraction (CFRAC) Dixon's Lenstra elliptic curve (ECM) Euler's Pollard's rho p − 1 p + 1 Quadratic sieve (QS) General number field sieve (GNFS) Special number field sieve (SNFS) Rational sieve Fermat's Shanks' square forms Trial division Shor's
Multiplication	Ancient Egyptian Long Karatsuba Toom–Cook Schönhage–Strassen Fürer's
Discrete logarithm	Baby-step giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve
Greatest common divisor	Binary Euclidean Extended Euclidean Lehmer's
Modular square root	Cipolla Pocklington's Tonelli–Shanks
Other algorithms	Chakravala Cornacchia LLL Integer square root Modular exponentiation Schoof's
Italics indicate that algorithm is for numbers of special forms

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