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, 2¹ + 1 = 3 is divisible by 3, so 3 is prime.
• for p = 5, 3² + 1 = 10 is divisible by 5, so 5 is prime.
• for p = 13, 5⁶ + 1 = 15626 is divisible by 13, so 13 is prime.
• for p = 9, which is not prime, there is no a such that a⁴ + 1 is divisible by 9.

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

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

As of 2009, the largest known Proth prime is 19249 · 2^13018586 + 1, found by Seventeen or Bust. It has 3,918,990 digits and is the largest known prime which is not a Mersenne prime.

History

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

See also

• Pépin's test (the special case k = 1, where one chooses a = 3)
• Sierpinski number

External links

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