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

${\displaystyle a^{\frac {p-1}{2}}\equiv -1{\pmod {p}},}$

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, furthermore, since the calculation is mod p, only values of a smaller than p have to be taken into consideration.

In practice, however, a quadratic nonresidue of p is found via a modified Euclid's algorithm and taken as the value of a, since if a is a quadratic nonresidue modulo p then the converse is also true, and the test is conclusive. For such an a the Legendre symbol is

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

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. Note that if a is chosen to be a quadratic nonresidue as described above, the runtime is constant, save for the time spent on finding such a quadratic nonresidue. Finding such a value is very fast compared to the actual test.