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Proth's theorem

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

Contents

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 ≡ − 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:

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 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 ⋅ 2 31172165 + 1 , and is 9,383,761 digits long. It was found by Szabolcs Peter in the PrimeGrid distributed computing project which announced it on 6 November 2016. It is also the largest known non-Mersenne prime. Second largest known Proth prime is 19249 · 213018586 + 1, found by Seventeen or Bust.

    History

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

    References

    Proth's theorem Wikipedia
    (Text) CC BY-SA