Baillie–PSW Primality Test

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In number theory, the **Baillie–PSW primality test** is a probabilistic primality testing heuristic algorithm: it determines if a number is composite or a probable prime.The authors of the test offered $30 for the discovery of a composite number that passed this test. , the value was raised to $620Guy, R. (1994). “Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes.” §A12 in *Unsolved Problems in Number Theory*. 2nd ed., p. 28, New York: Springer-Verlag. ISBN 0-387-20860-7. consequently this can be considered a sound primality test on numbers below that upper bound.

A primality testing software PRIMO uses this algorithm to check for probable primes, and no certification of this test has yet failed. The author, Marcel Martin, estimates by those results that the test is accurate for numbers below 10000 digits. There is a heuristic argument suggesting that there may be infinitely many counterexamples.

## The test

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A primality testing software PRIMO uses this algorithm to check for probable primes, and no certification of this test has yet failed. The author, Marcel Martin, estimates by those results that the test is accurate for numbers below 10000 digits. There is a heuristic argument suggesting that there may be infinitely many counterexamples.

- Optionally, perform trial division to check if the number isn't a multiple of a small prime number.
- Perform a base 2 strong pseudoprimality test. If it fails;
*n*is composite. - Find the first
*a*in the sequence 5, −7, 9, −11, ... for which the Jacobi symbol <math>left(fracright) = -1</math>. - Perform a Lucas pseudoprimality test with discriminant
*a*on*n*. If this test does not......
...

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