Baillie–PSW primality test

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. As of 1994, the value was raised to $620 and no pseudoprime was found up to 10¹⁷, 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 (Pomerance 1984) suggesting that there may be infinitely many counterexamples.

The test

1. Optionally, perform trial division to check if the number isn't a multiple of a small prime number.
2. Perform a base 2 strong pseudoprimality test. If it fails; n is composite.
3. Find the first a in the sequence 5, −7, 9, −11, ... for which the Jacobi symbol $\left(\frac{a}{n}\right) = -1$.
4. Perform a Lucas pseudoprimality test with discriminant a on n. If this test does not fail, n is likely a prime.

Further reading

• Nicely, Thomas R., The Baillie-PSW primality test., http://www.trnicely.net/misc/bpsw.html 
• Baillie, Robert; Samuel S. Wagstaff, Jr. (1980), "Lucas pseudoprimes", Math. Comp. 35: 1391–1417, http://mpqs.free.fr/LucasPseudoprimes.pdf 
• Pomerance, Carl; John L. Selfridge, and Samuel S. Wagstaff, Jr. (1980), "The pseudoprimes to 25 * 10⁹", Math. Comp. 35: 1003–1026, http://www.math.dartmouth.edu/~carlp/PDF/paper25.pdf