A Cubic Composite Test
Pierre Laurent, Paul Underwood
TL;DR
This paper develops a single-parameter cubic composite primality test for odd integers based on the cubic polynomial $f_a=x^3-ax-a$, exploiting the discriminant being a square to guide gcd pruning and a strengthened congruence condition. It defines a practical algorithm by combining gcd checks with a refined test on $B\equiv x^{n-1}\pmod{n,f_a}$ and a subsequent congruence $B^2+B+1\equiv -x^2+x+a$, using a parametric choice $a=7+k(k-1)$ and frequent prime-restart heuristics to filter candidates. The authors report extensive verifications with no counterexamples up to large bounds and compare the method’s computational cost to Baillie-PSW, finding competitive performance for cryptographic-sized inputs. Additionally, a related quadratic test based on the reducible case of $f_a$ and a connection to Selfridge’s challenge is discussed. The work highlights a robust, cost-efficient alternative framework for primality testing, while acknowledging that it is not a deterministic prime prover and inviting further exploration of potential pseudoprimes and optimizations.
Abstract
A single parameter cubic composite test for odd positive integers is given which relies on the discriminant always being a square integer. This test has no known counterexample despite extensive verifications. As well as a comparison with the Baillie-PSW tests, a related quadratic composite test is briefly examined which also has no known counterexample.