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Counterexamples to the Baillie-PSW primality test
Problem (1) Find a counterexample to Baillie-PSW primality test or prove that there is no one.
Problem (2) Find a composite
or
which divides both
(see Fermat pseudoprime) and the Fibonacci number
(see Lucas pseudoprime), or prove that there is no such
.
![$ n\equiv 3 $](/files/tex/493f3d9bc88a631c25371ab6138625d0cbf08f52.png)
![$ 7\pmod{10} $](/files/tex/1f85cdb6c4f3165da306e494803c7a7170153dfd.png)
![$ 2^{n-1} - 1 $](/files/tex/e9553e6870300b908f907d6edc486e5b87147e2d.png)
![$ F_{n+1} $](/files/tex/ada72b72f8ae80fb477cc7ba27cbb70b839959ff.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
Selfridge, Wagstaff, and Pomerance offered $500 + $100 + $20 for satisfying Problem 2, and $20 + $100 + $500 for a proof that there is no such
(R. Guy, 1994).
Bibliography
Carl Pomerance. "Are There Counterexamples to the Baillie-PSW Primality Test?"
Thomas R. Nicely. " The Baillie-PSW primality test."
R. K. Guy. "Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes". §A12 in "Unsolved Problems in Number Theory", 2nd ed. New York: Springer-Verlag, pp. 27-30, 1994.
* indicates original appearance(s) of problem.