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Extra Strong Lucas Pseudoprime

Given the Lucas sequence U_n(b,-1) and V_n(b,-1), define Delta=b^2+4. Then an extra strong Lucas pseudoprime to the base b is a composite number n=2^rs+(Delta/n), where s is odd and (n,2Delta)=1 such that either U_s=0 (mod n) and V_s=+/-2 (mod n), or V_(2^ts)=0 (mod n) for some t with 0<=t<r-1. An extra strong Lucas pseudoprime is a strong Lucas pseudoprime with parameters (b,1). Composite n are extra strong pseudoprimes for at most 1/8 of possible bases (Grantham 1997).

See also

Lucas Pseudoprime, Strong Lucas Pseudoprime

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References

Grantham, J. "Frobenius Pseudoprimes." http://www.pseudoprime.com/pseudo1.psGrantham, J. "A Frobenius Probable Prime Test with High Confidence." 1997. http://www.pseudoprime.com/pseudo2.psJones, J. P. and Mo, Z. "A New Primality Test Using Lucas Sequences." Preprint.Nicely, T. R. "The Baillie-PSW Primality Test." http://www.trnicely.net/misc/bpsw.html.

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Extra Strong Lucas Pseudoprime

Cite this as:

Weisstein, Eric W. "Extra Strong Lucas Pseudoprime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExtraStrongLucasPseudoprime.html

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