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### Strong Pseudoprime

A strong pseudoprime to a base is an odd composite number with (for odd) for which either(1)or(2)for some , 1, ..., (Riesel 1994, p. 91). Note that Guy (1994, p. 27) restricts the definition of strong pseudoprimes to only those satisfying (1).The definition is motivated by the fact that a Fermat pseudoprime to the base satisfies(3)But since is odd, it can be written , and(4)If is prime, it must divide at least one of the factors, but can't divide both because it would then divide their difference(5)Therefore,(6)so write to obtain(7)If divides exactly one of these factors but is composite, it is a strong pseudoprime. A composite number is a strong pseudoprime to at most 1/4 of all bases less than itself (Monier 1980, Rabin 1980). The strong pseudoprimes provide the basis for Miller's primality test and Rabin-Miller strong pseudoprime test.A strong pseudoprime to the base is also an Euler pseudoprime to the base (Pomerance et al. 1980)...

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