Pseudoprimeira

Pseudoprimeira is a term used in some informal or speculative discussions to denote a composite number that passes a specified primality test for a prescribed set of bases. The word blends pseudoprime with a feminine suffix and is not a standard term in mainstream number theory. In this sense, the concept serves as a generalized or contextual version of pseudoprimes.

Definition and scope: Let T be a primality test defined by congruences or related conditions, and let B be a set of integers with gcd(a,n)=1. A composite number n is a pseudoprimeira to T with respect to B if for every a in B, the test T produces a result that would be correct for a prime. If B equals the full set of units modulo n and T is the Fermat test, such numbers align with the classical Carmichael numbers. If T is a strong primality test (as in Miller-Rabin), the corresponding objects are sometimes discussed as strong pseudoprimes to a base set.

Relation to established concepts: Pseudoprimeiras generalize the idea of Fermat pseudoprimes and Carmichael numbers by varying the test and the base set. They are not standardized in the literature, and precise definitions can vary between sources. In formal contexts, a Carmichael number or a strong pseudoprime to a given base set serves a comparable role to what some discussions label a pseudoprimeira.

Properties and prevalence: The existence and abundance of pseudoprimeiras depend on the chosen test and base set; broader base sets yield fewer composite numbers that pass the test. They are primarily of theoretical interest in illustrating exceptions to primality tests.

See also: pseudoprime, Carmichael number, Fermat pseudoprime, Miller-Rabin primality test, strong pseudoprime.